METHOD AND DEVICE FOR DETERMINING THE PHYSICAL STATE OF AN OBJECT

Abstract

The present application relates to a computer-implemented method for determining the physical state of an object with a shape, comprising: dividing the shape of the object into several sub-shapes; determining, using the neural network (NN) model, local solutions for each sub-shape of the several sub-shapes based on the global boundary conditions for the shape and the several sub-shapes, wherein the local solution for each sub-shape expresses the physical state of the part of the object that has that sub-shape; and determining the global solution for the shape based on the local solutions for each sub-shape of the several sub-shapes, wherein the global solution for the shape expresses the physical state of the object.

Description

AREA OF INVENTION

[0001] The present application relates to artificial intelligence technology, more specifically to a method and a device for determining the physical state of an object. STATE OF THE ART

[0002] Neural network (NN) models (such as NN operators) are gaining popularity in solving partial differential equations (PDEs) because of their outstanding ability to capture complex mappings between function spaces in complex sets. However, current NN models, which are often focused on a specific shape, present limitations to their widespread application. For one, training NN models requires large amounts of training data. For example, in structural engineering, a product might have a specific shape, meaning that a NN model for that particular shape might be needed to represent its physical state (e.g., velocity, pressure, temperature, electric field, or magnetic field).) of the product with this shape, and accordingly, training data for this specific shape may need to be used to train this NN model. However, the amount of training data for this specific shape may sometimes be insufficient, resulting in poor performance of the NN model trained with this data for this specific shape.

[0003] On the other hand, it is difficult to extend neural network (NN) models used in the aforementioned current framework from the form used in the training data to a completely different form (e.g., one not included in the training data). For example, a first NN model may have been trained for a first form (e.g., of a first product). With regard to a second product with a second form, the first NN model, because the second form is completely different from the first, may not produce an ideal result when predicting the physical state of the second product. In this case, a second NN model must be trained again for the second form. This leads to relatively low efficiency and hinders the application of NN models to practical industrial problems.

[0004] Consequently, an improved system is needed to determine the physical state of an object, in which the NN model is not limited to the shape of a specific product and in which sufficient training data can be generated for this NN model; in this improved system, this NN model can be used to better extend to any arbitrary shape, so that the physical state of any object with different shapes can be determined, thereby increasing the efficiency of using NN models in practical industrial applications. REVELATION OF THE INVENTION

[0005] The following introduction is provided to briefly present some of the selected concepts that will be described in more detail in a later, more comprehensive description. This introduction is not intended to highlight essential or necessary features of the claimed subject matter, nor to limit the scope of the claimed subject matter.

[0006] With regard to the problem described above, the present application provides a novel computer-implemented method for generating training data and, accordingly, a novel system for determining the physical state of an object with a given shape using a neural network (NN) model. By using the method according to the embodiments of the present application, sufficient training data for the NN model can be generated to ensure optimal performance. Based on such a trained NN model, the physical state of an object with any shape can then be determined, thus increasing the training and usage efficiency of the NN model.

[0007] According to one aspect of the present application, a computer-implemented method for generating training data is provided, comprising: generating a basic form; assigning boundary conditions related to a first partial differential equation (PDE) to the boundary of the basic form; determining the solution of the first PDE for the basic form based on the basic form and the boundary conditions, wherein the solution expresses the physical state of the object with the basic form; and wherein the training data are obtained based on the basic form, the boundary conditions and the solution.

[0008] According to one aspect of the present application, a computer-implemented method for training a neural network (NN) model to determine the physical state of an object with a shape is provided, comprising: predicting, using the NN model, a solution for a basic shape in the training data based on the training data generated using the method according to the embodiments of the present disclosure, wherein the solution expresses the physical state of the object with the basic shape; determining the loss based on the solution predicted by the NN model and the solution in the training data; and updating the learnable parameters of the NN model based on the loss.

[0009] According to one aspect of the present application, a computer-implemented method for determining the physical state of an object with a shape is provided, comprising: subdividing the shape of the object into several sub-shapes, wherein the totality of the several sub-areas corresponding to the several sub-shapes covers the entire area of ​​the shape; determining, using the neural network (NN) model, local solutions for each sub-shape of the several sub-shapes based on the global boundary conditions for the shape and the several sub-shapes, wherein the local solution for each sub-shape expresses the physical state of the part of the object that has that sub-shape; and determining the global solution for the shape based on the local solutions for each sub-shape of the several sub-shapes, wherein the global solution for the shape expresses the physical state of the object.

[0010] According to one aspect of the present application, a computer-implemented method for performing a structural optimization of an object is provided, comprising: using the method according to the present application to predict the physical state of an object; updating the structural features of the object based on the predicted physical state of the object.

[0011] According to one aspect of the present application, a device for generating training data is provided, comprising: a shape generation module that generates a basic shape; a boundary condition module that assigns boundary conditions related to a first partial differential equation (PDE) to the boundary of the basic shape; a solution module that determines the solution of the first PDE for the basic shape based on the basic shape and the boundary conditions, wherein the solution expresses the physical state of the object with the basic shape; and wherein the training data are obtained based on the basic shape, the boundary conditions and the solution.According to one aspect of the present application, a device for training a neural network (NN) model to determine the physical state of an object with a shape is provided, comprising: an NN model that predicts a solution for the basic shape based on a basic shape in the training data and boundary conditions generated by the method according to the embodiments of the present disclosure, wherein the solution expresses the physical state of the object with the basic shape; a loss module that determines the loss based on the solution predicted by the NN model and the solution in the training data; and an update module that updates the learnable parameters of the NN model based on the loss.

[0012] According to one aspect of the present application, a device for determining the physical state of an object with a shape is provided, comprising: a subdivision module that subdivides the shape of the object into several sub-shapes, wherein the totality of the several sub-areas corresponding to the several sub-shapes covers the entire area of ​​the shape; a local solution module that, based on the global boundary conditions for the shape and the several sub-shapes, uses the neural network (NN) model to determine a local solution for each sub-shape of the several sub-shapes, wherein the local solution for each sub-shape expresses the physical state of the part of the object that has the sub-shape; and a global solution module that, based on the local solutions for each sub-shape of the several sub-shapes, obtains the global solution for the shape, wherein the global solution for the shape expresses the physical state of the object.

[0013] According to one aspect of the present application, a device for performing a structural optimization of an object is provided, comprising: a prediction module that predicts the physical state of the object using the method according to the embodiments of the present disclosure; an update module that updates the structural features of the object based on the predicted physical state of the object.

[0014] According to one aspect of the present application, a processing device is provided comprising: one or more processors; one or more memories, wherein computer-executable instructions are stored in these memories and, when these instructions are executed by the one or more processors, at least one of the operations according to the embodiments of the present disclosure is performed for generating training data, for training a NN model for determining the physical state of an object with a shape, for determining the physical state of an object with a shape and for performing a structural optimization of an object.

[0015] According to one aspect of the present application, a machine-readable storage medium is provided on which executable instructions are stored, and when these instructions are executed, one or more processors are caused to perform at least one of the operations according to the embodiments of the present disclosure for generating training data, for training a NN model to determine the physical state of an object with a shape, for determining the physical state of an object with a shape, and for performing a structural optimization of an object.

[0016] According to one aspect of the present application, a computer program product is provided which includes executable instructions, and when these instructions are executed, one or more processors are caused to perform at least one of the operations according to the embodiments of the present disclosure for generating training data, for training a NN model to determine the physical state of an object with a shape, for determining the physical state of an object with a shape, and for performing a structural optimization of an object. DESCRIPTION OF THE FIGURES

