Method and apparatus for determining the physical state of an object

Abstract

The present application relates to a computer-implemented method for the structure optimization of an object, comprising iteratively performing the following steps: receiving initial structure data representing an initial form of the object; determining initial calibrated physical data representing the physical state of the object with the initial form, by a numerical solver based on the initial structure data if an initial condition is met; determining the initial physical data representing the physical state of the object with the initial form, by the initial neural network model based on the initial structure data, reference structure data representing a reference form of the object, and physical reference data representing the physical state of the object with the reference form if the initial condition is not met;and updating the object's initial structural data based on the initial physical data to obtain updated initial structural data representing an updated initial form of the object.

Description

Technical field The present application relates to artificial intelligence technologies, in particular to a method and a device for determining the physical state of an object, a method and a device for optimizing the structure of an object, and a method and a device for training a neural network model. State of the art The structure of an object—for example, its shape, size, or material distribution—influences its properties. Examples of such objects include the wings of an airplane, pipes in a reactor, the body of a vehicle, and so on. For instance, the shape of a wing affects the pressure and velocity of the airflow relative to the wing, which in turn influences the properties of the wing's physical system. The physical system of an object can be described by partial differential equations (PDEs). Solving the PDE based on structural data allows us to obtain the physical data, such as the pressure and velocity, of the physical system. In a structural optimization process, the performance of an object is improved by modifying its structure. Structural optimization is applied in various fields, including science, engineering, and industry. For example, before a product is manufactured, its shape (such as chemical catalyst particles, automotive parts, etc.) can be optimized through a structural optimization process. Since most physical systems are described by PDEs, structural optimization can be performed using the relevant PDEs. For example, the relevant constraint PDEs are solved in each iteration of the structural optimization to determine the physical state of the current structure. For the structural optimization of an object, especially one with a complex structure, the efficiency and accuracy of solving the relevant constraint PDEs are crucial for improving the performance of the object's structural optimization. Neural network (NN) models for solving PDEs offer the advantage of fast inference, which can reduce computational costs while maintaining good accuracy. Compared to a numerical solver for solving PDEs, using NN models to predict the solutions of constrained PDEs and to optimize the object's structure based on these predicted solutions can significantly accelerate the PDE-constrained structure optimization process. However, errors in the NN models' predictions and the resulting accumulation of errors in the structure optimization process can lead to inaccuracies. Disclosure of the invention The following overview is provided to simplify the presentation of some concepts that will be discussed in more detail below. This overview is not intended to highlight the critical or essential features of the claimed subject matter, nor to limit the scope of the claimed subject matter. In light of the aforementioned problems, one aspect of the present application provides a computer-implemented method for determining the physical state of an object with a shape, comprising: receiving initial structural data representing an initial shape of the object; generating, based on the initial structural data, an initial feature sequence comprising N feature vectors; transforming the initial feature sequence into a second feature sequence comprising M feature vectors; performing Fourier-transform-based processing on the second feature sequence to obtain a third feature sequence comprising M feature vectors; transforming the third feature sequence into a fourth feature sequence comprising N feature vectors; and determining initial physical state data representing the physical state of the object with the initial shape, based on the fourth feature sequence. According to one aspect of the present application, a computer-implemented method for structural optimization of an object is provided, comprising iteratively performing the following steps: receiving initial structural data representing an initial form of the object; predicting initial physical state data representing the physical state of the object with the initial form, using a method according to various embodiments of the present disclosure; updating the initial structural data based on the initial physical state data to obtain updated initial structural data representing an updated initial form of the object. According to one aspect of the present application, a computer-implemented method for training a neural network model to determine a physical state of an object with a shape is provided, wherein the neural network model comprises a virtual Fourier layer, the virtual Fourier layer comprising a first transformation layer, a Fourier processing layer, and a second transformation layer, wherein the method comprises: receiving first structural data representing a first shape of a first object and first labeled physical state data representing the physical state of the first object with the first shape; generating, based on the first structural data, a first feature sequence comprising N feature vectors; transforming the first feature sequence into a second feature sequence comprising M feature vectors by the first transformation layer;Performing Fourier-transform-based processing on the second feature sequence by the Fourier processing layer to obtain a third feature sequence comprising M feature vectors; transforming the third feature sequence into a fourth feature sequence comprising N feature vectors by the second transformation layer; determining first predicted physical state data representing the physical state of the first object with the first shape, based on the fourth feature sequence; updating learnable parameters of the neural network model based on the first predicted physical state data and the first labeled physical state data. According to one aspect of the present application, a computer-implemented method for optimizing the structure of an object is provided, comprising iteratively performing the following steps: receiving initial structure data representing an initial form of the object; determining initial calibrated physical data representing the physical state of the object with the initial form by a numerical solver based on the initial structure data when an initial condition is satisfied, wherein the initial calibrated physical data are used as the initial physical data representing the physical state of the object with the initial form, or wherein the initial calibrated physical data are used as physical reference data for an initial neural network model to predict the initial physical data for the physical state of the object with the initial form;Determining the first physical data representing the physical state of the object with the first form, by the first neural network model based on the first structural data, reference structural data representing a reference form of the object, and physical reference data representing the physical state of the object with the reference form when the first condition is not met; and updating the first structural data of the object based on the first physical data to obtain updated first structural data representing an updated first form of the object. According to one aspect of the present application, a device for determining a physical state of an object with a shape is provided, comprising: a generation module configured to generate a first feature sequence comprising N feature vectors based on first structural data representing a first shape of the object; a virtual Fourier module configured to transform the first feature sequence into a second feature sequence comprising M feature vectors, to perform Fourier-transform-based processing on the second feature sequence to obtain a third feature sequence comprising M feature vectors, and to transform the third feature sequence into a fourth feature sequence comprising N feature vectors;and a determination module that, based on the fourth feature sequence, determines initial physical state data representing the physical state of the object with the first form. According to one aspect of the present application, a device for structural optimization of an object is provided, comprising: a physical state data determination module configured to determine, based on initial structural data representing an initial form of the object, initial physical state data representing the physical state of the object with the initial form, by the method according to various embodiments of the present disclosure; a structure update module configured to update the initial structural data based on the initial physical state data in order to obtain updated initial structural data representing an updated initial form of the object.According to one aspect of the present application, a device for training a neural network model to determine a physical state of an object with a shape is provided, comprising: a processing module with the neural network model that receives first structural data representing a first shape of a first object and first labeled physical state data representing the physical state of the first object with the first shape; that generates a first feature sequence comprising N feature vectors based on the first structural data; that transforms the first feature sequence into a second feature sequence comprising M feature vectors; that performs Fourier-transform-based processing on the second feature sequence to obtain a third feature sequence comprising M feature vectors;that transforms the third feature sequence into a fourth feature sequence comprising N feature vectors; that, based on the fourth feature sequence, determines the first predicted physical state data representing the physical state of the first object with the first shape; an update module that updates learnable parameters of the neural network model based on the first predicted physical state data and the first labeled physical state data. According to one aspect of the present application, a device for structural optimization of an object is provided, comprising: a physical state data determination module configured to receive initial structural data representing an initial form of the object; that, in the case where an initial condition is satisfied, it determines, by means of a numerical solver, initial calibrated physical data representing the physical state of the object with the initial form, based on the initial structural data, wherein the initial calibrated physical data are used as the initial physical data representing the physical state of the object with the initial form, or wherein the initial calibrated physical data are used as physical reference data for an initial neural network model to predict the initial physical data for the physical state of the object with the initial form;that, in the case where the first condition is not met, the first neural network model determines the first physical data representing the physical state of the object with the first form, based on the first structural data, reference structural data representing a reference form of the object, and physical reference data representing the physical state of the object with the reference form; a structure update module configured to update the first structural data of the object based on the first physical data to obtain updated first structural data representing an updated first form of the object. According to one aspect of the present application, a processing device is provided comprising: one or more processors; one or more memories on which computer-executable instructions are stored, wherein the instructions, when executed by the one or more processors, perform a method according to embodiments of the present disclosure. According to one aspect of the present application, a machine-readable storage medium is provided on which executable instructions are stored, wherein the instructions, when executed, cause one or more processors to execute a method according to embodiments of the present disclosure. According to one aspect of the present application, a computer program product is provided which comprises executable instructions, wherein the instructions, when executed, cause one or more processors to execute a method according to embodiments of the present disclosure. In the technical solutions according to various aspects of the present application, a virtual Fourier layer is used in a neural network (NN) model for solving physical problem entities (PDEs) to improve the NN model's ability to learn derivatives. This allows the physical data predicted by the NN model to exhibit lower derivative deviations in a gradient-based structural optimization process, thereby improving the performance of the gradient-based structural optimization. The technical solutions according to various aspects of the present application employ a hybrid inference method, combining neural operators and numerical solvers in the structural optimization process to suppress error accumulation and control deviation of the optimization trajectory. Further advantages of the technical solutions of the present application are explained below. Brief description of the characters A further understanding of the essence and advantages of the content of this application can be achieved by referring to the following drawings. Similar components or features may be designated with the same reference numerals in the drawings. Fig. 1 shows an exemplary process and framework for optimizing the structure of an object according to an embodiment. Fig. 2 shows a schematic diagram in which a gradient-based structure optimization process using neural operators according to an embodiment is affected by accumulated errors. Fig. 3 shows a schematic process for training a neural network model according to an embodiment. Fig. 4 shows an exemplary comparison between an ordinary neural operator and a neural reference operator according to an embodiment.Figures 5A to 5D each show an exemplary model architecture of a neural reference operator according to an embodiment. Figures 5E and 5F each show an exemplary structure of an integral operator layer according to an embodiment. Figure 6 shows an exemplary schematic view for determining a geometric deformation of a query domain relative to a reference domain according to an embodiment. Figures 7A and 7B show schematic diagrams of a process for performing a structural optimization of an object according to an embodiment. Figures 8 and 9 each show a schematic representation of an object and its physical system according to an embodiment. Figure 10 shows a method for determining a physical state of an object with a shape according to an embodiment. Figure 11 shows a method for structural optimization of an object according to an embodiment.Figure 12 shows a method for training a neural network model to determine the physical state of an object with a given shape according to an embodiment. Figure 13 shows a method for optimizing the structure of an object according to an embodiment. Figure 14 shows a device for determining the physical state of an object with a given shape according to an embodiment. Figure 15 shows a device for optimizing the structure of an object according to an embodiment. Figure 16 shows a device for training a neural network model to determine the physical state of an object with a given shape according to an embodiment. Figure 17 shows a device for optimizing the structure of an object according to an embodiment. Figure 18 shows a block diagram of a processing unit according to an embodiment. Detailed descriptions The items described herein will now be explained with reference to exemplary embodiments. It is understood that the discussion of these embodiments is intended solely to enable the person skilled in the art to better understand and implement the items described herein and is not intended to limit the scope of protection, applicability, or example set out in the claims. Fig. 1 shows an exemplary process and framework for optimizing the structure of an object according to an embodiment. In the example shown in Fig. 1, the object to be structurally optimized is an aircraft wing. As shown in Block 120, the shape of the wing is represented by a boundary λ, where the boundary λ is parameterized by structural data W. In this example, the structural data W can be a group of control points forming a spline curve representing the shape of the wing. In other words, the structure to be optimized is the shape of the wing, represented by a spline curve with a group of control points W. The symbol Ω denotes a problem