[0017] A further understanding of the nature and advantages of the present application can be achieved by reference to the following figures. Similar components or features in the figures may have the same reference symbols. Fig. Figure 1 shows a block diagram of a device for generating training data and for training a NN model according to an exemplary embodiment. Fig. Figure 2A shows a schematic representation of a basic shape generated according to an exemplary embodiment. Fig. Figure 2B shows a schematic representation of a basic shape generated and discretized according to an exemplary embodiment. Fig. Figure 2C shows a schematic representation of a shape generation module according to an exemplary embodiment. Fig. Figure 3A shows a schematic representation of a basic shape with a single assigned boundary condition according to an exemplary embodiment. Fig. Figure 3B shows a schematic representation of a basic shape with mixed assigned boundary conditions according to an exemplary embodiment. Fig. Figure 3C shows a schematic representation of a boundary condition module according to an exemplary embodiment. Fig. Figure 4A shows a schematic representation of a solution for a basic shape determined according to an exemplary embodiment. Fig. Figure 4B shows a schematic representation of a data-augmented basic form and solution according to an exemplary implementation. Fig. Figure 5 shows a schematic representation of training a NN model according to an exemplary implementation. Fig. Figure 6A shows a block diagram of a device for determining the physical state of an object with a shape according to an exemplary embodiment. Fig. Figure 6B shows a schematic representation for determining the physical state of an object with a shape according to an exemplary embodiment. Fig. Figure 7A shows a schematic representation of a shape divided into several non-overlapping sub-shapes according to an exemplary embodiment. Fig. Figure 7B shows a schematic representation of a shape divided into several overlapping sub-shapes according to an exemplary embodiment. Fig. Figure 7C shows a schematic representation of an overlapping subdividing module according to an exemplary embodiment. Fig. Figure 8 shows a schematic representation of a local solution module according to an exemplary embodiment. Fig. Figure 9 shows a schematic representation of a global solution module according to an exemplary embodiment. Fig. Figure 10 shows a schematic representation of shapes and the data efficiency of applying SNI to the individual shapes according to an embodiment of the present disclosure. Fig. Figure 11 shows a schematic representation of an object with a shape according to an exemplary embodiment. Fig. Figure 12 shows a flowchart of the procedure for performing a structural optimization of an object according to an exemplary embodiment. Fig. Figure 13 shows a flowchart of the computer-implemented procedure for generating training data according to an exemplary embodiment. Fig. Figure 14 shows a flowchart of the computer-implemented procedure for training a NN model to determine the physical state of an object with a shape according to an exemplary embodiment. Fig. Figure 15 shows a flowchart of the computer-implemented method for determining the physical state of an object with a shape according to an exemplary embodiment. Fig. Figure 16 shows a flowchart of the computer-implemented procedure for performing a structural optimization of an object according to an exemplary embodiment. Fig. Figure 17 shows a block diagram of a device for generating training data according to an exemplary embodiment. Fig. Figure 18 shows a block diagram of a device for training a NN model to determine the physical state of an object with a shape according to an embodiment. Fig. Figure 19 shows a block diagram of a device for determining the physical state of an object with a shape according to an exemplary embodiment. Fig. Figure 20 shows a block diagram of a device for performing a structural optimization of an object according to an exemplary embodiment. Fig. Figure 21 shows a block diagram of a processing device according to an exemplary embodiment. DETAILED DESCRIPTION OF THE EXECUTION FORMS

[0018] The subject matter described herein will now be discussed with reference to exemplary embodiments. It is understood that the explanation of these embodiments serves only to improve the understanding of those skilled in the art and to realize the subject matter of this document, without limiting the scope of protection, the applicability, or the examples in the claims. The functions and arrangement of the described elements can be modified without altering the scope of protection of the content of this application. In each example, various processes or components can be omitted, replaced, or added as needed. For example, the described methods can be carried out in a different sequence than described, and individual steps can be added, omitted, or combined.Furthermore, features described in relation to some examples can also be combined in other examples.

[0019] The term "comprehensive" and its variants used herein can be interpreted as an open concept meaning "comprehensive but not limited to." The term "based on" can mean "at least partially based on." The terms "one embodiment," "some embodiments," etc., can mean "at least one embodiment." The term "another embodiment" means "at least one other embodiment." The terms "first," "second," etc., can refer to different or the same objects. Other definitions may be included below, either explicitly or implicitly. The definition of a term is consistent throughout this specification unless the context clearly indicates otherwise.

[0020] Solving partial differential equations (PDEs) is essential in numerous industries; for example, solving PDEs in engineering systems can be used for structural optimization. Stationary PDE problems can be defined as follows: in Ω we haveL(u)=f on D we have u=uD on N we have∂u∂n=g where L a partial differential operator, for example an elliptic partial differential operator, and ΓD∪ΓN=∂Ω for the Dirichlet boundary and the Neumann boundary, respectively. It can be assumed that all sets Ω in Euclidean space ℝ nThese are embedded orientable manifolds with boundary. It is understood that the aforementioned stationary PDE problem is merely an example; the solution of the present application can also be applied to the treatment of stationary PDE problems based on other models as well as dynamic PDE problems (for example, time-related PDE problems).

[0021] Using training data, neural networks (NNs) can be trained for PDEs to learn the mapping between two spaces. G:A→U, where U The solution space of the PDE and the function space for determining the solution of the PDE can be represented. Examples include PDE parameter functions (e.g., comprising coefficients (e.g., a coefficient field) and / or a source term), boundary conditions / initial conditions for defining the PDE, or parameters for defining the form in the set.

[0022] According to the individual embodiments of the present application, the generation of the PDE in the training data need not be restricted to a specific PDE parameter function, boundary condition, or shape, so that sufficient training data can be generated. Based on the training data from the individual embodiments of the present application, a neural network (NN) model can be sufficiently trained and achieve good generalizability. Accordingly, the geometric shape used when using the NN model for inference can be defined in the set Ω. inf from the form used in training the NN model in the set Ω train be independent, in other words, the form must be independent in the set Ω. int It does not fall within the range of forms used in training or resemble them; it can be any form.

[0023] Fig. Figure 1 shows a block diagram of a device 100 for generating training data and training a neural network model according to an exemplary embodiment. In the example from Fig. 1. The device 100 comprises a shape generation module 110, a boundary condition module 120, (optionally) a PDE parameter function module 160, a solution module 130, (optionally) a data augmentation module 140, and a NN model 150. It is understood that the device 100 may include further modules, in Fig. 1 only those modules are shown which relate to the embodiments of the present disclosure.

[0024] The shape generation module 110 can generate the basic shape S 115. In the embodiments of this disclosure, the choice of the basic shape must ensure that the NN model can solve PDE problems for any shape. It can be the probability space (P,μ) are specified, where µ represents the probability distribution in the shape space P In one embodiment, the generated basic shape preferably satisfies two conditions: (1) Sampling feasibility: Sampling from µ and solving boundary value problems of the shapes in P must be simple in order to generate data for training the NN model; (2) Complete coverage: The basic shapes in P must flexibly cover any shape in the domain.

[0025] In one embodiment, Ω ⊆ ℝ 2 It can be the space () of simple polygons that pass through a specific region in ℝ 2 are uniformly bounded and have a maximum of n vertices (that is, planar polygons without self-intersections and without holes), are used Ps(n). For example, this particular area could be the one in Fig. The defined area 215 shown in Figure 2A can be used, but it can also be any other suitable area. A simple polygon is a Lipschitz domain with a direct sampling method and is flexible enough to form any discretized planar region. It goes without saying that if the two aforementioned criteria are met, any other basic shapes can also be generated, such as convex polygons, star polygons, etc.

[0026] Fig. Figure 2A shows a schematic representation of a basic shape 205 generated according to an embodiment. In one embodiment, the basic shape can be a planar polygon without self-intersections and without holes. For example, this planar polygon can have n vertices (for example, 3 ≤ n ≤ 12). For example, the basic shape 205 can be bounded by a defined region 215. For example, this defined region 215 can be a unit square [0.5, 0.5]. 2 ⊂ ℝ 2 act.

[0027] In one example, the basic form can be represented discretely. Fig. Figure 2B shows a schematic representation of a basic shape 225 generated and discretized according to an exemplary embodiment. For example, the basic shape can be discretized by sampling. The discretized basic shape can be expressed by coordinate points on the boundary (and the interior).

[0028] Fig. Figure 2C shows a schematic representation of a shape generation module 110 according to an exemplary embodiment. In the exemplary embodiment from Fig. 2C includes the shape generation module 110, a plane polygon generation module 210, and optionally a sampling module 220. It is understood that the shape generation module 110 can include further modules, in Fig. 2C shows only the modules that relate to the embodiments of the present disclosure.

[0029] The shape generation module 110 can generate the basic shape S 235 using the plane polygon generation module 210. For example, the plane polygon generation module 210 can be given a number n of vertices, and it can randomly generate a planar polygon S 235 with n vertices within the specified area. Alternatively, the plane polygon generation module 210 can randomly specify the number n of vertices and randomly generate a planar polygon S 235 with n vertices within the specified area. In one embodiment, let 3 ≤ n ≤ 12.

[0030] The shape generation module 110 can optionally include the sampling module 220. For example, the sampling module can subject the basic shape S 235 generated by the plane polygon generation module 210 to sampling in order to generate the discrete basic shape S' 245.

[0031] Back to Fig. 1: The shape generation module 110 can output the generated basic shape S 115, and the boundary condition module 120 can assign boundary conditions B 125 related to a first PDE to the boundary of the basic shape S 115. According to the embodiments of the present application, the boundary conditions can be related to the type of boundary conditions and to the range of values ​​of the boundary conditions, and the assignment of the boundary conditions related to the first PDE to the boundary of the basic shape can include the assignment of the following elements to the boundary of the basic shape: (1) Type of boundary conditions: The boundary of the basic shape may be assigned a single boundary condition or mixed boundary conditions. In the case of a single boundary condition, the entire boundary of the basic shape may be assigned the same type of boundary condition. In the case of mixed boundary conditions, the boundary of the basic shape may be assigned several types of boundary conditions. For example, different parts of the boundary of the basic shape may each be assigned corresponding types of boundary conditions. In one embodiment, the basic shape may be discretized, and the type of boundary conditions assigned to the boundary of the basic shape may include the type of boundary conditions assigned to the discrete points on the boundary of the basic shape. (2) Boundary condition range: The boundary condition values ​​can be in numerical form or in the form of functions. During the inference process in the domain, any arbitrary range of values ​​may occur in the boundary conditions, but values ​​for the no-boundary condition in the training data used to train the NN model are generally not feasible. Consequently, ranges of values ​​in numerical form or in the form of functions can be assigned to the boundary conditions in the training data. In one embodiment, the basic shape can be discretized, and the range of values ​​of the boundary conditions assigned to the boundary of the basic shape can include the range of values ​​of the boundary conditions assigned to the discrete points on the boundary of the basic shape.

[0032] Fig. Figure 3A shows a schematic representation of a basic shape 345 with a single assigned boundary condition according to an exemplary embodiment. In this exemplary embodiment, in the case of a single boundary condition, the entire boundary of the basic shape 345 can be assigned the same type of boundary condition 315. For example, the basic shape 345 can be discretized, and a type of boundary condition can be assigned to the discrete points on the boundary of the basic shape 345. For example, the boundary condition can be one of the Dirichlet boundary conditions, the Neumann boundary condition, and the Robin boundary condition. It is understood that any other suitable type of boundary condition can be assigned to the boundary of the basic shape.

[0033] Fig. Figure 3B shows a schematic representation of a basic shape 355 with mixed assigned boundary conditions according to an embodiment. In an embodiment, in the case of mixed boundary conditions, several types of boundary conditions can be assigned to the boundary of the basic shape 355. For example, the basic shape 355 can be discretized, and several types of boundary conditions can be assigned to the discrete points on the boundary of the basic shape 355. For example, the boundary of the basic shape can be divided into a first part and a second part, and a boundary condition of the first kind 325 can be assigned to the first part of the basic shape 355, and a boundary condition of the second kind 335 can be assigned to the second part of the basic shape 355. In an embodiment, the boundary condition of the first kind can be a Dirichlet boundary condition, and the boundary condition of the second kind can be a Neumann boundary condition.For example, the multiple types of boundary conditions can include at least two of the Dirichlet boundary condition, Neumann boundary condition, and Robin boundary condition. It is understood that any other suitable types of multiple boundary conditions can also be assigned to the boundary of the basic form.

[0034] Fig. Figure 3C shows a schematic representation of a boundary condition module 120 according to an exemplary embodiment. In the exemplary embodiment from Fig. 3C comprises the boundary condition module 120, an optional subdivision module 310, a boundary condition type assignment module 320, and a boundary condition value assignment module 330. It is understood that the boundary condition module 120 may include further modules, in Fig. 3C shows only those modules that relate to the embodiments of the present disclosure.

[0035] If the boundary of the basic shape is assigned a single boundary condition, it is not necessary to subdivide the boundary of the basic shape using the subdivision module 310. If the boundary of the basic shape is assigned mixed boundary conditions, the subdivision module 310 can subdivide the boundary of the basic shape S 115 into multiple parts. For example, the subdivision module 310 can randomly subdivide the boundary of the basic shape S 115 into several (for example, 2) interconnected parts. The subdivision module 310 can output the subdivided parts to the boundary condition type assignment module 320 and the boundary condition value assignment module 330.

[0036] If a single boundary condition is assigned to the boundary of the basic form, the boundary condition type assignment module 320 can assign a boundary condition type BT 365 to the boundary of the basic form 305. For example, the PDE of the Laplace2d-Dirichlet type is a two-dimensional (2D) Laplace equation with a single Dirichlet boundary condition, which can be used to solve problems related to the distribution of electric or magnetic fields, in the area of ​​problems relating to the structural design or materials engineering of industrial products, such as batteries (e.g., bipolar plates of fuel cells), electric heaters, electromagnets, etc. For PDEs of the Laplace2d-Dirichlet type, the boundary condition type assignment module 320 can assign a single Dirichlet boundary condition to the basic form, where the Laplace2d-Dirichlet is bounded by the following equation: In Ω, Δu=0; on ∂Ω, u=uD.

[0037] As another example, the PDE of type Darcy2d is a 2D Darcy flow with a single Dirichlet boundary condition, with coefficient field a(x) and source term f(x), which can be used to solve problems related to flows (such as fluid velocity and pressure) in the field of problems relating to the structural design or materials engineering of industrial products, such as hydraulic structures or water collection systems, etc. For PDEs of type Darcy2d, the boundary condition type assignment module 320 of the basic form can assign a single Dirichlet boundary condition, where the Darcy2d is bounded by the following equation: In Ω, −∇(a(x)∇u)=f(x) on ∂Ω, u=uD holds.

[0038] As another example, the PDE of type Heat2d is a 2D time-dependent heat conduction equation that has a coefficient for thermal diffusivity α, time-dependent boundary conditions, and initial conditions. It can be used to solve problems related to temperature distribution, specifically in the structural design or materials engineering of industrial products such as electric heaters, batteries, building material design, cooling and insulating materials, etc. For PDEs of type Heat2d, the boundary condition type assignment module 320 can assign Dirichlet boundary conditions to the basic form, where Heat2d is bounded by the following equation: in Ω×[0,T]applies to ∂u∂t=αΔuon ∂Ω×[0,T]u(x,t)=uD(x,t)applies to Ω×{0}applies to (x,0)=u0(x)

[0039] In the case where mixed boundary conditions are assigned to the boundary of the basic shape, the boundary condition type assignment module 320 can assign several types of boundary conditions BT 365 to the boundary of the basic shape 305. For example, the boundary condition type assignment module 320 can assign one type of boundary condition from the several types of boundary conditions to each part of the basic shape, which is subdivided into several parts. For example, the PDE of type Laplace2d-Mixed is a two-dimensional (2D) Laplace equation with mixed Dirichlet and Neumann boundary conditions on ∂Ω=D∪N, which can be used to solve problems related to the distribution of electric or magnetic fields, specifically in the area of ​​problems relating to the structural design or materials engineering of industrial products such as batteries (e.g., bipolar plates of fuel cells), electric heaters, electromagnets, etc. For PDEs of the Laplace2d-Mixed type, the boundary condition type assignment module 320 can assign a Dirichlet boundary condition or a Neumann boundary condition to the different parts of the basic form, where the Laplace2d-Mixed is bounded by the following equation: In Ω, Δu=0; in D, u=uD; in N, ∂u∂n=g

[0040] The boundary conditions can be in numerical form or in the form of functions. The boundary condition value assignment module 330 can assign values ​​BV 375 in numerical form or in the form of functions for the boundary conditions. For example, the boundary condition value assignment module 330 can generate random values ​​in a defined range (e.g., within [0,1]) for the (subdivided) basic form (e.g., using a random function), and the boundary condition values, which are in numerical form or in the form of functions, can be based on these random values ​​(e.g., they can correspond to these random values, or they can be certain piecewise linear functions using these random values, etc.).

[0041] In one example, the boundary condition module 120 can output a boundary condition B 125 assigned to the basic form, which includes the boundary condition type BT 365 and the boundary condition value BV 375.

[0042] Back to Fig. 1, where the device 100 may optionally include the PDE parameter function module 160, which assigns to the basic form S 115 a PDE parameter function P 165 related to a first PDE, wherein the PDE parameter function P 165 includes a coefficient related to the first PDE (for example, the coefficient field a(x) from equation (3), α) from equation (4)) and / or a source term (for example, f (x) from equation (3)). In one example, the PDE parameter function module 160 can generate random values ​​in a defined range (for example, within [0,1]) (for example, by means of a random function), and the values ​​of the PDE parameter function P 165 related to the first PDE can be based on these random values ​​(for example, they can correspond to these random values, or they can be certain piecewise linear functions using these random values, etc.).