domain, more precisely a geometric space of a physical system related to a wing of a specific shape.It is understood that the structural data W is a discretized representation of the boundary λ and that the two equivalently represent the shape of the object, which is why in the description here no longer strictly distinguishes between the two, but usually uses the symbol λ to denote the structural data of the shape of the object. As shown in Block 110, the lines with arrows represent the airflow over the wing. The physical state of the airflow over the wing can be represented by physical data of the physical system associated with the wing. In this example, the physical data can be a velocity u = u(x) and a pressure p = p(x). The symbol x represents spatial coordinates in the domain Ω. As shown in Figure 140, the relationship between the velocity u = u(x) and the pressure p = p(x), as well as the wing shape represented by the limit λ, can be characterized by a partial differential equation (PDE). It is understood that a PDE is a known system of physical equations that can be used to describe the three-dimensional motion of viscous fluid substances. A second-order PDE can be used to model weather, ocean currents, heat conduction, airflow around wings, water flow in pipes or reactors, and in many other applications. A PDE for characterizing a physical system can be derived according to physical laws relating to the physical system of the object. As shown in Figure 140,As shown in Figure 1, PDEs that characterize the physical system of the wing can be derived according to the physical laws relating to pressure and velocity of the airflow. Since these PDEs are used during the structural optimization process to control the optimization of the wing structure, they can be referred to as controlling PDEs or constraining PDEs. As shown by symbol 130 in Fig. 1, the goal of wing structure optimization is to achieve a desired pressure distribution on the wing surface by changing the structural parameters W or λ. An objective function J can be formulated as shown by symbol 130, where the goal of structure optimization is to minimize the objective function J, in this case being a function of the pressure p. For the wing shape parameterized by the structural data W and corresponding to the limit λ, the objective function J can be minimized by iterative optimization to achieve the desired pressure distribution on a wing surface. A PDE-constrained structural optimization problem, exemplified at 130 and 140 in Fig. 1, can generally be represented as equation (1): where “const.” represents a constraint, where Λ represents a structural space, which can also be called a design space, for example a high-dimensional design space, where the objective function is defined as Ĵ(λ) := J((u(λ),λ)), such as the objective function shown at 130 in Fig. 1, and where C(u, λ) represents the constraining PDE, such as the constraining PDE shown at 140 in Fig. 1. A gradient-based method for structural optimization seeks a gradient of the objective function with respect to design parameters, which is also commonly referred to as sensitivity, as shown in equation (2): In each cycle of structural optimization, the structural parameters or the design parameters λ can be iteratively updated by , where η represents a learning rate. Returning to the example from Fig. 1, the PDE solver 170, based on the current structural data λ = λt, solves the PDE 140 to determine the corresponding physical data u = u(λ), where in this example the physical data u include the velocity u = u(x) and the pressure p = p(x). After determining a current value of the objective function J = J ((u(λ), λ) at 130, the gradient of the structural optimization objective function J with respect to the structural data λ is determined at 150, for example, using equation (2). Then, at 160, the updated structural data are obtained. The above optimization process is performed iteratively to obtain the optimized structural data λ of the object. The PDE solver 170 can be implemented using conventional numerical solvers, such as the finite element method (FEM), or by a neural network model (also known as a neural operator). Numerical solvers can accurately solve the physical parameters of the PDE, but they have the disadvantage of requiring significant computational effort and time. For example, the large-scale numerical calculations required to solve the PDE can cause the structural optimization process to take hours or even days to converge to the optimal design. A neural operator implemented by a neural network model learns the mapping (GS) from geometric space to the PDE solution space, where G represents a geometry and S represents a solution. This allows the solution to be predicted much faster, which can, for example, reduce the time of the structural optimization process by orders of magnitude and thus offer advantages in the structural design of objects.However, the use of a NN model to implement the PDE solver 170 can lead to deviations in the structure optimization process. For example, for a trained NN model as represented by equation (3): where θ is a learning parameter of the NN model, and Λ and represent a structure space and a solution space, respectively. Then the gradient calculation shown in equation (2) is represented as .This can lead to several problems: First, there is a discrepancy between the predicted solution of the NN model and the true solution u, resulting in a different objective function value for J(u); second, similarly, this leads to an error between the gradient and the true gradient; third, these errors accumulate continuously through multiple iterations of the gradient-based structure optimization process, which can cause the optimization trajectory to enter an out-of-distribution (OoD) region of the neural operator, resulting in unreliable predictions by the neural operator and a deviation in the structure optimization process. Figure 2 shows a schematic diagram in which a gradient-based structure optimization process using a neural operator 170 is affected by accumulated errors. In this figure, λ, u, and û represent structure data input to the neural operator 170, an output solution prediction, and a true value of the solution, respectively. The curve with arrows on the left side of Figure 2 is a structure optimization trajectory or a structure optimization process using the neural operator 170, while the curve with arrows on the right side is a structure optimization trajectory based on the true solution. Comparing the two trajectories reveals that the structure optimization trajectory using the neural operator 170 exhibits a significant deviation from the starting point (λ0, u0) to the final optimization point (λT, uT), and its accumulated error becomes uncontrolled.The technical solution presented in this disclosure addresses the aforementioned problems from several perspectives. Firstly, it enables the neural network (NN) model to effectively learn the gradient of the optimization goal with respect to the structure. Secondly, it effectively samples and utilizes data close to the optimum. Thirdly, it manages the accumulated error of the neural operator during the optimization process. According to the first aspect, the technical solution introduced into the NN model enhances the gradient learning capability of the neural operator. Secondly, it utilizes data from historical optimization trajectories in various structure optimization projects as a training set.Furthermore, according to another aspect, the technical solution of the present disclosure uses a NN model architecture of a Reference Neural Operator (RNO) to achieve more effective learning of the NN model from these training data. Moreover, by introducing a virtual physical layer into the RNO model architecture, the gradient of the structure optimization target with respect to the structure can be effectively learned using historical optimization trajectories from various structure optimization projects. According to the third aspect, the technical solution of the present disclosure uses a hybrid inference method in which the neural operator and the numerical solver are used in combination during structure optimization, thereby suppressing error accumulation and controlling the deviation of the optimization trajectory. It is understood that the three aspects mentioned above constitute the solution shown in Fig.The two problems of deviation or out-of-control of the optimization trajectory shown can be solved or mitigated individually, in partial or complete combination. In a particular embodiment of the present disclosure, not all of the above-mentioned aspects need be included, and the advantages of the technical solution of the present disclosure are not limited to the three aspects mentioned above; further advantages are presented in the following description. Fig. 3 shows a schematic process for training a NN model according to an exemplary embodiment. The NN model 370 with learning parameters θ is, for example, the neural operator 170 shown in Fig. 1, whose input is the spatial structure data λ and whose output is the physical data u as a PDE solution. Since the gradient of NN model 370 is crucial for gradient-based structure optimization, derivative learning is performed on NN model 370. The sensitivity is defined by equation (2). Derivative learning means that the sensitivity of NN model 370 is monitored. For gradient-based structure optimization trajectories in the training dataset (e.g.,In this gradient-based structural optimization process, the PDE solution is determined by a numerical solver, or the optimization result of this gradient-based structural optimization process is an already verified optimal result. Given a sample in this structural optimization trajectory comprising the input structural data λ and the corresponding true value û of the PDE solution, the true value of the optimization target, and the true value of the gradient of the optimization target with respect to the structural data Ĵ (here, .̂ is used to denote quantities with true values), a sensitivity loss solution can be imposed on the NN model 370: where represents the PDE solution u predicted by the NN model 370 based on the input data λ, and ∇λJ represents the derivative of the optimization target J with respect to the structural data λ. The prediction loss of the NN model 370 is therefore the total loss: where a > 0 is a hyperparameter used to control the weight of the sensitivity loss. As shown by the dashed line in Fig. 3, the model parameters θ are updated during a model training process based on a loss L. Any suitable model parameter optimization method can be used to update the model parameters θ based on the loss L; for example, the Adam algorithm is used to update the model parameters θ. It is understood that the model training process shown in Fig. 3 is applicable to various neural operators, including the neural reference operator (RNO) introduced below. Fig. 4 shows an exemplary comparison between an ordinary neural operator (NO, Neural Operator) and a neural reference operator (RNO) according to an embodiment. As shown in Fig. 4 (left), the ordinary neural operator NO maps the geometric shape / structure data λ directly to the solution u of the PDE, which typically requires a large dataset to cover various geometric shapes so that the neural operator can learn this mapping relationship. As shown in Fig. 4 (right), unlike the GS, the RNO does not predict the PDE solution of an arbitrary geometric structure, but rather, given a reference design with a geometric reference structure λ and a reference solution ur, predicts the change δu of the PDE solution. The underlying principle is that similar changes across different geometric shapes have similar effects on the solution of the PDE.Consider, for example, the flow of a fluid in a channel with one hole: regardless of how many holes the channel has, enlarging, reducing, or moving a hole located near the channel's entrance will have similar effects on the fluid. The RNO is able to predict changes in the solution caused by changes in its geometric shape. Specifically, as shown in Fig. 4 (right), the RNO uses the reference solution ur, the corresponding geometric reference form λr, and the shape change φ of a geometric query form λq compared to the geometric reference form as input and outputs a prediction δu for the solution change magnitude between the reference solution and the query solution. The predicted solution uq for the geometric query form λq can then be obtained based on the reference solution ur and the solution change magnitude δu. The RNO shown in Fig. 4 (right) can be expressed by equation (6): where θ represents a learning parameter of the NN model, Λ and λ represent a structure space and a solution space, respectively, and λ represents a deformation on the structure space Λ, i.e., a deformation between a geometric reference structure λ and a geometric query structure λq, with s ≥ 1. It is understood that the mapping relationship shown in equation (6) is not the only implementation of RNO. In another embodiment, RNO can be implemented as shown in equation (7): In the RNO shown in equation (7), the RNO takes the geometric query form λq, the reference solution ur, and the shape change φ between the geometric query form λq and the geometric reference form λ as input and outputs the predicted solution uq. In this embodiment, by setting uq as the prediction target of the NN model, the solution variation magnitude δu can be learned indirectly, since jump connections exist in a residual structure of the neural network. Therefore, in a practical application, uq can be used directly as the prediction target of the NN model, which can simplify the implementation process. Through such a modification, the RNO can be understood as a new extension of the ordinary neural operator. This extension is key to increasing the learning effectiveness of the RNO, as it enables learning from intrinsic relationships between the solution data. Fig. 5A shows an exemplary model architecture of an RNO according to one embodiment. For ease of explanation, rounded blocks are used in Fig. 5A to represent processing units and rectangular blocks to represent data. A model architecture 500 comprises a preprocessing module 510 and an RNO module 520. The preprocessing module 510 performs preprocessing of input data, which includes query data Q and reference data R. The query data Q comprises structural data, where xqi represents the coordinates of a point qi in a structural query domain λq. For a one-dimensional domain, xqi represents, for example, a one-dimensional value. For a two-dimensional domain, xqi represents a two-dimensional pair of values. For a three-dimensional domain, xqi represents a three-dimensional triple of values. In general, for an n-dimensional domain, xqi represents an n-dimensional tuple of values. The reference data R includes reference structural data, where xri represents the coordinates of a point ri in the structural reference domain λr, and a reference solution, where u(xri) represents a solution at a point xri, which can be ds-dimensional.If the solution of the PDE includes, for example, three physical quantities, e.g., velocities u1, u2 and pressure p, then u(xri) is a three-dimensional triple of values. In one embodiment, the input sequences ur, λrund λq are tensors of the form Nr× ds, Nr× n and Nq× n, respectively. The preprocessing module 510 includes a deformation determination module 5110, which determines a geometric deformation φ of the structural query domain λq relative to the structural reference domain λr based on the input structural data. Fig. 6 shows an exemplary schematic view for determining a geometric deformation of a query domain relative to a reference domain according to an embodiment. Boundaries 610 and 620 represent the boundaries of the query domain λq and the reference domain λr, respectively. Using the example of Fig. 5A, the query domain λq and the reference domain λr each contain discrete points. To construct the geometric deformation φ of the query domain relative to the reference domain, displacement vectors of points on the query domain boundary 610 to corresponding points on the reference domain boundary 620 can first be determined, as represented by the vectors from circle 610 to circle 620; then, based on the displacement vectors of the points on the boundaries, the displacement vectors of all points in the query domain can be determined. The displacement vectors between boundaries 610 and 620 can be determined by any suitable method. For example, boundary data representing domain boundaries can be used to determine the displacement vectors between the domain boundaries. The boundary data can be geometric parameters. In this example, the geometric parameters can be a circle center and a radius, and the displacement vectors between the two circles 610 and 620 can be found using a matching procedure based on the coordinates of the circle center and the points on the circles. The boundary data can also be a group of boundary points; accordingly, displacement vectors between the boundary points of the two circles 610 and 620 can be found using a matching procedure. After obtaining the displacement vectors between the two circles 610 and 620, for each point xqi630 in the query domain, a displacement vector of the boundary point on boundary 610 that is closest to that point xqi30 is selected. This displacement vector is weighted according to a