[0043] The device 100 can include the solution module 130, which, based on the basic shape S 115 and the boundary condition B 125 (and optionally the PDE parameter function P 165), determines the solution u 130 of the first PDE for the basic shape, wherein the solution u 130 expresses the physical state of the object with the basic shape. In some embodiments, the solution module 130 for the first PDE can determine the solution u 130 for the basic shape by means of a numerical solution. For example, a numerical solution method, such as the finite element method (FEM), can be used for the first PDE to determine the solution u 130 for the basic shape.The first PDE can be a stationary PDE (for example, the previously described Laplace2d-Dirichlet, Darcy2d or Laplace2d-Mixed or any other stationary PDE), but it can also be a dynamic (such as a time-dependent) PDE (for example, the previously described Head2d or any other dynamic PDE).

[0044] For example, in the case of a dynamic PDE, the solution can express the physical state of the object with the basic form over time. For example, the first PDE can be a time-dependent PDE (like the previously described Head2d), where the time-dependent PDE in an example can be bounded by the following equation: in Ω×[0,T]giltut−Lu=fon ∂Ω×[0,T]giltu(x,t)=uD(x,t)on Ω×{0}giltu(x,0)=u0(x) where L a partial differential operator, for example an elliptic partial differential operator. The method according to the embodiments of the present disclosure can be applied to a dynamic PDE, as expressed, for example, in the equation (6) described above, in a space-time dimension. For example, the method according to the embodiments of the present disclosure can be applied to dynamic PDEs in a space-time dimension by partitioning the space-time domain Ω × [0, T]. The partitioning of the space-time domain can take the form of Ω i × [t j-1 - δ T , t j + δ T ] assume, where δ TThe temporal depth is denoted and represents the overlap in the time domain. Once such a decomposition of the space-time domain has been performed, the method according to the embodiments of the present disclosure can be applied to the space-time domain to obtain a global solution on the space-time domain. According to one embodiment, parallel iteration in the space and time domains is possible. In an embodiment applied to a PDE for heat dissipation, as shown, for example, in Fig. (8), a fixed time step t can be used. s and the rollout length k is used to divide the time domain into the times t = 0, t s , ..., (k - 1)t s Accordingly, the solutions u(x, 0), u(x, t) can be s ), ..., u(x, (k - 1) t s ) of solution module 130 the physical state of the object with the basic form at times t = 0, t s , ..., (k - 1) t s to express.

[0045] Fig. Figure 4A shows a schematic representation of a solution 425 determined according to an embodiment for a basic shape 415. For example, the basic shape 415 can be discretized, and the solution module 130 can, within the constraints of the first PDE and its boundary conditions, determine the solution 425 at the discrete points within the boundary of the basic shape 415.

[0046] According to one aspect of the present application, data augmentation can be performed to improve the generalizability of the NN model, so that the training data includes more sufficient data (which, for example, provide more comprehensive information). The symmetry group of a general partial differential operator L This refers to a series of transformations in which one solution is mapped to another. Thanks to the use of symmetries, multiple new solutions can be generated based on a given solution.

[0047] Back to Fig. 1, where the device 100 may optionally include the data augmentation module 140, which can perform data augmentation based on at least one of the elements basic form S 115, boundary condition values ​​of the boundary condition B 125, PDE parameter function P 165, and solution u 135 in order to obtain, according to at least one of the elements, a data-augmented basic form, data-augmented boundary condition values, a data-augmented PDE parameter function, and a data-augmented solution. The training data D 145 are based on at least one of the elements data-augmented basic form, data-augmented boundary condition values, data-augmented PDE parameter function, and data-augmented solution, as well as on the basic form, the boundary conditions, the PDE parameter function, and the solution.

[0048] In one embodiment, the data augmentation module 140 can perform data augmentation based on the basic shape. For example, the data augmentation module 140 can subject the basic shape to at least one of the operations spatial translation, spatial scaling and / or spatial rotation to obtain the data-augmented basic shape.

[0049] For example, with respect to any arbitrary point X (x1, x2) in the basic form S, the execution of a spatial translation of the basic form can include the following transformation of point X: (x1, x2) → (x1+t1, x2+t2), the execution of a spatial rotation of the basic form can include the following transformation of point X: (x1, x2) → (x1 cos θ - x2 sin θ, x1 sin θ + x2 cos θ), and the execution of a spatial scaling of the basic form can include the following transformation of point X: (x1, x2) → (sx1, sx2).

[0050] In one embodiment, if the basic form is subjected to spatial scaling, no corresponding data augmentation is required for the boundary condition values, the solution, and the PDE parameter function, and they also match a basic form that has undergone data augmentation (for example, if the first PDE is the Laplace2d-Dirichlet PDE described above in Equation (2)). Furthermore, in another embodiment, if the basic form is subjected to spatial scaling, the data augmentation module 140 can subject at least one of the elements—boundary condition values, solution, and PDE parameter function—to data augmentation so that the at least one element—boundary condition values, solution, and PDE parameter function—that has undergone data augmentation can match the data-augmented basic form.For example, if the basic form is subjected to a spatial scaling from (x1, x2) → (sx1, sx2), the data augmentation module 140 can further perform the following transformations (e.g. value scaling) for the solution u and the boundary condition u with respect to a PDE of the type Darcy2d previously described in equation (3). D and perform the coefficient field a(x) and the source term f(x) in the PDE parameter function: x → su, u D → su D , a(x) → a(x), f (x) → f (x); with respect to a PDE of the type Heat2d described above in equation (4), the data augmentation module 140 can further perform the following transformations (e.g. value scaling) for the solutions u, u0, the boundary condition u D and perform the coefficient in the PDE parameter function: u → u, u0 → u0, u D → u D , α → s 2α; with respect to a previously described PDE of the type Laplace2d-Mixed in equation (5), the data augmentation module 140 can further perform the following transformations (e.g. value scaling) for the solution and the boundary conditions: u → su, u D → su D , g → g.

[0051] For example, the data augmentation module 140 can perform data augmentation based on at least one of the elements boundary condition values, PDE parameter function, and solution. Specifically, the data augmentation module 140 can apply a value offset and / or value scaling to at least one of these elements to obtain corresponding data-augmented boundary condition values, a data-augmented PDE parameter function, and a data-augmented solution.

[0052] In one example, the data augmentation module 140, by subjecting the solution and the boundary condition to the subsequent transformation, can apply the value offset to the solution and the boundary condition: u → u + t, u D → u D + t. In one example, the data augmentation module 140 can be used by providing the solution u and the boundary condition u. D the following transformation, for the solution u and the boundary condition u D Perform the value scaling: u → su, u D → su D In one example, the data augmentation module 140 can further enhance a previously described PDE of the type Heat2d within the framework of equation (4) by providing the solutions u, u0 and the boundary condition u D The following transformation is performed, including value scaling: u → su, u0 → su0, u D → su D; with respect to a previously described PDE of the type Laplace2d-Mixed in equation (5), the data augmentation module 140 can further by providing the solution and the boundary conditions u D , subjecting g to the following transformation, perform the value scaling: u → su, u D → su D , g → sg.

[0053] It is understood that the preceding transformations and combinations for performing data augmentation are merely exemplary; the data augmentation module 140 can perform any other suitable data augmentation for any of the elements boundary condition values, PDE parameter function and solution.

[0054] Fig. Figure 4B shows a schematic representation of a data-augmented basic shape 415' and solution 425' according to an embodiment. For example, the data-augmented basic shape 415' can be obtained by a spatial rotation of the basic shape 415, and the data-augmented solution 425' can be determined by applying a value offset and / or a value scaling to the solution 425. In another embodiment, only the basic shape 415 can be subjected to a spatial rotation to obtain the data-augmented basic shape 415', while the solution 425 of the corresponding discrete point remains unchanged.

[0055] Back to Fig. 1: Based on the embodiments of the present application, training data can be generated for training the NN model 150, and the NN model 150 can be used to determine the physical state of an object with a shape.

[0056] The NN model 150 can encompass neural network models of any suitable structure. For example, the NN model 150 can adapt to flexible input / output formats and possess sufficient expressiveness to solve local problems with randomly changing shapes and PDE parameter functions. For instance, the NN model 150 could be a neural network model such as GNOT, FNO, DeepONet, etc.

[0057] Fig. Figure 5 shows a schematic representation of training a NN model 150 according to an exemplary embodiment. Fig. Figure 5 shows the NN model 150, the loss module 510, and the update module 520. It is understood that further modules may be included. Fig. Figure 5 shows only modules that relate to the embodiments of the present disclosure.

[0058] The NN model 150 can predict the solution P 525 for the basic shape S 505 in the training data D 145 generated using the method according to the embodiments of this disclosure and the boundary conditions B 545, where the solution P 525 expresses the physical state of the object with the basic shape. The loss module 510 can include any suitable loss functions, including the L1 loss function, the MSE loss function, or the cross-entropy function, etc. The loss module 510 can determine the loss L 535 based on the solution P 525 predicted by the NN model and the solution GT 515 in the training data. The update module 520 can update the learnable parameters of the NN model 150 based on the loss L 535. It is understood that any suitable optimization methods can be used to update the learnable parameters of the NN model.

[0059] By using the training data described in the present application to train the NN model 150, the NN model 150 can learn the solutions for the basic form. In particular, if the training data includes data for which data augmentation has been performed, the NN model 150 can train more comprehensively in order to capture complex details and changes in the basic form.

[0060] When determining the physical state of an object with a specific shape, the shape of that object may be relatively complex. The present application introduces a novel method and apparatus for neural inference (SNI, Schwarz Neural Inference) in which a basic shape (e.g., a relatively complex one) is first subdivided into several sub-shapes; then, the NN model is applied to each sub-shape to obtain the partial solution for that sub-shape; and based on the local solutions for each sub-shape, the global solution for the shape is then determined.Using the SNI method from local to global, it is possible for the NN model to solve PDE problems in the inference phase based on any shape (and not necessarily on a specific shape used in the training phase) and to determine the physical state of the object with that shape, thereby significantly increasing the efficiency of using NN models in practical industrial applications.

[0061] Fig. Figure 6A shows a block diagram of a device 600 for determining the physical state of an object with a shape according to an exemplary embodiment. In the example from Fig. 6A The device 600 comprises a subdivision module 610, a local solution module 620, and a global solution module 630. It is understood that the device 600 may include further modules, in Fig. 6A shows only the modules that relate to the embodiments of the present disclosure.

[0062] The subdivision module 610 can subdivide the form S 6015 of the object into several subforms SS 6025, whereby the totality of the several sub-areas corresponding to the several subforms SS 6025 covers the complete area of ​​the form S 6015.

[0063] For example, the subdivision module 610 can subdivide the shape S 6015 of the object into several sub-shapes SS 6025 based on the data in the structure data that expresses the shape S 6015 of the object. In one embodiment, the shape of the object is expressed in the structure data by at least one of the following elements: circle center, radius, width, height, length, grid cell, discrete point, or geometric shape.

[0064] See the example from Fig. 6B. The object can have the form 6095 and the subdivision module 610 can divide the form 6095 into several (for example, K, where K is an integer greater than 1) subforms SS1, SS2, ... SS Ksubdivide. For the sake of simplicity, in Fig. Figure 6B shows only two subforms, 6085 and 6105. Each subform of the multiple subforms can be located in a defined area. For example, subform 6105 can be located in the defined area 6115.

[0065] In the present application, the boundary of the object's shape can be referred to as the global boundary, and the boundaries of the sub-shapes can be referred to as local boundaries. See the examples from Fig. 6B. In an example, the boundary of the form 6095 with the global boundary conditions 6125 (as in, for example, Fig. 6B (represented by the red and green parts) can be connected. Accordingly, at least one part of the multiple subforms resulting from subdivision can be connected to a part of global boundary conditions 6125. For example, subform 6105 can be connected to part of global boundary conditions 6145 (as represented, for example, by the red and green parts in SS1) of global boundary conditions 6125. The values ​​of part of global boundary conditions 6145 of subform 6105 can correspond to the values ​​of the part of global boundary conditions 6125 of form 6095 corresponding to subform 6105.

[0066] In one example, the boundary of shape 6105 can be connected to the global boundary conditions 6135 (as represented, for example, by the gray part in SS1). For example, the virtual boundary conditions 6135 of the sub-shape can be boundary conditions on the boundary of sub-shape 6105 that is not part of the boundary of shape 6095. For example, the virtual boundary conditions 6135 can include boundary conditions on the boundary formed by the points in the interior (for example, not on the boundary of the shape) of the shape. In one embodiment, the local solutions for each sub-shape and the global solution are determined by several iteration rounds, and the values ​​of the virtual boundary conditions 6135 of the sub-shape in round n+1 can be based on the global solution gu(n) of the previous round for shape 6095.

[0067] In one embodiment, the local boundary conditions of the sub-shape can include a part of the global boundary conditions and / or virtual boundary conditions for the sub-shape. For example, the local boundary conditions of sub-shape 6105 can include a part of the global boundary conditions 6145 and the virtual boundary conditions 6135.

[0068] Based on the global boundary conditions B 6035 for the shape S 6015 and the multiple sub-shapes SS 6025 (and optionally the PDE parameter function 6065), the local solution module 620 can use the neural network (NN) model to determine the local solutions for each sub-shape of the multiple sub-shapes SS 6025, where the local solution for each sub-shape expresses the physical state of the part of the object that possesses that sub-shape. The data used to express the shape S 6015 of the object, the global boundary conditions B 6035, and the optional PDE parameter function 6065 can be referred to as structural data, which represent the structure of the object and the constraints of the physical system to which it belongs.In one embodiment, the local solutions for each subform and the global solution are determined through several iteration rounds, and the local solution module 620 can further obtain the local solution lu(n+1) 6045 of round n+1 based on the global solution gu(n) 6075 of round n. For example, the global solution gu(0) can be initialized before the first local solution round. For example, the global solution gu(0) can be set to the default value (for example, 0 or 1) or this global solution gu(0) can be initialized randomly.

[0069] For example, based on each sub-shape and the local boundary conditions for each sub-shape, the local solution module 620 can use the NN model to determine a local solution for each sub-shape of the multiple sub-shapes. For example, based on sub-shape 6105 and the local boundary conditions associated with it, the local solution module 620 can use the NN model to determine the local solution for the points inside sub-shape 6105 (as represented, for example, by the light blue points). While in Fig. Figure 6B uses the light blue points inside shape 6105 to illustrate that the NN model obtains the local solution for the points inside sub-shape 6105. However, in another embodiment, based on sub-shape 6105 and its associated local boundary conditions, the NN model can determine the local solution for all points (including those on the local boundary and those inside) on sub-shape 6105. During the iterative process, the values ​​on the global boundary remain unchanged, and the values ​​on the virtual boundary are updated with each iteration.

[0070] Back to Fig. 6A, where the global solution module 630 of the device 600 can determine the global solution for the shape based on the local solutions for each of the multiple sub-shapes, the global solution for the shape expressing the physical state of the object. In one embodiment, the local solutions for each sub-shape and the global solution are determined by several iteration rounds, and the global solution gu(+1 n) 6055 of round n+1 can be based on the local solution lu(n+1) 6045 for each sub-shape.

[0071] See the example from Fig. 6B: The global solution module 630 can extend and link the local solutions for each subform to obtain the global solution 6165 for subform 6095 (as shown, for example, by the dark blue dots in Fig. 6B (shown).

[0072] In one embodiment, the PDE associated with the shape can be a dynamic PDE. Accordingly, the solution for the shape expresses the physical state of the object with the basic shape over time. In one embodiment, the object can be at least one of the following: bipolar plate of a fuel cell, automotive part, aircraft part, part of a building, reactor tube, and flow deflector, wherein the physical state includes at least one of the elements: velocity, pressure, temperature, electric field, and magnetic field.

[0073] In one example, the subdivision module 610 can include a non-overlapping subdivision module that divides the shape into several non-overlapping sub-shapes in order to form several non-overlapping sub-shapes, the totality of the several sub-areas corresponding to the several sub-shapes covering the entire area of ​​the shape. Fig. Figure 7A shows a schematic representation of a shape divided into several non-overlapping sub-shapes according to an exemplary embodiment. For example, for the sake of simplicity, in Fig. Figure 7A shows only two subforms, 7015 and 7025, of the multiple non-overlapping subforms. In the example where the form was discretized, no two subforms of the multiple non-overlapping subforms contain any overlapping points. The subdivision module 610 can subdivide the form into K non-overlapping forms, based on a graph partitioning algorithm such as METIS, where K is an integer greater than 1. In one example, K can be predefined; for instance, K can be a hyperparameter.