distance between that point xqi630 and its nearest boundary point. The selected and weighted displacement vector is then used as the displacement vector 640 of that point xqi630. For the query domain λq, the geometric deformation φ can be represented by the set of displacement vectors of the individual points in the query domain. The geometric deformation φ containing these displacement vectors can be represented as a tensor of the form Nq×n. It is understood that the specific procedure for determining the geometric deformation φ described above in connection with Fig. 6 is merely an example, and that any suitable procedure for determining the deformation φ between the query domain and the reference domain can be used. For example, a cut-off function can be applied to the displacement vector 640 mentioned above to prevent points in the domain from being shifted beyond the geometric boundary. For example, in an example where the number of grid points in the query domain and the reference domain is the same, the geometric deformation φ between the two domains can be represented by the set of displacement vectors between corresponding points in the two domains, e.g., φ = λr - λq or φ = λq - λr. It is understood that, although the example of Fig. 6 shows one boundary 610 of a domain, in practical applications there may be multiple boundaries and the process for determining the displacement vectors representing the geometric deformation φ is applicable to geometric domains with multiple boundaries. It is understood that the query domain and the reference domain shown in Fig. 6 correspond to the query form and the reference form of an object, respectively, where the form of the object can also be equivalently referred to as the geometric form or geometric structure of the object. Therefore, in the context of this description, the terms "domain," "form," and "geometric structure" can be used interchangeably. As shown in Fig. 5A, coordinate points of the query domain can be transformed into the reference domain using the geometric deformation φ. Specifically, the displacement vectors δxq, which represent the geometric deformation φ, can be added to the coordinate points to obtain the transformed coordinate points in the reference domain. The preprocessing module 510 includes a solution estimation module 5120, which, based on the coordinate points representing the reference domain, the reference solution at the coordinate points, and the transformed coordinate points, determines an estimate of the PDE solution at the transformed coordinate points. The estimate of the PDE solution at the transformed coordinate points can be represented as ur·φ(xqi), i.e., a tensor of the form Nq× ds. Specifically, the reference solution at the coordinate points is interpolated to an interpolated solution ur·φ(xqi) at the transformed coordinate points, which serves as the estimate of the solution at the transformed coordinate points. The solution estimation module 5120 can be referred to as an interpolation module. It is understood that any suitable interpolation method can be used to perform such an interpolation. The architecture of RNO 520 can be represented as: where represents an encoder that lifts input values ​​into a latent space, where represents an integral operator layer with (I = 1,..., L), where represents a decoder that projects latent variables back into a target space, where s represents the dimension of the features in the latent space. The RNO 520 comprises an encoder 5210, several integral operator layers 5220, and a decoder 5230. The encoder 5210 includes a first sub-encoder 5211, a second sub-encoder 5212, and a third sub-encoder 5213. The first sub-encoder 5211 takes the interpolated solution ur·φ(xqi) as input and outputs a first feature sequence v1 ∈ ℝNq ×s. This feature sequence can also be called a feature map. The second sub-encoder 5212 takes the geometric deformation φ = δxq as input and outputs a second feature sequence v2 ∈ ℝNq ×s. The third sub-encoder 5213 uses the transformed coordinate points φ(xqi) as input and outputs a third feature sequence v3∈ ℝNq×s. It is understood that the encoder 5210 can be implemented by any suitable neural network, with, for example, these sub-encoders being implementable using a multilayer perceptron (MLP) neural network. The integral operator layer 5220 comprises an integral kernel KN for processing the aforementioned first, second, and third feature sequences, corresponding to the reference solution ur, the geometric deformation φ, and the query structure data λq, respectively, to ultimately obtain a feature sequence zL ∈ ℝNq×s processed by integration. The decoder 5230 takes the feature sequence ZL as input and outputs a change in the solution δu, which is a tensor of the form Nq× dsist. A predicted value of the solution uq = ur·φ(xq) + δu can then be obtained. In another embodiment, as shown by the dashed line at the output of the decoder 5230 in Fig. 5A, the decoder 5230 can directly output the predicted value of the solution uq = u, which is a tensor of the form Nq× dsist. Fig. 5B shows an exemplary model architecture of an RNO according to one embodiment. In Fig. 5A and Fig. 5B, identical or corresponding symbols are used to denote identical or corresponding units. In the embodiment shown in Fig. 5B, the third sub-encoder 5213 uses the coordinate points of the query domain as input and outputs the third feature sequence v3∈ ℝNq×s. In contrast, in the embodiment shown in Fig. 5A, the third sub-encoder 5213 uses the transformed coordinate points φ(xqi) as input and outputs the third feature sequence v3∈ ℝNq×s. The remaining operations of the RNO in the embodiment shown in Fig. 5B are identical to those of the RNO in the embodiment shown in Fig. 5A and are therefore not described again. Fig. 5C shows an exemplary model architecture of an RNO according to one embodiment. In Figs. 5A to 5C, identical or corresponding symbols are used to denote identical or corresponding units. The embodiment shown in Fig. 5C differs from the embodiment shown in Fig. 5B in that the solution estimation module 5120 determines an estimate of the PDE solution ur·xqian at the coordinate points based on the coordinate points of the reference domain, the reference solution at the coordinate points, and the coordinate points of the query domain. Specifically, the reference solution at the coordinate points is interpolated to an interpolated solution ur·xqian at the coordinate points, which serves as the estimate of the solution at the coordinate points. The remaining operations of the RNO in the embodiment shown in Fig. 5C are the same as those of the RNO in the embodiment shown in Fig. 5B and are therefore not described again. Fig. 5D shows an exemplary model architecture of an RNO according to one embodiment. In Figs. 5A to 5D, identical or corresponding symbols are used to denote identical or corresponding units. In the embodiment shown in Fig. 5D, the preprocessing module does not include a solution estimation module 5120, but instead uses the reference solution directly as input for the first sub-encoder 5211 to obtain the first feature sequence v1∈ ℝNq×s, where the number of grid points of the reference domain Nr and the number of grid points of the query domain Nq can be equal. Furthermore, the geometric deformation can be simplified as the difference between the coordinate points of the reference domain and the corresponding coordinate points of the query domain, which can be represented as φ = δxq = λr - λq or φ = δxq = λq - λr. The remaining operations of the RNO in the embodiment shown in Fig. 5D are the same as those of the RNO in the embodiment shown in Fig. 5C and are therefore not described again. Fig. 5E shows an exemplary structure of an integral operator layer 5220 according to an embodiment. In this embodiment, an attention-based network structure is used to implement the integral kernel KN, which can also be referred to as a transformer-based network architecture. Although not shown in Fig. 5E, in one embodiment a third feature sequence v3 is processed using a Q-matrix (query matrix) and a K-matrix (key matrix) to obtain a Q-feature sequence (query feature sequence) q and a K-feature sequence (key feature sequence) k, where the Q-matrix and the K-matrix are learnable parameters of the NN model. The Q-matrix and the K-matrix are s×s matrices, while the Q-feature sequence q and the K-feature sequence k are Nq×s matrices. Thus, for a series of inputs, Q and K feature sequences q(X), k(X) ∈ ℝNq×s and M input value feature sequences vj(X) ∈ ℝNq×s(j = 1,..., M) are obtained.In this example, M=2, and the corresponding value feature sequences vjs are a first and second sequence v1, v2. As shown in Fig. 5E, attention can be expressed as follows: attn(x,yi) = softmax(q(x)kT(yi)), which can also be called the attention vector or attention matrix. A distance weight helps to enhance attention according to the spatial relationship between elements of the Q and K feature sequences q(X), k(X). Given that in certain problems (e.g., in fluid dynamics) the change in the solution may be closely related to the position of the deformation, a distance weight can be applied to the attention operation, thereby implementing the integral operator as a distance-aware cross-attention layer (DACA), which can be expressed by equation (8): In this example, N = Nq and M = 2, where d(x, yi) represents a distance function, e.g., the Euclidean distance, where r represents a hyperparameter. The attention attnj for each value-feature sequence vj can have its own learnable Q-matrix and K-matrix to learn different core parameters for different value-feature sequences vj. As shown in Fig. 5E, for the I-th integral operator layer or attention layer 5220-1 (i.e., the first integral operator layer when I = 1), the input zl is the third feature sequence v3; when I > 1, the input zl is the output of the previous integral operator layer 5220-(1-1). The output of integral operator layer 5220 is the output of the last integral operator layer 5220-L. The first and second feature sequences v1 and v2, which serve as value feature sequences vj, are used as input for each integral operator layer or attention layer 5220-1. In this example, each integral operator layer is constructed as: where Wl is defined by equation (8), where f is a composite function consisting of layer normalization and a multilayer perceptron (MLP) with a nonlinear activation function. It is understood that the attention-based integral operator layer shown in Fig. 5E represents one implementation, and in concrete applications there may be various specific implementations that realize an attention-based integral operator layer. Looking back at the sensitivity learning of the training process shown in Fig. 3, the nonlinearity of the attentional unit in a transformer structure leads to complex derivation, which is reflected in the following aspects: (i) the derivative of the softmax function in an attentional mechanism: let s = softmax(z) and z ∈ ℝⁿ, then (ii) the chain rule and product rule for the derivatives with respect to Q, K, V. Even in a linear transformer model, the derivation of the normalization of linear attention is quite complex. These components of the transformer structure introduce a large derivation bias into the structure of the neural operator, which is detrimental to sensitivity learning of the neural operator. On the other hand, the Fourier transform in a Fourier transform-based neural network layer is indeed linear, so it does not introduce any additional distortion with respect to the derivative. However, the limitation of the Fourier-based neural network layer is that it can only process a fixed number of uniform grid points, while many applications, especially structural optimization applications, need to process irregular grids with variable scores. The technical solution of the present disclosure proposes a virtual Fourier layer that transforms irregular grids into a virtual physical space with a fixed number of detection points in order to perform Fourier transform-based processing on these fixed number of detection points within the virtual physical space.The use of the virtual Fourier layer on the network structure enables the neural operator to perform better sensitivity learning while simultaneously processing irregular grids with variable scores. Fig. 5F shows an exemplary structure of an integral operator layer 5220 according to an embodiment. In this embodiment, an integral kernel KN is implemented using a network structure based on Fourier transforms. For the I-th integral operator layer or Fourier layer 5220-1 (i.e., the first integral operator layer when I = 1), the input zl is a feature sequence obtained based on the first to third feature sequence v1, v2, v3, e.g., when l > 1, the input zl is the output of the previous integral operator layer 5220-(l-1). The output of the integral operator layer 5220 is the output 5220-L of the last integral operator layer 5220-L. As shown in the figure, the integral kernel KN-I comprises a first transformation layer P1, a Fourier transform-based integral kernel FKN, and a second transformation layer P2. The first transformation layer P1 transforms the feature sequence zl, which contains Nq feature vectors and corresponds to Nq grid points in physical space, into a virtual physical space with a fixed number of detection points (i.e., M detection points) to obtain a feature sequence zv containing M feature vectors. The Fourier integral kernel FKN performs Fourier transform-based processing on the feature sequence zvi in ​​the virtual physical space to obtain an integrated feature sequence containing M feature vectors.The second transformation layer, P2, transforms the feature sequence in virtual physical space back into physical space to obtain an integrated feature sequence containing Nq feature vectors, corresponding to the Nq grid points in physical space. In the depicted structure of the integral operator layer, Fourier-based integral processing is performed on the feature sequence in virtual physical space, which is why the integral operator layer 5220-I can be referred to as a virtual physical layer or virtual Fourier layer. Since the number of grid points Nq in physical space is variable in various application examples, and the Fourier integral kernel FKN can only process a fixed number of grid points, the use of the virtual Fourier layer shown in the figure enables the NN model to process various applications with a variable number of grid points.At the same time, as already mentioned, the use of the virtual Fourier layer on the network structure enables the neural operator to perform better sensitivity learning. In one embodiment, the first transformation layer P1 performs a pointwise transformation of each feature vector in the feature sequence zl. For example, for each feature vector zli ∈ ℝ1 × sin of the input feature sequence zl ∈ ℝNq × s(i = 1,..., Nq), a pointwise transformation projection is performed by the first transformation layer P1 to obtain Project(zli) ∈ ℝ1 × M. This transformation projection can be implemented by any linear transformation, e.g., by an s × M-dimensional projection matrix, so that, using Nq projection matrices, the Nq feature vectors can each be linearly transformed into Nq vectors Project(zli) ∈ ℝ1 × M(i = 1,..., Nq). The Nq projection matrices can be learnable parameters. Let this result in a distribution matrix l ∈ ℝNq×M, where the element li,je is a probability, e.g.a log probability indicates that the i-th point in the feature sequence z is linearly classified as the j-th detection point in the virtual physical space. Then, the first transformation layer P1 applies a normalized exponential function softmax() in one direction of the first dimension of the distribution matrix I to obtain a weight matrix ω(1)∈ ℝNq×M. Specifically, the elements in the weight matrix are with i = 1,..., Nq. Then, based on the weight matrix ω(1), a weighted aggregation of the Nq feature vectors in the feature sequence zl is performed to obtain the feature sequence zv∈ ℝM×s in the virtual physical space. For example, the feature vectors zvj(j = 1,...,M) in the feature sequence zv are obtained by weighted aggregation, as shown in equation (10): The feature sequence zv∈ ℝM×s can be intuitively understood as a one-dimensional signal that is sampled by M detection points in virtual physical space, where each detection point has s dimensions or comprises s channels. As shown in Fig. 5F, the Fourier integral kernel FKN performs Fourier transform-based processing on the feature sequence zv, where the feature sequence zvM comprises feature vectors as M detection points in virtual physical space. In one embodiment, the Fourier integral kernel FKN comprises a Fourier transform layer, a linear transform layer Rθ, and an inverse Fourier transform layer. The Fourier transform layer performs a Fourier transform of the feature sequence zv∈ ℝM×s and truncates a high-frequency component to obtain a Fourier sequence, where k represents the number of frequency modes of the Fourier transform. The linear transform layer Rθ performs a linear transformation of the Fourier sequence to obtain a weighted Fourier sequence, where for 1 ≤ i ≤ k, 1 ≤ j, and .The inverse Fourier transform layer performs an inverse Fourier transform on the weighted Fourier sequence to obtain an integrated feature sequence in virtual physical space. The Fourier transform and inverse Fourier transform mentioned above can be performed in any suitable