[0074] In one example, the subdivision module 610 can comprise an overlapping subdivision module that divides the shape into several overlapping sub-shapes to form multiple overlapping sub-shapes, wherein the totality of the multiple sub-areas corresponding to the multiple sub-shapes covers the entire area of ​​the shape. In the example where the shape has been discretized, at least two sub-shapes of the multiple overlapping sub-shapes can comprise overlapping points. For example, the overlapping subdivision of the shape can assist in convergence based on the SNI method according to the embodiments of this disclosure, so that the physical state of the object with the shape can be determined more accurately.

[0075] Fig. Figure 7C shows a schematic representation of an overlapping subdividing module 710 according to an exemplary embodiment. In the example from Fig. 7C comprises the overlapping subdividing module 710, the intermediate subdivision module 720, and the shape extension module 730. It is understood that the overlapping subdividing module 710 may include further modules, in Fig. 7C only show modules relating to the embodiments of the present disclosure.

[0076] The intermediate subdivision module 720 can subdivide the shape S 7085 into several non-overlapping sub-shapes to form several non-overlapping intermediate shapes ISS 7095. In one embodiment, the intermediate subdivision module 720 can comprise the aforementioned non-overlapping subdividing module, and the entirety of the several sub-areas corresponding to the several intermediate shapes ISS 7095 covers the entire area of ​​the shape. In another embodiment, the entirety of the several sub-areas corresponding to the several non-overlapping intermediate shapes ISS 7095 subdivided by the intermediate subdivision module 720 may not cover the entire area of ​​the shape, so that the several non-overlapping intermediate shapes may differ from the several non-overlapping sub-shapes.In an example where the form S 7085 has been discretized, there are no overlapping points in any two intermediate forms of the multiple non-overlapping intermediate forms, and at least one point of the form cannot be contained in any intermediate form.

[0077] See Fig. 7B: The subdivision module 720 can subdivide the shape 7035 into several non-overlapping intermediate shapes. For the sake of simplicity, in Fig. 7B shows only the first intermediate part form 7045 and the second intermediate part form 7055 of the several intermediate part forms.

[0078] Back to Fig. 7C, where the overlapping subdividing module 710 can include the shape extension module 730 to extend some or all of the intermediate shapes of the multiple non-overlapping intermediate shapes, thus creating multiple overlapping sub-shaped shapes. In the example where the shape has been discretized, the shape extension module 730 can, for instance, iteratively incorporate adjacent points into some or all of the intermediate shapes of the multiple intermediate shapes, thus creating multiple overlapping sub-shaped shapes. In one example, this iteration can occur in d rounds, where d can be called the extension depth d and is predefined; for example, d can be a hyperparameter.In another example, this iteration can be performed until the totality of the multiple sub-areas corresponding to the several overlapping sub-forms covers the entire area of ​​the form.

[0079] See Fig. 7B, where the mold extension module 730 can extend the first intermediate mold 7045 to the first partial mold 7065 and the second intermediate mold 7055 to the second partial mold 7075.

[0080] Fig. Figure 8 shows a schematic representation of a local solution module 620 according to an exemplary embodiment. According to the exemplary embodiment from Fig. 8. The local solution module 620 can include a subform operator generation module 8010, a subform preprocessing module 8020, a NN model 8050, a postprocessing module 8060, a local boundary conditions module 8070, a value operator generation module 8080, and a value preprocessing module 8090. Fig. Figure 8 shows, in addition to the local solution module 620, an optional PDE parameter function operator generation module 8030 and an optional PDE function parameter preprocessing module 8040. In one embodiment, the PDE parameter function operator generation module 8030 and the PDE function parameter preprocessing module 8040 can also be integrated into the local solution module 620. It is understood that the local solution module 620 can include further modules, as described in Figure 8. Fig. Figure 8 shows only the modules that relate to the embodiments of the present disclosure.

[0081] During the inference process with respect to any subdivided subform, the domain of the subform and its boundary conditions may not coincide with the domain of the training data (for example, the training data generated according to the method described in the embodiments of this disclosure). To ensure that the NN model trained on the training data is better suited for the subdivided subforms during the inference phase, the transformation T: P × H → P × H can be applied to transform local problems with shapes or values ​​outside the domain of the training data into those within the domain of the training data, where P represents the subform space and H represents the boundary conditions and optionally the parameter of the PDE function. In the embodiments of this disclosure, the transformation T can be referred to as preprocessing.For example, the preprocessing can include at least one of the following operations: spatial displacement, spatial scaling, value offset, and value scaling, where the spatial displacement and spatial scaling can be applied to the subforms, and the value offset and value scaling can be applied to the PDE parameter function and / or the values ​​for the local boundary conditions. After inference using the NN model G: P × H → U, the resulting solution function can be transformed back using a suitable inverse transformation T̃: U → U. In the embodiments of this disclosure, the transformation T̃ can be referred to as postprocessing.In examples where the domain of the subforms and boundary conditions may coincide with the domain of the training data, the aforementioned preprocessing and postprocessing of the subforms and boundary conditions can be omitted, or the preprocessing and postprocessing transformations can be performed using standard preprocessing and postprocessing operators that do not actually transform the subforms and boundary conditions. For example, the preprocessing / postprocessing operators could include preprocessing / postprocessing operators for the subforms, preprocessing / postprocessing operators for the PDE parameter functions, and value preprocessing / value postprocessing operators.

[0082] The subform operator generation module 8010 can obtain subform preprocessing operators SO 8015 and subform postprocessing operators SO' 8025 for each subform of the multiple subforms SS 8005 based on these subforms. For example, the subform range (for example, range 6115 for subform 6105) is available. Fig. 6B) possibly outside the specified range (for example, 215 from Fig. 2A) of the base form in the training data, and the subform operator generation module 8010 can determine the preprocessing operator SO 8015 and the postprocessing operator SO' 8025 based on the subform's region 6115 and the base form's fixed region 215. For example, this subform preprocessing operator SO 8015 can correspond to the spatial shift and / or spatial scaling of the subform to transform the subform region 6115 to within the fixed region 215, and the subform postprocessing operator SO' 8025 can be the inverse transformation corresponding to the subform preprocessing operator SO 8015. For example, the fixed region 215 is the unit square [0.5, 0.5]. 2 ⊂ℝ 2, the subform area 6115 is [0, 2] * [0, 2], the subform preprocessing operator SO 8015 can correspond to the spatial shift [-1, -1] and the spatial scale 0.5 and the corresponding subform postprocessing operator SO' 8025 can correspond to the spatial shift [+1, +1] and the spatial scale 2, so that the preprocessed subform is transformed into the specified area of ​​the basic form serving as training data.

[0083] Accordingly, the subform operator generation module 8020 can apply the obtained subform preprocessing operator SO 8015 to each subform of the multiple subforms SS 8005 to obtain the preprocessed subform TS 8035. For example, in an example where the subform is discretized, for any point X (x1, x2) in the form, a spatial translation of the subform includes the following transformation of the points X: (x1, x2) → (x1+t1, x2+t2), and a spatial scaling of the subform includes the following transformation of the points X: (x1, x2) → (sx1, sx2) or (x1+t1, x2+t2) → (s(x1+t1), s(x2+t2)). In one embodiment, the NN model 8050 can obtain an intermediate value of the local solution for each of the multiple sub-forms based on the pre-processed sub-form TS 8035.In one embodiment, the post-processing module 8060 can apply the subform post-processing operator SO' 8025 to the intermediate value of the local solution to determine the local solution for each subform of the multiple subforms. For example, the post-processing module 8060 can apply the subform post-processing operator SO' 8025 to the pre-processed subform TS 8035 to transform it back into the original subform 8005, thus obtaining the local solution for each subform of the multiple subforms. It is understood that the above transformation is merely an example, and any other suitable transformation can also be used to perform pre-processing / post-processing on the subform.

[0084] For example, the local solutions for each subform and the global solution for the form can be determined through several iteration rounds, and the intermediate value of the local solution of the current round (n+1) can be expressed as Ilu(n+1) 8085, where Ilu(n+1) 8085 can include the intermediate value of the local solution for each subform of the multiple subforms, and the local solution of the subform can be expressed as lu(n+1) 8145, where lu(n+1) 8145 can include the local solution for each subform of the multiple subforms.