manner. For example, the Fourier transform and inverse Fourier transform can be performed using the Fast Fourier Transform (FFT) and Fast Inverse Fourier Transform (FFT-1), respectively. Since the Fourier transform-based processing performed by the Fourier integral kernel (FKN) is inherently linear, its derivative does not introduce any additional distortion to the FKN. This is highly advantageous for learning the derivative of neural operators. The second transformation layer P2 transforms the feature sequence in virtual physical space back into physical space to obtain an integrated feature sequence containing Nq feature vectors and corresponding to the Nq grid points in physical space. According to one embodiment, in one direction of the second dimension of the distribution matrix l ∈ ℝNq×M above, the normalized exponential function softmax() is applied to obtain a weight matrix ω(2) ∈ ℝNq×M. Specifically, the elements in the weight matrix are j = 1, ..., M. Then, based on the weight matrix ω(2), a weighted aggregation of the M feature vectors in the feature sequence is performed to obtain the feature sequence in virtual physical space. For example, the feature vectors in the feature sequence are obtained by weighted aggregation, as shown in equation (11): As shown in the lower branch of Fig. 5F, the weighting layer W of the integral operator layer 5220-1, using the weight matrix W ∈ ℝs×s, performs its linear transformation on the input feature sequence zl∈ ℝNq×s to obtain a weighted feature sequence Wzl∈ ℝNq×s, which, after addition with an integral feature sequence, undergoes an activation function σ to obtain an output feature sequence of the integral operator layer 5220-I. The activation function σ is a nonlinear activation function, such as a Gaussian error linear unit (GeLU). It is understood that the integral operator layer shown in Fig. 5F, based on Fourier transforms, is an implementation according to this embodiment, whereby in specific applications obvious modifications can be made to this implementation in order to realize an integral operator layer based on Fourier transforms. The following section, referring to Fig. 3 and using the RNO shown in Figs. 5A to 5F as an example, explains the training of a neural network (NN) model. The RNO essentially learns the relationship between the deformation of the geometric structure and the magnitude of the change in the PDE solution. Since similar changes to different geometric shapes have similar effects on the PDE solution, the RNO can be better generalized to predict PDE solutions for different objects with varying shapes and, consequently, learn more efficiently from historical optimization trajectories in different structural optimization projects. Intuitively, the RNO can learn to make high-quality predictions by referencing neighboring data within the same trajectory.In fact, the intrinsic relationships between data points along the same trajectory are important, and the RNO is able to detect changes in the solution based on subtle transformations between input structures. To train the RNO, a pair of data points must be provided for each query; this can be referred to as the reference query data pair. According to one example, each data point in the same trajectory can be paired with its nearest neighboring data points, since the training data is derived from historical optimization trajectories of structural optimization. According to one embodiment, this pairing process can be performed using a user-defined data loader, where the data loader randomly selects a reference data point from the same trajectory for each data point. Specifically, in a data trajectory, represents the number of optimization steps in that trajectory, i.e., the number of data points. The data loader randomly assigns a reference data point j to an i-th data point within a given index distance d. That is, it selects an index j ∈ [I,r], where I = max(0, id) and r = min(Ns, i + d). In this example, the index j of the reference data can be randomly set to i; this can regulate the predictive behavior of the RNO when the query data is close to the reference data. Since the reference data and the query data are selected from the same structural optimization trajectory, they also exhibit the same grid structure. Even with deformed grids in the shape optimization, there is a one-to-one correspondence between reference data and query data, thus reducing the complexity of constructing the deformation φ and interpolating between different grids, as described in the embodiments shown in Figures 5A to 5F. This is an advantage of using the structural optimization trajectories as the training dataset. Table 1 below describes a training process according to an exemplary implementation in the form of pseudocode. In this embodiment, Ne represents the number of training rounds, where the objective function value J can be calculated based on a structural optimization goal corresponding to a historical optimization trajectory, and where a gradient can be calculated using a gradient computation tool. Table 2 below describes a training process according to an exemplary implementation in the form of pseudocode. In this embodiment, the reference inputs λ and φ are randomly discarded during the training process, causing the RNO to degenerate into an ordinary neural operator that maps from λ to u. For example, the discard rate rdrop can be set to 0.3 to compensate for training performance. Enabling this random discard operation during training is advantageous because it allows the RNO to flexibly perform inference without reference data. This is particularly useful in the initial stages of the optimization process, as the RNO can operate without dependence on any ground-truth solution. Thus, unnecessary computations by the numerical solver can be avoided in the initial phase of the structural design process. Fig. 7A and Fig. 7B show schematic diagrams of a process for carrying out a structural optimization of an object according to an exemplary embodiment. Since the RNO predicts the query solution based on the reference solution, it can recalibrate its own prediction and thus recalibrate the direction of the structure optimization if a new ground-truth solution is provided during the structure optimization. Utilizing the aforementioned properties of the RNO, according to an embodiment in the structure optimization process shown, for example, in Fig. 1, where the neural operator 170 is an RNO, a numerical solver determines the PDE solution uq as the calibrated ground-truth solution based on the current structure data λq when a certain condition is met. In the next prediction by the RNO, this calibrated ground-truth solution is used as the reference solution, thereby continuously calibrating the RNO's prediction to suppress accumulated errors.For example, the specified condition could be that the difference between the last calibrated ground-truth solution determined by the numerical solver and the last predicted solution determined by the RNO is greater than a threshold. The difference between the calibrated ground-truth solution and the predicted solution could be their distance, where any suitable distance representing the difference between two data sets can be used, such as the Euclidean distance. When the specified condition is met, it means that the reference degree of the calibrated ground-truth decreases, so a new calibrated ground-truth is generated to recalibrate the RNO calculation.Another example is that the specific condition can include a fixed number of rounds having passed since the last start of the numerical solver, so that after each fixed number of structure optimization rounds a calibrated ground-truth solution is determined by the numerical solver. Figures 7A and 7B show the same optimization trajectory, with the two diagrams showing, for clarity, trajectories projected onto a plane and a plane J-Λ, respectively. In Figure 7A, the solid curve shows the trajectory of the RNO-predicted solution u during a structure optimization process, while the dashed curve shows the trajectory of the ground-truth solution u during a structure optimization process. In Figure 7B, the solid curve shows the trajectory of an optimization target J determined during a structure optimization process based on the RNO-predicted solution u, while the dashed curve shows the trajectory of an optimization target J determined during a structure optimization process based on the ground-truth solution u. As shown in Figure 7B, the trajectory of the optimization target J is determined during a structure optimization process based on the ground-truth solution u.As shown in Fig. 7A, starting from the input structure λ0, the PDE solution u is predicted using the RNO in iterative structure optimization rounds to iteratively optimize the structure or form λ of the object. If, in a given round, the current form is λ1, the numerical solver is called due to the fulfillment of the trigger condition to determine the calibrated ground-truth solution of the PDE. As shown by the short arrow on the dashed line at λ1, the prediction accuracy of the RNO is corrected in a timely manner by utilizing the RNO's property of predicting the query solution based on the reference solution. Simultaneously, as shown in Fig. 7B, the accuracy of the optimization target is corrected, effectively suppressing accumulated errors. As shown by the short arrows on the dashed lines at λ1, λ2...λn on the abscissa in Fig. 7A and Fig. 7B, the accuracy of the optimization target is also corrected.As shown in Figure 7B, when the trigger condition is met, the numerical solver is called to perform the calibration process. According to one embodiment, in a structure optimization round, noise can be added to current structure data λ to obtain multiple (i.e., N1) structure data instances, where λ represents a normal distribution with a mean of 0 and a standard deviation of σ, where σ can be set to, for example, 5%. Based on the N1 structure data instances, the RNO generates N1 corresponding PDE solution instances. Based on the N1 PDE solution instances, N1 optimization target value instances are generated, thereby generating N1 gradient instances of the optimization target with respect to the current structure. Then, based on the N1 gradient instances, a gradient value for the current structure optimization round is obtained. For example, the N1 gradient instances are averaged to obtain a final gradient value, as shown in equation (12): In this embodiment, by obtaining the gradient value of the current structure optimization round by adding noise to the current structure data, the gradient can be further smoothed during the structure optimization process, thereby improving the performance of the structure optimization. According to one embodiment, when processing one of the input structure data λ by the RNO, several temporary solutions can be determined, each based on several (N2) reference data, where φj is obtained based on the current structure data λ and the reference structure λrj. Then, a predicted solution u is obtained based on the N2 temporary solutions. For example, the N2 temporary solutions are averaged to obtain a final predicted solution u, as shown in equation (13): In this embodiment, the structural data λi and the corresponding PDE solutions ui can be cached in the last several optimization steps, with the cached data being used as the reference data mentioned above in the current optimization round. It is understood that in certain optimization rounds the number of cached reference data points is less than N2, and processing analogous to equation (13) is then performed based on at least a subset of the cached reference data points. In this embodiment, by predicting the PDE solution based on multiple reference data points, the gradient can be further smoothed during the structure optimization process, thereby improving the performance of the structure optimization. When using a combination of the embodiments shown in equations (12) and (13), the maximum number of forward inferences of the RNO in a single structure optimization round is N1N2. Since the derivation operations of the optimization target J with respect to the individual structure instances (λ+εi) are independent of each other and can therefore be performed simultaneously, the N1N2 forward inferences of the RNO do not significantly slow down the derivation calculation of equation (12). Table 3 below describes a structural optimization process of an object according to an exemplary implementation in the form of pseudocode. In this embodiment, dist (ut, ugt) represents the distance between the latest predicted solution and the latest calibrated ground-truth solution. The operation of resetting cache B = [ugt] causes historical values ​​in cache B to be deleted, leaving only ugt stored. In subsequent optimization steps, the solutions predicted by the RNO are stored sequentially. In one embodiment, after storing N2 solutions in cache B, the oldest value is replaced by the newest value in a FIFO (First-In, First-Out) order. It is understood that, although not shown in Table 3, the structure λ corresponding to the cached solutions u must also be cached. It is understood that the embodiment shown in Table 3 provides a specific optimization process, and obvious modifications to the process shown can be made in various embodiments. For example, the procedure shown in equations (12) and (13) can be omitted. Instead, in one structure optimization step, only a forward inference from the RNO is performed to obtain the PDE solution ut, and accordingly, the optimization target value J and its gradient δJ with respect to the structure are calculated based on this PDE solution. For example, only the procedure of equation (12) or only the procedure of equation (13) can be used.For example, if an evaluation condition is met, the ground-truth solution obtained by the numerical solver can be directly used as the solution for the current round; and accordingly, the optimization target value J and its gradient δJ with respect to the structure are calculated based on this PDE solution. As another example, the optimization process can omit the use of the parameter warm_up_steps and the corresponding evaluation condition i > warm_up_steps. As yet another example, the order of the individual steps in the processing process shown in Table 3 can be changed, with more or fewer steps being included. Fig. 8 shows an exemplary schematic representation of an object and its physical system according to an embodiment. The physical system shown in Fig. 8 corresponds to a guide vane, where the boundaries γ2 to γ5 are guide vanes in the corresponding regions. Limiting PDEs used to describe the physical system associated with the guide vanes can be as follows: where x = (x1, x2) is a spatial coordinate, u = (u1, u2) is a velocity, k is a given constant, f = f(x) is a given function, γ1,left is a left part of γ1, γ1,left,top is an upper part of γ1,left, and γ1,left,bottom is a lower part of γ1,left, where γWall = γ1,top ∪ γ1,bottom ∪ γ2 ∪ γ3 ∪ γ4 ∪ γ5 ∪ γ1,left,bottom. According to one embodiment, the solution of the PDE of the physical system of the guide vane can be predicted by the RNO according to various embodiments of the present disclosure (e.g., the NN model described in connection with Figs. 5A to 5F). According to one embodiment, the RNO can be trained by the training method according to various embodiments of the present disclosure (e.g., the training method shown in connection with Fig. 3). According to one embodiment, the objective of the guide vane structure optimization is to optimize the shape and position of the guide vanes γ2 to γ5 in order to obtain a uniform fluid flow at the outlet γ1, on the right, while simultaneously minimizing the energy consumed by the fluid. The objective function of the structure optimization can be derived based on fluid dynamics, as shown in equation (15): According to one embodiment, the structural optimization process described above in conjunction with Figs. 7A and 7B can be used to perform the optimization design of the guide vane. In iterative structural optimization steps, the RNO can be used to predict the corresponding physical data u based on the current structural data λ, the reference structure, and the physical reference data. If necessary, the numerical solver is triggered to determine the corresponding physical data u based on the current structural data λ. The geometric structural parameters of the object can include the coordinates of the center point and the radius of the boundaries γ2 to γ5. After lattice discretization of the geometric structural domain Ω formed by the boundaries γ1 to γ5, discretized structural data λ can be obtained, e.g.,with λ={( x1, x2, m1)}N, where N represents a number of discretized grid points, x1 and x2 represent the spatial coordinates, and m1E {0,1} represents a mask used to indicate whether the point (x1, x2) belongs to an interior boundary, i.e., γ2∪γ3∪γ4∪γ5. Fig. 9 shows an exemplary schematic representation of an object and its physical system according to an embodiment. The physical system shown in Fig. 9 corresponds to a two-dimensional battery pack. Restricting PDEs used to describe the physical system associated with the two-dimensional battery pack can be as follows: where x = (x1, x2) is a spatial coordinate, t is a time coordinate, T(x, t) is a time-dependent temperature, and k is a thermal conductivity, where h is a heat transfer coefficient. VT = (∂T / ∂x1, ∂T / ∂x2). Here, Ta, Tc, and Tw represent the temperature of the air, the temperature of a battery cell (nc = 11 battery cells with a radius of rc), and the temperature of a cooling tube (nw = 6 tubes with a radius of rw), respectively, and are set to Ta = 0.1, Tc = 5, and Tw = 1. T0 represents an initial temperature, which in this example is set to T0 = 0.1. In Fig. 9, you represents an outer boundary of the battery pack, γc,i represents a boundary of a battery cell, and γp,i represents a boundary of the cooling tube.The temperature T(x, t) is the unknown physical quantity that must be solved according to this PDE. According to one embodiment, for a time-dependent PDE of the physical system of the two-dimensional battery pack, the solution of the PDE can be predicted by the RNO according to various embodiments of the present disclosure (e.g., the NN model described in connection with Figs. 5A to 5F) by using time t as an additional spatial dimension. According to one embodiment, the RNO can be trained by the training method according to various embodiments of the present disclosure (e.g., the training method shown in connection with Fig. 3). According to one embodiment, the boundaries of the battery cells and cooling tubes within the region Ω are structures that need to be optimized. The goal of the battery pack's structural optimization is to optimize the shape and position of the battery cells and cooling tubes to achieve a uniform distribution of the time-dependent temperature. The structural optimization goal of the two-dimensional battery pack can be represented by equation (17): where T represents a temperature and Treffür a reference temperature. The structural optimization aims to minimize the objective function J. According to one embodiment, the structural optimization process described above in conjunction with Figs. 7A and 7B can be used to perform the optimization design of the two-dimensional battery pack. In iterative structural optimization steps, the RNO can be used to predict the corresponding physical data u based on the current structural data λ, the reference structure, and the physical reference data. If necessary, the numerical solver is triggered to determine the corresponding physical data u based on the current structural data λ. The geometric structural parameters of the object can include the coordinates of the center point and the radius of the boundaries γc,i and γp,ige. After lattice discretization of the geometric structural domain Ω, formed by the outer boundary you and the inner boundaries γc,iso and γp,i, discretized structural data λ can be obtained, e.g.,with λ={( x1, x2, m1, m2)}N, where N represents a number of discretized grid points, where x1 and x2 represent the spatial coordinates, and where m1 and m2∈ {0,1} represent masks, which are each used to indicate whether the point (x1, x2) belongs to the inner boundary γc,ib or γp,i. It should be understood that, although the prediction of the PDE solution, the training of RNO, and the structural optimization process in the embodiments of this disclosure have been explained using specific objects such as guide vanes and two-dimensional battery packs, these methods can also be applied to other objects. For example, the object may be one of the following to be manufactured or designed: a bipolar plate of a fuel cell, a specific component of a car, a specific component of an aircraft, a specific component of a building, a pipe of a reactor, a guide vane, etc. For example, the structural data associated with the structure of the object may include at least some of its position, radius, width, height, length, anchor point, etc.include, where the physical data associated with a physical state of the object may include at least some aspects such as velocity, pressure, temperature, electric field, magnetic field, etc. Fig. 10 shows a method for determining a physical state of an object with a shape according to an exemplary embodiment. In step 1010, initial structural data is received, representing an initial form of the object. In step 1020, a first feature sequence comprising N feature vectors is generated based on the initial structural data. This first feature sequence is, for example, the feature sequence zlim physical space, which is described in conjunction with Fig. 5F. In step 1030, the first feature sequence is transformed into a second feature sequence comprising M feature vectors. This second feature sequence is, for example, the feature sequence zvim virtual physical space, described in conjunction with Fig. 5F. In step 1040, a Fourier transform-based processing is performed on the second feature sequence to obtain a third feature sequence comprising M feature vectors. This third feature sequence is, for example, the feature sequence in virtual physical space described in conjunction with Fig. 5F. In step 1050, the third feature sequence is transformed into a fourth feature sequence comprising N feature vectors. This fourth feature sequence is, for example, the feature sequence in physical space described in conjunction with Fig. 5F. In step 1060, based on the fourth feature sequence, initial physical state data are determined, representing the physical state of the object with the first shape. As explained in the embodiments described in conjunction with Figures 5A to 5D, in one embodiment, the decoder 5230 can directly output the initial physical state data of the object. In another embodiment, the decoder 5230 can output a change magnitude of the physical state data relative to physical reference state data; and then, based on this change magnitude and the physical reference state data, the initial physical state data are obtained.It is understood that the procedure 1000 does not necessarily have to be executed using the network architecture of the RNO, but can also be executed using a non-RNO network architecture that includes the virtual Fourier layer exemplified in connection with Fig. 5F. According to one embodiment, in step 1010, reference structure data representing a reference shape of the object, as well as physical reference data representing the physical state of the object with the reference shape, are received. In step 1020, the first feature sequence is generated based on the initial structure data, the reference structure data, and the physical reference data. According to one embodiment, in step 1020, a fifth feature sequence comprising N feature vectors is generated based on the first structural data; a sixth feature sequence comprising N feature vectors is generated based on the first structural data and the reference structural data; a seventh feature sequence comprising N feature vectors is generated based on the physical reference data; and the first feature sequence is generated based on the fifth, sixth, and seventh feature sequences. This fifth feature sequence is, for example, a feature sequence related to the query structure, such as feature sequence v3 in the embodiments shown in Figures 5A to 5D. This sixth feature sequence is, for example, a feature sequence related to the deformation, such as feature sequence v2 in the embodiments shown in Figures 5A to 5D.This seventh feature sequence is, for example, a feature sequence related to the reference solution, e.g., the feature sequence v1 in the embodiments of Fig. 5A to 5D. According to one embodiment, in step 1030 a first weight matrix is ​​obtained based on the first feature sequence; wherein, based on the first weight matrix, a weighted aggregation of the N feature vectors in the first feature sequence is performed to obtain the second feature sequence, which comprises M feature vectors. This first weight matrix is, for example, the weight matrix ω(1) described in connection with the embodiment of Fig. 5F, wherein, for example, based on M column vectors in the first weight matrix, a weighted aggregation of the N feature vectors in the first feature sequence is performed to obtain the second feature sequence, which comprises M feature vectors. According to one embodiment, in step 1050 a second weight matrix is ​​obtained based on the first feature sequence; wherein, based on the second weight matrix, a weighted aggregation of the M feature vectors in the third feature sequence is performed to obtain the fourth feature sequence, which comprises N feature vectors. This second weight matrix is, for example, the weight matrix ω(2) described in connection with the embodiment of Fig. 5F, wherein, for example, based on the N row vectors in the second weight matrix, a weighted aggregation of the M feature vectors in the third feature sequence is performed to obtain the fourth feature sequence, which comprises N feature vectors. According to one embodiment, obtaining the first weight matrix based on the first feature sequence in step 1030 comprises: transforming the N feature vectors in the first feature sequence into N distribution vectors, wherein the N distribution vectors each comprise M distribution elements and form a distribution matrix; and obtaining the first weight matrix based on the distribution matrix. This distribution matrix is, for example, the distribution matrix l described in connection with the embodiment of Fig. 5F, wherein for each of the N feature vectors in the first feature sequence, the M distribution elements in the corresponding distribution vector each specify a probability that the point in physical space corresponding to this feature vector is distributed among the corresponding M detection points in virtual physical space.According to one embodiment, obtaining the second weight matrix based on the first feature sequence in step 1050 includes: obtaining the second weight matrix based on the distribution matrix. According to one embodiment, obtaining the first weight matrix based on the distribution matrix in step 1030 comprises: for each distribution element in the distribution matrix, determining a weight element corresponding to that distribution element in the first weight matrix based on the elements in a column in which that element is located. According to one embodiment, obtaining the second weight matrix based on the distribution matrix in step 1050 comprises: for each distribution element in the distribution matrix, determining a weight element corresponding to that distribution element in the second weight matrix based on the elements in a row in which that element is located. According to one embodiment, the object is provided to be at least one of the following components: a bipolar plate of a fuel cell, a component of a car, a component of an aircraft, a component of a building, a pipe of a reactor, a guide plate; and that the physical data include at least one of the following quantities: velocity, pressure, temperature, electric field, magnetic field. Fig. 11 shows a method for structural optimization of an object according to an exemplary embodiment. Procedure 1100 comprises steps 1110 to 1130, which are performed iteratively. In step 1110, initial structural data is received, representing a first form of the object. This initial structural data is the current structural data that was output in step 1130 during the last optimization step or optimization round. In step 1120, based on the initial structural data, initial physical state data are determined, representing the physical state of the object with the first form. In this step, the methods provided in the various embodiments of this disclosure can be used to determine the initial physical state data. In step 1130, the initial structure data is updated based on the initial physical state data to obtain updated initial structure data representing an updated initial form of the object. This updated initial structure data is then provided for step 1110. According to one embodiment, in step 1130, a first gradient of the optimization goal of the structure optimization with respect to the first structure data is determined based on the first physical data; wherein, based on the first structure data and the first gradient, the updated first structure data are obtained. According to one embodiment, step 1120 provides that, if a first condition is met, a numerical solver determines first calibrated physical data representing the physical state of the object with the first form based on the first structural data, wherein the first calibrated physical data are used as the first physical data representing the physical state of the object with the first form, or wherein the first calibrated physical data are used as physical reference data to predict the first physical data representing the physical state of the object with the first form using a neural network model; that, if the first condition is not met, the first physical state data representing the physical state of the object with the first form are determined by the neural network model. According to one embodiment, the first condition is that the difference between the first calibrated physical data determined by the numerical solver when the first condition was last satisfied and the first physical data determined most recently by the NN model is greater than a first threshold. Fig. 12 shows a method for training a neural network (NN) model to determine the physical state of an object with a given shape according to an exemplary embodiment. The NN model comprises a virtual Fourier layer, wherein the virtual Fourier layer includes a first transformation layer, a Fourier processing layer, and a second transformation layer. In step 1210, initial structural data, representing an initial form of a first object, and initial labeled physical state data, representing the physical state of the first object with the first form, are received. In step 1220, a first feature sequence is generated based on the first structural data, comprising N feature vectors. In step 1230, the first feature sequence is transformed by the first transformation layer into a second feature sequence comprising M feature vectors. In step 1240, a Fourier-transform-based processing is performed on the second feature sequence by the Fourier processing layer to obtain a third feature sequence comprising M feature vectors. In step 1250, the third feature sequence is transformed by the second transformation layer into a fourth feature sequence comprising N feature vectors. In step 1260, the first predicted physical state data, representing the physical state of the first object with the first shape, are determined based on the fourth feature sequence. In step 1270, the learnable parameters of the neural network model are updated based on the first predicted physical state data and the first labeled physical state data. According to one embodiment, in step 1210, reference structure data representing a reference shape of the first object, as well as physical reference data representing the physical state of the object with the reference shape, are received. In step 1220, the first feature sequence is generated based on the first structure data, the reference structure data, and the physical reference data. According to one embodiment, the neural network model further comprises an encoder and a decoder. In step 1220, a fifth feature sequence comprising N feature vectors is generated by the encoder based on the first structural data; wherein a sixth feature sequence comprising N feature vectors is generated by the encoder based on the first structural data and the reference structural data; wherein a seventh feature sequence comprising N feature vectors is generated by the encoder based on the physical reference data; and wherein the first feature sequence is generated based on the fifth feature sequence, the sixth feature sequence, and the seventh feature sequence. According to one embodiment, in step 1230 a first weight matrix is ​​obtained based on the first feature sequence; a weighted aggregation of the N feature vectors in the first feature sequence is performed based on the first weight matrix to obtain the second feature sequence, which comprises M feature vectors. According to another embodiment, in step 1250 a second weight matrix is ​​obtained based on the first feature sequence; a weighted aggregation of the M feature vectors in the third feature sequence is performed based on the second weight matrix to obtain the fourth feature sequence, which comprises N feature vectors. According to one embodiment, obtaining the first weight matrix based on the first feature sequence in step 1230 comprises: transforming the N feature vectors in the first feature sequence into N distribution vectors, wherein the N distribution vectors each comprise M distribution elements and form a distribution matrix; obtaining the first weight matrix based on the distribution matrix. According to one embodiment, obtaining the second weight matrix based on the first feature sequence in step 1250 comprises: obtaining the second weight matrix based on the distribution matrix. According to one embodiment, obtaining the first weight matrix based on the distribution matrix in step 1230 comprises: for each distribution element in the distribution matrix, determining a weight element corresponding to that distribution element in the first weight matrix based on the elements in a column in which that element is located. According to one embodiment, obtaining the second weight matrix based on the distribution matrix in step 1250 comprises: for each distribution element in the distribution matrix, determining a weight element corresponding to that distribution element in the second weight matrix based on the elements in a row in which that element is located. According to one embodiment, the method 1200 further comprises: receiving a first labeled gradient of an optimization target of a structure optimization for the first object with respect to the first structure data, corresponding to the first structure data and the first labeled physical state data; determining a first gradient of the optimization target of the structure optimization with respect to the first structure data based on the first predicted physical state data. In step 1270, the learnable parameters of the neural network model are updated based on the first predicted physical state data, the first labeled physical state data, the first gradient, and the first labeled gradient.According to one embodiment, in step 1270 a first loss is determined based on the first predicted physical state data and the first labeled physical state data; wherein a second loss is determined based on the first gradient and the first labeled gradient; and wherein the learnable parameters of the neural network model are updated based on the first loss and the second loss. According to one embodiment, the first structural data, the first identified physical state data, and the first identified gradient are derived from historical optimization trajectory data used to perform the structural optimization of the first object. According to one embodiment, method 1200 further comprises: generating a random value. The generation of the first feature sequence based on the first structural data, the reference structural data, and the physical reference data in step 1220 comprises: generating the first feature sequence based on the first structural data, the reference structural data, and the physical reference data if a comparison of the random value with a reference rejection threshold satisfies a second condition; generating the first feature sequence based only on the first structural data if the comparison of the random value with the reference rejection threshold does not satisfy the second condition. According to one embodiment, method 1200 further comprises: receiving second structural data representing a second form of a second object, and second characterized physical state data representing the physical state of the second object with the second form; generating a first feature sequence comprising K feature vectors based on the second structural data of the second object; transforming the first feature sequence comprising K feature vectors by the first transformation layer into a second feature sequence comprising M feature vectors; performing Fourier-transform-based processing on the second feature sequence by the Fourier processing layer to obtain a third feature sequence comprising M feature vectors; transforming the third feature sequence by the second transformation layer into a fourth feature sequence.the fourth feature sequence, which includes K feature vectors; determining second predicted physical state data representing the physical state of the second object with the second shape, based on the fourth feature sequence, which includes K feature vectors; updating the learnable parameters of the neural network model based on the second predicted physical state data and the second labeled physical state data. In this embodiment, the second object is different from the first object. For example, the first object and the second object are the circuit board described in Fig. 8 and the two-dimensional battery pack shown in Fig. 9, respectively. By introducing the virtual Fourier layer into the NN model, the NN model is enabled to process structural data of different objects.while simultaneously improving gradient learning capability. Furthermore, the NN model using the RNO architecture can learn the intrinsic relationship between shape changes and solution transformations, thereby more effectively learning the relationship between structural changes of various objects and PDE solutions. Thus, by using the aforementioned virtual Fourier layer and / or the RNO framework, the NN model is enabled to effectively learn the prediction of PDE solutions, as well as the gradient of the structural optimization target with respect to the structure, from historical optimization trajectories of various structural optimization projects for different objects. According to one embodiment, method 1200 further comprises: receiving a second characterized gradient of an optimization target of the structure optimization for the second object with respect to the second structure data, which corresponds to the second structure data and the second characterized physical state data; determining a second gradient of the optimization target of the structure optimization with respect to the second structure data based on the second predicted physical state data;where updating the learnable parameters of the neural network model based on the second predicted physical state data and the second labeled physical state data includes: updating the learnable parameters of the neural network model based on the second predicted physical state data, the second labeled physical state data, the second gradient, and the second labeled gradient. Fig. 13 shows a method for structural optimization of an object according to an exemplary embodiment. Procedure 1300 comprises the iterative execution of the following steps 1310 to 1350. In step 1310, initial structural data is received, representing an initial form of the object. Step 1320 assesses whether a first condition is met. In step 1330, if the first condition is met, a numerical solver determines first calibrated physical data representing the physical state of the object with the first form based on the first structural data, using the first calibrated physical data as the first physical data representing the physical state of the object with the first form, or using the first calibrated physical data as physical reference data for a first NN model to predict the first physical data representing the physical state of the object with the first form.As indicated by the arrows from 1330 to 1350, if the first condition is met, the first calibrated physical data are provided as the first physical data representing the physical state of the object with the first shape for step 1350. Alternatively, if the first condition is met, as indicated by the dashed arrow from 1330 to 1340, the first calibrated physical data are provided for step 1340 so that they can be used by the first NN model as physical reference data to predict the first physical data representing the physical state of the object with the first shape; and the predicted first physical data are provided for step 1350. In step 1340, if the first condition is not met, the first NN model determines the first physical data representing the physical state of the object with the first shape, based on the first structural data, reference structural data representing a reference shape of the object, and physical reference data representing the physical state of the object with the reference shape. In step 1350, the object's initial structural data is updated based on the initial physical data to obtain updated initial structural data representing an updated initial form of the object. This updated initial structural data is then provided to step 1310 to cyclically execute procedure 1300 in the next structural optimization step. According to one embodiment, the first condition is that the difference between the first calibrated physical data determined by the numerical solver when the first condition was last satisfied and the first physical data determined most recently by the first NN model is greater than a first threshold. The difference is, for example, the distance between the first calibrated physical data and the first physical data, e.g., dist(ut, ugt), as illustrated in Table 3 above. It is understood that the first condition can only include that the difference is greater than the first threshold; and that it can alternatively include other additional conditions. According to one embodiment, the reference structure data or the physical reference data comprise the first structure data processed by the numerical solver when the first condition was last satisfied, or the correspondingly determined first calibrated physical data. As illustrated, for example, in Table 3 above, the first calibrated physical data ugt determined by the numerical solver when the first condition was last satisfied can be stored in cache B, e.g., B = [ugt]. According to one embodiment, the reference structure data or the physical reference data comprise one or more of the first structure data points processed by the first NN model in one or more iteration rounds after the first condition was last satisfied, or one or more of the correspondingly determined first physical data points. As illustrated, for example, in Table 3 above, in one iteration round, the first physical data points are obtained by the RNO and stored in cache B, which can contain at most the N2 previous solutions. According to one embodiment, the reference structure data or the physical reference data comprise several of the first structure data processed in several iteration rounds, or the several correspondingly determined first physical data. In step 1340, the first NN model determines several first temporary physical data representing the physical state of the object with the first shape, based on each of the first structure data, each of the several first structure data, and each of the several correspondingly determined first physical data; and wherein the first physical data are determined based on the several first temporary physical data. For example, the several first temporary physical data are averaged to obtain the first physical data. According to one embodiment, in step 1350 a first gradient of the optimization goal of the structure optimization with respect to the first structure data is determined based on the first physical data; wherein the updated first structure data are obtained based on the first structure data and the first gradient. According to one embodiment, in step 1340, noise is added to the initial structure data to obtain multiple initial structure data instances. The initial neural network model then determines multiple initial physical data instances, representing the physical state of the object with the initial form, as the initial physical data, based on each of these multiple initial structure data instances, the reference structure data, and the physical reference data. In step 1350, multiple initial gradient instances of the optimization target of the structure optimization with respect to the initial structure data are determined, based on each of these multiple initial physical data instances. The initial gradient is then determined based on these multiple initial gradient instances. For example, the multiple initial gradient instances are averaged to obtain the initial gradient. According to one embodiment, the reference structure data or the physical reference data comprise several of the initial structure data that were processed in several iteration rounds, or the several correspondingly determined initial physical data. As illustrated by Table 3 above, for example, several PDE solutions and several corresponding structure data are stored in cache B in the last several iteration rounds. In this example, the several PDE solutions in cache B comprise at most N2 solutions and may or may not contain the calibrated solution ugt. For example, if more than N2 rounds have passed since the round in which the first condition was last satisfied, cache B contains the solutions from the last N2 rounds and no longer the calibrated solution ugt.Determining the multiple first physical data instances representing the physical state of the object with the first form in step 1340 comprises: for each of the multiple first structure data instances: determining multiple first temporary physical data representing the physical state of the object with the first form, by the first NN model, based on each of these multiple first structure data instances, each of the multiple first structure data, and each of the multiple correspondingly determined first physical data; determining one of these first structure data instances as one of the multiple first physical data instances based on the multiple first temporary physical data. According to one embodiment, the first NN model comprises a virtual Fourier layer, wherein the virtual Fourier layer includes a first transformation layer, a Fourier processing layer, and a second transformation layer. The first transformation layer transforms a first feature sequence comprising N feature vectors into a second feature sequence comprising M feature vectors, wherein the first feature sequence is obtained based on the first structural data, the reference structural data, and the physical reference data, where N is not equal to M. N is the number of points in a physical space representing the structure of the object, where M is the number of detectors in a virtual physical space, where N is variable and M is fixed.The Fourier processing layer performs Fourier transform-based processing on the second feature sequence to obtain a third feature sequence comprising M feature vectors. The second transformation layer transforms the third feature sequence into a fourth feature sequence comprising N feature vectors. According to one embodiment, transforming the first feature sequence, comprising N feature vectors, into the second feature sequence, comprising M feature vectors, by the first transformation layer includes: obtaining a first weight matrix based on the first feature sequence; and performing a weighted aggregation of the N feature vectors in the first feature sequence based on the first weight matrix to obtain the second feature sequence, comprising M feature vectors. For example, a weighted aggregation of the N feature vectors in the first feature sequence is performed based on M column vectors in the first weight matrix to obtain the second feature sequence, comprising M feature vectors. According to one embodiment, transforming the third feature sequence into the fourth feature sequence, which comprises N feature vectors, by the second transformation layer includes: obtaining a second weight matrix based on the first feature sequence; performing a weighted aggregation of the M feature vectors in the third feature sequence based on the second weight matrix to obtain the fourth feature sequence, which comprises N feature vectors. For example, the weighted aggregation of the M feature vectors in the third feature sequence is performed based on N row vectors in the second weight matrix to obtain the fourth feature sequence, which comprises N feature vectors. According to one embodiment, obtaining the first weight matrix based on the first feature sequence comprises: transforming the N feature vectors in the first feature sequence into N distribution vectors, where each N distribution vector comprises M distribution elements and forms a distribution matrix; and obtaining the first weight matrix based on the distribution matrix. Obtaining a second weight matrix based on the first feature sequence comprises: obtaining the second weight matrix based on the distribution matrix. According to one embodiment, obtaining the first weight matrix based on the distribution matrix comprises: for each distribution element in the distribution matrix, determining a corresponding weight element in the first weight matrix based on the elements in a column containing that element. Obtaining the second weight matrix based on the distribution matrix comprises: for each distribution element in the distribution matrix, determining a corresponding weight element in the second weight matrix based on the elements in a row containing that element. According to one embodiment, the first NN model further comprises an encoder and a decoder. Based on the first structural data, the reference structural data, and the physical reference data, the encoder generates a fifth feature sequence, a sixth feature sequence, and a seventh feature sequence, each containing N feature vectors, with the first feature sequence being generated based on the fifth, sixth, and seventh feature sequences. Based on the fourth feature sequence, the decoder predicts the first physical data or a change in physical data used to determine the first physical data. Fig. 14 shows a device for determining a physical state of an object with a shape according to an exemplary embodiment. The device 1400 comprises: a generation module 1410 configured to generate a first feature sequence comprising N feature vectors based on initial structural data representing a first form of the object; a virtual Fourier module 1420 configured to transform the first feature sequence into a second feature sequence comprising M feature vectors, to perform Fourier-transform-based processing on the second feature sequence to obtain a third feature sequence comprising M feature vectors, and to transform the third feature sequence into a fourth feature sequence comprising N feature vectors; and a determination module 1330 configured to determine initial physical state data representing the physical state of the object with the first form based on the fourth feature sequence. According to one embodiment, the generation module 1410 generates the first feature sequence based on the first structural