[0085] If the structure data contains a PDE parameter function, the PDE parameter function operator generation module 8030 can obtain the PDE parameter function preprocessing operator PO 8055 and the PDE parameter function postprocessing operator PO' 8065 based on the PDE parameter function P 8045.For example, the range of values ​​(e.g., the maximum and / or minimum of the coefficients (such as the coefficient field) and / or the source term) of the PDE parameter function (e.g., the PDE parameter function 6065) in the structure data may be outside the range of values ​​(e.g., [0,1]) of the PDE parameter function in the training data, and the PDE parameter function operator generation module 8030 can determine the PDE parameter function preprocessing operator PO 8055 and the PDE parameter function postprocessing operator PO' 8065 based on the range of values ​​of the PDE parameter function in the structure data and the range of values ​​of the PDE parameter function in the training data.For example, this PDE parameter function preprocessing operator PO 8055 can correspond to the value offset and / or value scaling to transform the value range of the PDE parameter function in the structure data to within the value range of the PDE parameter function in the training data, and the PDE parameter function postprocessing operator PO' 8065 can be the inverse transformation corresponding to the PDE parameter function preprocessing operator PO 8055.

[0086] Accordingly, the PDE parameter function preprocessing module 8040 can apply the obtained PDE parameter function preprocessing operator PO 8055 to the PDE parameter function in the structure data to obtain the preprocessed PDE parameter function TP 8075. For example, the value offset with respect to the coefficient α (or the coefficient field a(x)) and / or the source term f(x) in the PDE parameter function can include the transformations a → a + t, a(x) → a(x'), f(x) → f(x'), and the value scaling can include the transformations a → sa, a(x) → a(x'), f(x) → f(x'). For example, in an example where the subform is discretized, x and x' each represent the coordinates of a point in the aforementioned subform 8005 and the coordinates of the corresponding point in the preprocessed subform TS 8035, respectively.In one embodiment, the NN model 8050 can determine an intermediate value Ilu(n+1) 8085 of the local solution for each of the multiple subforms based on the preprocessed PDE parameter function TP 8075. In another embodiment, the postprocessing module 8060 can apply the PDE parameter function postprocessing operator PO' 8065 to the intermediate value of the local solution to obtain the local solution lu(n+1) 8145 for each of the multiple subforms. It is understood that the above transformation serves only as an example, and any other suitable transformation can also be used to perform preprocessing / postprocessing on the PDE parameter function.

[0087] The Local Boundary Conditions Module 8070 can, for each subform of the multiple subforms SS 8005, determine the local boundary conditions LB 8105 for that subform based on at least one of the elements global boundary conditions B 8085 and global solution gu(n) 8095 of the previous round for the form. For example, the Local Boundary Conditions Module 8070 can determine for each subform of the multiple subforms SS 8005 that the local boundary conditions of that subform correlate only with a part of the global boundary conditions, and can accordingly set the local boundary conditions LB 8105 of that subform as the boundary condition value of that boundary part in the global boundary conditions B 8085;or the Local Boundary Conditions module 8070 can determine that the local boundary conditions of this subform correlate only with the virtual boundary (for example, the virtual boundary does not belong to the global boundary of the form and may be located in the interior of the form), and can accordingly set the local boundary conditions LB 8105 of this subform to the value of the global solution on the virtual boundary of the global solution gu(n) 8095;or the Local Boundary Conditions module 8070 can determine that the local boundary of the first part of this subform correlates with the global boundary of the first part and the local boundary of the second part of this subform correlates with the virtual boundary of the second part, and can accordingly set the first local boundary condition of the local boundary conditions LB 8105 of this subform to the value of the boundary condition on the global boundary of the first part of the global boundary conditions B 8085 and accordingly set the second local boundary condition of the local boundary conditions LB 8105 of this subform to the value of the global solution on the virtual boundary of the second part of the global solution gu(n) 8095.

[0088] In one example, the local solutions for each subform of the multiple subforms are determined based on the local boundary conditions for that subform and that subform.

[0089] The value operator generation module 8080 can obtain, for each subform of the multiple subforms SS 8005, the value preprocessing operator VO 8115 and the value postprocessing operator VO' 8125 for this subform based on the local boundary conditions LB 8105 for this subform. For example, with respect to each subform of the multiple subforms SS 8005, the value range (for example, based on its maximum and / or minimum) of the local boundary conditions LB 8105 of this subform may lie outside the specified range (for example, [0,1]) of the boundary conditions outside the training data, and the value operator generation module 8080 can determine the value preprocessing operator VO 8115 and the value postprocessing operator VO' 8125 based on the value range (for example, based on its maximum and / or minimum) of the local boundary conditions LB 8105 and the specified range of the boundary conditions in the training data.For example, this value preprocessing operator VO 8115 can correspond to the value offset and / or value scaling to transform the value range of the local boundary conditions LB 8105 to within the specified value range of the boundary conditions in the training data, and the value postprocessing operator VO' 8125 can be the inverse transformation corresponding to the value preprocessing operator VO 8115.

[0090] The value preprocessing module 8090 can apply the value preprocessing operator VO 8115 to the local boundary conditions LB 8105 for each subform of the multiple subforms SS 8005 to obtain the preprocessed local boundary conditions Tlb 8135. For example, the value offset with respect to the value of the local boundary conditions LB 8105 can include the transformation: b → b + t, and the value scaling with respect to the value of the local boundary conditions LB 8105 can include the transformation: b → sb or b + t → s(b + t). It is understood that the above transformation is merely an example, and any other suitable transformation can be used to perform preprocessing / postprocessing on the value of the local boundary conditions LB 8105.

[0091] In one embodiment, the NN model 8050, based on the preprocessed subform TS 8035 and the preprocessed local boundary conditions Tlb 8135 (and optionally the preprocessed PDE parameter function TP 8075), can determine the intermediate value Ilu(n+1) 8085 of the local solution for each subform of the multiple subforms. In another embodiment, the post-processing module 8060 can apply the value post-processing operator VO' 8125 to the intermediate value of the local solution to determine the local solution lu(n+1) 8145 for each subform of the multiple subforms.In one embodiment, if the structure data contains a PDE parameter function TP 8045, it is possible to generate only the PDE parameter function preprocessing operator PO 8055, but not the PDE parameter function postprocessing operator PO' 8065; accordingly, the postprocessing module 8060 does not need to take the PDE parameter function postprocessing operator PO' 8065 into account.

[0092] In one example, the NN model 8050 can be trained according to the procedure described in this text. For instance, the NN model 8050 can be trained based on the training data generated according to the procedure described in this text.

[0093] Fig. Figure 9 shows a schematic representation of a global solution module 630 according to an exemplary embodiment. In the example from Fig. The global solution module 630 can include the limit and extension operator generation module 910, the linking module 920, and the update module 930. It is understood that the global solution module 630 can include further modules, in Fig. Only the modules relating to the embodiments of the present disclosure are shown in Figure 9.

[0094] The boundary and extension operator generation module 910 can obtain the boundary and extension operator for each subform of the multiple subforms SS 915 based on each subform. For example, the boundary and extension operator generation module 910 can determine for each subform of the multiple subforms SS 915 which part of the form S 905 this subform SS 915 correlates with (for example, which discrete points of the form this subform includes), and accordingly determine the boundary operator RO 925 to delimit the area of ​​this subform SS 915 within the area of ​​the form S 905, as well as the extension operator RO' 935 to extend the area of ​​this subform SS 915 to the area of ​​the form S 905.For example, the limiting operator RO 925 and the extension operator RO' 935 can be transposed matrices, where the dimensions of the matrix (number of rows and columns) of the extension operator RO' 935 are the total number of points on the form S 905 or the number of local points on the subform SS 915, respectively. The elements in this matrix corresponding to the points of position in the subform SS 915 are set to 1, and the other elements are set to 0. Consequently, by multiplying this matrix by the solution vector of the subform SS 915, the solution vector of the subform SS 915 can be extended to the dimension of the solution vector of the form S 905. Conversely, the transposition of the matrix of the extension operator RO' 935 can serve as the limiting operator RO 925.