data, reference structural data representing a reference shape of the object, and physical reference data representing the physical state of the object with the reference shape. According to one embodiment, the generation module 1410 generates a fifth feature sequence comprising N feature vectors based on the first structural data; it generates a sixth feature sequence comprising N feature vectors based on the first structural data and the reference structural data; it generates a seventh feature sequence comprising N feature vectors based on the physical reference data; and it generates the first feature sequence based on the fifth feature sequence, the sixth feature sequence, and the seventh feature sequence. According to one embodiment, the virtual Fourier module 1420 receives a first weight matrix based on the first feature sequence, and, based on the first weight matrix, performs a weighted aggregation of the N feature vectors in the first feature sequence to obtain the second feature sequence, which comprises M feature vectors. According to another embodiment, the virtual Fourier module 1420 receives a second weight matrix based on the first feature sequence; and, based on the second weight matrix, performs a weighted aggregation of the M feature vectors in the third feature sequence to obtain the fourth feature sequence, which comprises N feature vectors. According to one embodiment, the virtual Fourier module 1420 transforms the N feature vectors in the first feature sequence into N distribution vectors, wherein the N distribution vectors each comprise M distribution elements and form a distribution matrix; and it obtains the first weight matrix and the second weight matrix based on the distribution matrix. According to one embodiment, the virtual Fourier module 1420 determines, for each distribution element in the distribution matrix, a corresponding weight element in the first weight matrix based on the elements in a column containing that element. According to another embodiment, the virtual Fourier module 1420 determines, for each distribution element in the distribution matrix, a corresponding weight element in the second weight matrix based on the elements in a row containing that element. Fig. 15 shows a device for structural optimization of an object according to an exemplary embodiment. The device 1500 comprises: a physical state data determination module 1510, configured to determine initial physical state data representing the physical state of the object with the initial form based on initial structural data representing an initial form of the object; a structure update module 1520, configured to update the initial structural data based on the initial physical state data to obtain updated initial structural data representing an updated initial form of the object. The physical state data determination module 1510 can apply the methods provided in the various embodiments of this disclosure to determine the initial physical state data. The physical state data determination module 1510 and the structure update module 1520 can operate iteratively to perform the structure optimization of the object. The initial structure data on which the physical state data determination module 1510 relies in an optimization step or optimization round is the current structure data output by the structure update module 1520 in the previous optimization step or optimization round. According to one embodiment, the structure update module 1520 determines a first gradient of the optimization goal of the structure optimization with respect to the first structure data based on the first physical data; wherein it obtains the updated first structure data based on the first structure data and the first gradient. According to one embodiment, the physical state data determination module 1510, in the case where a first condition is met, determines first calibrated physical data representing the physical state of the object with the first form by means of a numerical solver based on the first structural data, wherein the first calibrated physical data are used as the first physical data representing the physical state of the object with the first form, or wherein the first calibrated physical data are used as physical reference data to determine the first physical state data representing the physical state of the object with the first form by means of a NN model;If the first condition is not met, the physical state data determination module 1510 uses the NN model to determine the first physical state data representing the physical state of the object with the first form. According to one embodiment, the first condition is that the difference between the first calibrated physical data determined by the numerical solver when the first condition was last met and the first physical data last determined by the NN model is greater than a first threshold value. Fig. 16 shows a device for training a NN model to determine a physical state of an object with a shape according to an exemplary embodiment. The device 1600 comprises: a processing module 1610 containing the NN model and receiving initial structural data representing the initial shape of a first object, and initial labeled physical state data representing the physical state of the first object with the initial shape; generating a first feature sequence comprising N feature vectors based on the initial structural data; transforming the first feature sequence into a second feature sequence comprising M feature vectors; performing Fourier-transform-based processing on the second feature sequence to obtain a third feature sequence comprising M feature vectors; transforming the third feature sequence into a fourth feature sequence comprising N feature vectors; and determining initial predicted physical state data based on the fourth feature sequence.which represent the physical state of the first object with the first shape; and an update module 1620 that updates the learnable parameters of the NN model based on the first predicted physical state data and the first labeled physical state data. According to one embodiment, the processing module 1610 further receives reference structure data representing a reference shape of the first object and physical reference data representing the physical state of the object with the reference shape; and it generates the first feature sequence based on the first structure data, the reference structure data and the physical reference data. According to one embodiment, the processing module 1610 generates a fifth feature sequence comprising N feature vectors based on the first structural data; it generates a sixth feature sequence comprising N feature vectors based on the first structural data and the reference structural data; it generates a seventh feature sequence comprising N feature vectors based on the physical reference data; and it generates the first feature sequence based on the fifth feature sequence, the sixth feature sequence, and the seventh feature sequence. According to one embodiment, the processing module 1610 receives a first weight matrix based on the first feature sequence; based on the first weight matrix, it performs a weighted aggregation of the N feature vectors in the first feature sequence to obtain the second feature sequence, which comprises M feature vectors. According to another embodiment, the NN model 1610 receives a second weight matrix based on the first feature sequence; based on the second weight matrix, it performs a weighted aggregation of the M feature vectors in the third feature sequence to obtain the fourth feature sequence, which comprises N feature vectors. According to one embodiment, the processing module 1610 transforms the N feature vectors in the first feature sequence into N distribution vectors, wherein the N distribution vectors each comprise M distribution elements and form a distribution matrix; and it obtains the first weight matrix and the second weight matrix based on the distribution matrix. According to one embodiment, the processing module 1610 determines, for each distribution element in the distribution matrix, a corresponding weight element in the first weight matrix based on the elements in a column in which that element is located. According to another embodiment, the processing module 1610 determines, for each distribution element in the distribution matrix, a corresponding weight element in the second weight matrix based on the elements in a row in which that element is located. According to one embodiment, the processing module 1610 receives a first labeled gradient of an optimization target of a structure optimization for the first object with respect to the first structure data, corresponding to the first structure data and the first labeled physical state data; and it determines a first gradient of the optimization target of the structure optimization with respect to the first structure data based on the first predicted physical state data. The update module 1620 updates the learnable parameters of the neural network model based on the first predicted physical state data, the first labeled physical state data, the first gradient, and the first labeled gradient.According to one embodiment, the device 1600 further comprises a loss module that determines a first loss based on the first predicted physical state data and the first labeled physical state data; and that determines a second loss based on the first gradient and the first labeled gradient. The update module 1620 updates the learnable parameters of the neural network model based on the first loss and the second loss. According to one embodiment, the first structural data, the first identified physical state data, and the first identified gradient are derived from historical optimization trajectory data used to perform the structural optimization of the first object. According to one embodiment, the processing module 1610 generates a random value; wherein, in the case where a comparison of the random value with a reference rejection threshold satisfies a second condition, it generates the first feature sequence based on the first structural data, the reference structural data and the physical reference data; wherein, in the case where the comparison of the random value with the reference rejection threshold does not satisfy the second condition, it generates the first feature sequence only based on the first structural data. According to one embodiment, the processing module 1610 receives second structural data representing a second shape of a second object, and second labeled physical state data representing the physical state of the second object with the second shape; it generates a first feature sequence comprising K feature vectors based on the second structural data of the second object; it transforms the first feature sequence comprising K feature vectors into a second feature sequence comprising M feature vectors; it performs Fourier-transform-based processing on the second feature sequence to obtain a third feature sequence comprising M feature vectors; and it transforms the third feature sequence into a fourth feature sequence comprising K feature vectors.where, based on the fourth feature sequence comprising K feature vectors, it determines second predicted physical state data representing the physical state of the second object with the second shape. Update module 1620 updates the learnable parameters of the neural network model based on the second predicted physical state data and the second labeled physical state data. According to one embodiment, the processing module 1610 receives a second labeled gradient of an optimization goal for the structure optimization for the second object with respect to the second structure data, corresponding to the second structure data and the second labeled physical state data; and it determines a second gradient of the optimization goal with respect to the second structure data based on the second predicted physical state data. The update module 1620 updates the learnable parameters of the neural network model based on the second predicted physical state data, the second labeled physical state data, the second gradient, and the second labeled gradient. Fig. 17 shows a device for structural optimization of an object according to an exemplary embodiment. The device 1700 comprises a physical state data determination module 1710 and a structure update module 1720. The physical state data determination module 1710 receives initial structure data representing an initial form of the object; it assesses whether an initial condition is met; if the initial condition is met, it determines, using a numerical solver based on the initial structure data, initial calibrated physical data representing the physical state of the object with the initial form, wherein the initial calibrated physical data are used as the initial physical data representing the physical state of the object with the initial form, or wherein the initial calibrated physical data are used as physical reference data for an initial neural network model to predict the initial physical data for the physical state of the object with the initial form;In the case where the first condition is not met, the first NN model determines the first physical data representing the physical state of the object with the first form, based on the first structural data, reference structural data representing a reference form of the object, and physical reference data representing the physical state of the object with the reference form. The structure update module 1720 updates the object's first structural data based on the first physical data to obtain updated first structural data representing an updated first form of the object. The updated first structural data is provided to the physical state data determination module 1710 in the next structure optimization step to iteratively perform the object's structure optimization. According to one embodiment, the first condition is that the difference between the first calibrated physical data determined by the numerical solver when the first condition was last satisfied and the first physical data determined most recently by the first NN model is greater than a first threshold. According to one embodiment, the reference structure data or the physical reference data comprise the first structure data processed by the numerical solver when the first condition was last satisfied, or the correspondingly determined first calibrated physical data. According to one embodiment, the reference structure data or the physical reference data comprise one or more of the first structure data that were processed by the first NN model in one or more iteration rounds after the last fulfillment of the first condition, or one or more of the correspondingly determined first physical data. According to one embodiment, the reference structure data or the physical reference data comprise several of the first structure data processed in several iteration rounds, or the several correspondingly determined first physical data. The physical state data determination module 1710 uses the first NN model to determine several first temporary physical data representing the physical state of the object with the first form, based on each of the first structure data, each of the several first structure data, and each of the several correspondingly determined first physical data; and it determines the first physical data based on the several first temporary physical data. For example, the several first temporary physical data are averaged to obtain the first physical data. According to one embodiment, the structure update module 1720 determines a first gradient of the optimization goal of the structure optimization with respect to the first structure data based on the first physical data; wherein it obtains the updated first structure data based on the first structure data and the first gradient. According to one embodiment, the physical state data determination module 1710 adds noise to the initial structure data to obtain multiple initial structure data instances. It then uses the first neural network model to determine multiple initial physical data instances representing the physical state of the object with the initial form, as the initial physical data, based on each of the multiple initial structure data instances, the reference structure data, and the physical reference data. The structure update module 1720 determines multiple initial gradient instances of the optimization target of the structure optimization with respect to the initial structure data, based on each of the multiple initial physical data instances. It then determines the initial gradient based on the multiple initial gradient instances. For example, the multiple initial gradient instances are averaged to obtain the initial gradient. According to one embodiment, the reference structure data or the physical reference data comprise several of the first structure data that were processed in several iteration rounds, or the several correspondingly determined first physical data. For each of the several first structure data instances, the physical state data determination module 1710 is configured to determine the first NN model several first temporary physical data representing the physical state of the object with the first form, based on each of the first structure data instances, each of the several first structure data, and each of the several correspondingly determined first physical data; and to determine one first physical data instance corresponding to this first structure data instance as one of the several first physical data instances based on the several first temporary physical data. According to one embodiment, the first NN model comprises a virtual Fourier layer, which includes a first transformation layer, a Fourier processing layer, and a second transformation layer. The first transformation layer transforms a first feature sequence comprising N feature vectors into a second feature sequence comprising M feature vectors, where the first feature sequence is obtained based on the first structural data, the reference structural data, and the physical reference data, where N is not equal to M. The Fourier processing layer performs Fourier transform-based processing on the second feature sequence to obtain a third feature sequence comprising M feature vectors. The second transformation layer transforms the third feature sequence into a fourth feature sequence comprising N feature vectors. According to one embodiment, transforming the first feature sequence, which comprises N feature vectors, into the second feature sequence, which comprises M feature vectors, by the first transformation layer includes: obtaining a first weight matrix based on the first feature sequence; performing a weighted aggregation of the N feature vectors in the first feature sequence based on the first weight matrix to obtain the second feature sequence, which comprises M feature vectors. According to one embodiment, transforming the third feature sequence into the fourth feature sequence, which comprises N feature vectors, by the second transformation layer includes: obtaining a second weight matrix based on the first feature sequence; performing a weighted aggregation of the M feature vectors in the third feature sequence based on the second weight matrix to obtain the fourth feature sequence, which comprises N feature vectors. According to one embodiment, obtaining the first weight matrix based on the first feature sequence comprises: transforming the N feature vectors in the first feature sequence into N distribution vectors, where each N distribution vector comprises M distribution elements and forms a distribution matrix; and obtaining the first weight matrix based on the distribution matrix. Obtaining a second weight matrix based on the first feature sequence comprises: obtaining the second weight matrix based on the distribution matrix. According to one embodiment, obtaining the first weight matrix based on the distribution matrix comprises: for each distribution element in the distribution matrix, determining a corresponding weight element in the first weight matrix based on the elements in a column containing that element. Obtaining the second weight matrix based on the distribution matrix comprises: for each distribution element in the distribution matrix, determining a corresponding weight element in the second weight matrix based on the elements in a row containing that element. According to one embodiment, the first NN model further comprises an encoder and a decoder. Based on the first structural data, the reference structural data, and the physical reference data, the encoder generates a fifth feature sequence, a sixth feature sequence, and a seventh feature sequence, each containing N feature vectors, with the first feature sequence being generated based on the fifth, sixth, and seventh feature sequences. Based on the fourth feature sequence, the decoder predicts the first physical data or a change in physical data used to determine the first physical data. Fig. 18 shows a block diagram of a processing unit 1800 according to an exemplary embodiment. The processing device or processing system 1800 comprises one or more control units or processing units 1810, which execute one or more machine-readable instructions stored or encoded in a machine-readable storage medium (i.e., memory 1820). In one embodiment, the processing unit 1810, when executing program instructions, is configured to perform the various operations and functions described above in conjunction with Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 to 17. The person skilled in the art can understand that the device described in the embodiments of the present disclosure may also include various other components, such as various communication modules, bus modules and possible user interface modules, etc. According to one embodiment, a machine-readable storage medium is provided. The readable medium can contain instructions which, when executed by the processing unit 1810, can perform various operations and functions described above in connection with Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 to 17 in the various embodiments of this disclosure. According to one embodiment, a computer program product is provided. The computer program product comprises machine-executable instructions which, when executed by the processing unit 1810, can perform various operations and functions described above in conjunction with Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 to 17 in the various embodiments of this disclosure. The specific embodiments described above in conjunction with the drawings represent exemplary embodiments, but do not constitute all embodiments that can be realized or that may fall within the scope of the claims. The term "exemplary," used throughout the description, means "serving as an example" and does not mean "preferred" or "advantageous" over other embodiments. For the purpose of understanding the described techniques, the detailed embodiments include specific details. However, these techniques can be implemented without these specific details. To avoid confusion regarding the concepts of the described embodiments, some examples show known structures and devices in block form. The foregoing description of the content of this application is provided to enable any person skilled in the art to realize or use the content of this application. The various modifications to the content of this application are obvious to the person skilled in the art, and the general principles defined herein can also be applied to other variants without altering the scope of protection of this application. Therefore, the content of this application is not limited to the examples and designs described herein, but is consistent with the broadest scope that corresponds to the principles and novel features disclosed herein.

Claims

A computer-implemented method for optimizing the structure of an object, comprising iteratively performing the following steps: Step a: Receiving initial structure data representing an initial form of the object; Step b: Determining initial calibrated physical data representing the physical state of the object with the initial form, by a numerical solver based on the initial structure data if an initial condition is satisfied, wherein the initial calibrated physical data are used as the initial physical data representing the physical state of the object with the initial form, or wherein the initial calibrated physical data are used as physical reference data for an initial neural network model to predict the initial physical data for the physical state of the object with the initial form;Step c: Determining the first physical data representing the physical state of the object with the first form, using the first neural network model, based on the first structural data, reference structural data representing a reference form of the object, and physical reference data representing the physical state of the object with the reference form when the first condition is not met; and Step d: Updating the first structural data of the object based on the first physical data to obtain updated first structural data representing an updated first form of the object. Method according to claim 1, wherein the first condition is that the difference between the first calibrated physical data determined by the numerical solver when the first condition was last satisfied and the first physical data determined most recently by the first neural network model is greater than a first threshold. The method of claim 1, wherein the reference structure data or the physical reference data comprise the first structure data processed by the numerical solver when the first condition was last satisfied, or the correspondingly determined first calibrated physical data. Method according to claim 1 or 3, wherein the reference structure data or the physical reference data comprise one or more of the first structure data that were processed by the first neural network model in one or more iteration rounds after the last fulfillment of the first condition, or one or more of the correspondingly determined first physical data. The method of claim 4, wherein the reference structure data or the physical reference data comprise several of the first structure data processed in several iteration rounds or the several correspondingly determined first physical data, and wherein step c comprises: determining several first temporary physical data representing the physical state of the object with the first form, by the first neural network model based on each of the first structure data, each of the several first structure data, and each of the several correspondingly determined first physical data; and determining the first physical data based on the several first temporary physical data. The method of claim 1, wherein step d comprises: determining a first gradient of an optimization target of the structure optimization with respect to the first structure data based on the first physical data; and obtaining the updated first structure data based on the first structure data and the first gradient. The method of claim 6, wherein step c comprises: adding noise to the first structure data to obtain multiple first structure data instances of the first structure data; and determining multiple first physical data instances representing the physical state of the object with the first form as the first physical data by the first neural network model, each based on each of the multiple first structure data instances, the reference structure data, and the physical reference data; wherein determining the first gradient of the optimization target of the structure optimization with respect to the first structure data based on the first physical data comprises: determining multiple first gradient instances of the optimization target of the structure optimization with respect to the first structure data, each based on each of the multiple first physical data instances;and determining the first gradient based on the multiple first gradient instances. The method according to claim 7, wherein the reference structure data or the physical reference data are several of the first structure data that were processed in several iteration rounds, respectively.comprising several appropriately determined first physical data instances, and wherein determining several first physical data instances representing the physical state of the object with the first form comprises: for each of the several first structure data instances, determining several first temporary physical data instances representing the physical state of the object with the first form, by the first neural network model based on each of these first structure data instances, each of the several first structure data instances, and each of the several appropriately determined first physical data instances; and determining one of these first structure data instances corresponding to a first physical data instance as one of the several first physical data instances based on the several first temporary physical data instances. The method of claim 1, wherein the first neural network model comprises a virtual Fourier layer, the virtual Fourier layer comprising a first transformation layer, a Fourier processing layer, and a second transformation layer, wherein the first transformation layer transforms a first feature sequence comprising N feature vectors into a second feature sequence comprising M feature vectors, wherein the first feature sequence is obtained based on the first structural data, the reference structural data, and the physical reference data, where N is not equal to M; wherein the Fourier processing layer performs Fourier transform-based processing of the second feature sequence to obtain a third feature sequence comprising M feature vectors; and wherein the second transformation layer transforms the third feature sequence into a fourth feature sequence comprising N feature vectors. The method of claim 9, wherein the transformation of the first feature sequence comprising N feature vectors into the second feature sequence comprising M feature vectors by the first transformation layer comprises: obtaining a first weight matrix based on the first feature sequence; and performing a weighted aggregation of the N feature vectors in the first feature sequence based on the first weight matrix to obtain the second feature sequence comprising M feature vectors; wherein the transformation of the third feature sequence into the fourth feature sequence comprising N feature vectors by the second transformation layer comprises: obtaining a second weight matrix based on the first feature sequence; and performing a weighted aggregation of the M feature vectors in the third feature sequence based on the second weight matrix to obtain the fourth feature sequence comprising N feature vectors. The method of claim 10, wherein obtaining the first weight matrix based on the first feature sequence comprises: transforming the N feature vectors in the first feature sequence into N distribution vectors, wherein the N distribution vectors each comprise M distribution elements and form a distribution matrix; and obtaining the first weight matrix based on the distribution matrix; wherein obtaining the second weight matrix based on the first feature sequence comprises: obtaining the second weight matrix based on the distribution matrix. The method of claim 11, wherein obtaining the first weight matrix based on the distribution matrix comprises: for each distribution element in the distribution matrix, determining a weight element corresponding to that distribution element in the first weight matrix based on the elements in a column in which that element is located; wherein obtaining the second weight matrix based on the distribution matrix comprises: for each distribution element in the distribution matrix, determining a weight element corresponding to that distribution element in the second weight matrix based on the elements in a row in which that element is located. A method according to any one of claims 9 to 12, wherein the first neural network model further comprises an encoder and a decoder, wherein the encoder generates a fifth feature sequence, a sixth feature sequence and a seventh feature sequence, each containing N feature vectors, based on the first structural data, the reference structural data and the physical reference data, wherein the first feature sequence is generated based on the fifth feature sequence, the sixth feature sequence and the seventh feature sequence; and wherein the decoder predicts the first physical data or a change in physical data used to determine the first physical data based on the fourth feature sequence. Device for structural optimization of an object, comprising: a physical state data determination module configured to receive initial structural data representing an initial form of the object; wherein, in the case where an initial condition is satisfied, it determines, by means of a numerical solver, initial calibrated physical data representing the physical state of the object with the initial form, based on the initial structural data, wherein the initial calibrated physical data are used as the initial physical data representing the physical state of the object with the initial form, or wherein the initial calibrated physical data are used as physical reference data for an initial neural network model to predict the initial physical data for the physical state of the object with the initial form;and wherein, in the case where the first condition is not met, the first neural network model determines the first physical data representing the physical state of the object with the first form, based on the first structural data, reference structural data representing a reference form of the object, and physical reference data representing the physical state of the object with the reference form; and a structure update module configured to update the first structural data of the object based on the first physical data to obtain updated first structural data representing an updated first form of the object. Processing device comprising: one or more processors; and one or more memories on which computer-executable instructions are stored, wherein the instructions, when executed by the one or more processors, perform operations according to any one of claims 1 to 13. Machine-readable storage medium on which executable instructions are stored, wherein the instructions, when executed, cause one or more processors to implement the method according to any one of claims 1 to 13. Computer program product comprising executable instructions, wherein the instructions, when executed, cause one or more processors to implement the method according to any one of claims 1 to 13.

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