[0095] For example, the area of ​​shape S 905 could be a square [-1, 1] 2and in the area of ​​the first subform SS 915 of the several subforms obtained by subdividing the form by a square [-1, 0] 2 act. Accordingly, the limiting and augmenting operator generation module 910 can determine the limiting operator RO 925 for the first subform, so that by applying the limiting operator RO 925 to the values ​​in the region [-1, 1] 2 the first subform of the form in the area [-1, 0] 2 can be obtained, the values ​​of a part have in the values ​​of the form S 905, which corresponds to the area [-1, 0] 2 the first subform SS 915. The limiting and extension operator generation module 910 can further determine the extension operator RO' 935 for the first subform SS 915, so that by applying the extension operator RO' 935 to the values ​​in the region [-1, 0] 2 the first subform SS 915 which covers the area ([-1, 1] 2) the extended subform covering form S 905 can be obtained, the extended subform being in the region [-1, 0] 2 has values ​​matching the first subform SS 915, but in the area [-1, 1] 2 in the other parts outside the area of ​​the first subform SS 915, standard values ​​(for example, 0) may be.

[0096] In one embodiment, the logic module 920 can obtain the intermediate value Igu(n+1) 955 of the global solution for the form S 905 based on the local solution lu(n+1) 945 for each subform SS 915 and the extension operator RO' 935 for each subform SS 915. For example, the logic module 920 can obtain the extension operator RO' for the subform k of the K subforms SS 915 obtained by subdividing the form S 905. k '935 for this subform k to the local solution lu(n+1) k apply to this subform (for example RO') k (lu(n + 1) k)) and the linking module 920 can provide the extended local solutions RO' k (lu(n + 1) k ) sum the individual subforms to obtain the intermediate value Igu(n+1) 955 of the global solution of the form (for example) Igu(n+1)=∑k=1KRO'k(lu(n+1)k)).

[0097] In another embodiment, the linking module 920 can obtain the intermediate value Igu(n+1) 955 of the global solution for the form S 905 based on the local solution lu(n+1) 945 for each subform SS 915, the extension operator RO' 935 and the limiting operator RO 925 for each subform SS 915 and the global solution gu(n) of the previous round for the form S 905 (for example, Igu(n+1)=∑k=1K(RO'k(lu(n+1)k)+(I−RO'kROk)gu(n), where l denotes the identity matrix). In one embodiment, RO' means k (lu(n + 1) k ) + (I - RO' k RO k)gu(n) for each subform k, such that the solution within the entire range of this form lies on the subset of the subform k lu(n + 1) k is and that the solution lies on the subset outside this subform k gu(n).

[0098] The update module 930 can obtain the global solution Gu(n+1) 975 for the form based on the intermediate value Igu(n+1) 955 of the global solution for the form S 905 and the global solution gu(n) of the previous round for the form S 905.

[0099] For example, update module 930 can obtain the global solution gu(n+1) 975 for the form S 905 based on a linear combination of the intermediate value Igu(n+1) 955 of the global solution for the form S 905 and the global solution gu(n) of the previous round for the form S 905. For example, update module 930 can compute the global solution gu(n+1) 975 for the form S 905 in the following way: gu(n+1)=(1−τK)gu(n)+τ Igu(n+1) where K is the number of subforms obtained by subdivision and τ is the step size, a hyperparameter used to control the convergence rate, for which the following holds: 0<τ<1K.

[0100] According to the embodiments of the present disclosure, the SNI process described in this text can be carried out based on the following algorithm 1:

[0101] In algorithm 1 in Table 1, the form S can also be called the domain Ω, and accordingly the subforms SS can be called SS. k also as subfields Ω k can be expressed. In an example, the convergence criterion C can be gu(n + 1) and change no more or less than a threshold value.

[0102] Fig. Figure 10 shows a schematic representation of shapes and the data efficiency of applying SNI to the individual shapes according to an embodiment of the present disclosure. Fig. In the first row, from left to right, the forms SA, SB and SC respectively are shown.

[0103] In the curve diagrams in the second and third rows of Fig. Figure 10 shows the size of the training dataset on the x-axis and the relative l2 error on the y-axis. The second row of Fig. Figure 10, from left to right, compares the relative l2 error between the result of direct inference with GNOT (orange), the result of inference based on SNI (blue), and the validation (red) when relating the PDEs of type Laplace2d-Dirichlet described in this text within the framework of equation (2) to the forms SA, SB, and SC, respectively. In the third row of Fig. Figure 10, from left to right, compares the relative l2 error between the results of direct inference with GNOT (orange), inference based on SNI (blue), and validation (red) when applying the Darcy2d PDEs described in equation (3) of this text to the forms SA, SB, and SC, respectively. The results of inference based on SNI and direct inference with GNOT are presented based on an inference process with 100 different boundary conditions. Furthermore, the best validation error from the training phase is provided as a reference.

[0104] From the in Fig. From the results presented in Figure 10 regarding the data efficiency of SNI, the following conclusions can be drawn: (1) In the case of large datasets, the relative l2 error of SNI is significantly lower than the relative error of direct inference using GNOT; (2) In the case of large datasets, the error of SNI is equal to or even lower than the validation error; (3) SNI requires a smaller dataset to achieve a comparable result to direct inference using GNOT. Overall, these results demonstrate that SNI offers significant advantages in terms of data efficiency. The framework presented by the embodiments of this disclosure has the remarkable ability to extract more information from limited data and scales more effectively with increasing datasets.

[0105] Compared to the initial approach, where, for example, GNOT-based neural network models are used to directly train and infer the entire shape of the object, the method presented in the embodiments of this disclosure shows better results for all stationary problems. For all shapes, the prediction error was reduced by 34.8% to 96.8%. The effectiveness of the method presented in the embodiments of this disclosure when processing arbitrary shapes not included in the training data clearly demonstrates its outstanding performance. In particular, since a simple polygon used in the training data may not be able to fully represent a complex test shape, the method presented in the embodiments of this disclosure is generally superior for more complex shapes.Furthermore, the method presented in the embodiments of this disclosure exhibits the same performance in the inference process for different shapes. With respect to all types of PDE in the test dataset, the deviation of the prediction errors for the different shapes was within 3.25%, demonstrating the ability to solve PDEs of any shape with consistent accuracy using a single trained NN model.

[0106] Fig. Figure 11 shows a schematic representation of an object with a shape according to an exemplary embodiment. The in Fig. The physical system shown in Figure 11 corresponds to a flow deflector, where the edges located in the corresponding grid cells 2 to 5 are the flow deflectors. The PDE used to describe the physical system corresponding to the flow deflector, based on the physical laws, can be formulated as follows: Δu(x)+k2u(x)=f,x∈Ω u(x)=u1(x),x∈γ1,left,up u(x)=0,x∈γwall where x = (x1, x2) denotes the spatial coordinates, u = (u1, u2) denotes the velocity, k is a given constant, f = f (x) is a given source term, γ 1, left denotes the left part of γ1, γ 1, left,up the upper part of γ 1, left denoted, γ 1, left,down the lower part of γ 1, left designated, γwall=γ1,up∪γ1,down∪γ2∪γ3∪γ4∪γ5∪γ1,left,down,and u1(x)=[(4−x2)(8−x2)16.0]T.

[0107] In one embodiment, the shape or area of ​​an object can be expressed by geometric parameters, where these geometric parameters can include the center point and radius of the boundaries γ2 to γ5. In another embodiment, the shape or area of ​​an object can be expressed by a set of discrete points, where the individual subsets of this set of discrete points include discrete points on the corresponding boundaries, and these discrete points can be expressed by coordinates. In yet another embodiment, the shape or area of ​​an object can be expressed by both the aforementioned geometric points and the set of discrete points.

[0108] For the in Fig. The exemplary flow deflector shown in Figure 11 and the physical system described by the PDE of equations 8(a) to 8(c) can be described using the embodiments described in the present disclosure together with Fig. Training data are generated according to the procedure described in section 1, and based on the generated training data, the NN model can be trained to solve the PDE of equations 8(a) to 8(c) for the basic form in order to preserve the physical state u, where this NN model can be called a local neural operator and this local neural operator is implemented in the local solution module 620. Fig. 6A can be used to predict the local solution u for the subform or subarea. Furthermore, for the in Fig. 8 exemplary flow deflector shown and the physical system described by the PDE of equations 8(a) to 8(c) using the embodiments of the present disclosure together with Fig. The physical state u of the flow deflector is determined in the process described in 6A.

[0109] Fig. Figure 12 shows a flowchart of the procedure for performing a structural optimization of an object, such as a flow deflector, according to an exemplary embodiment. As in Fig. As can be seen in Figure 11, the edges γ2 to γ5 are a flow deflector for which structural optimization is to be carried out, the aim of which is to optimize the shape and position of the flow deflector γ2 to γ5 in order to at the outlet γ 1, right to achieve a uniform distribution of the fluid flow.

[0110] The objective function of the structural optimization can be formulated based on fluid mechanics as shown in the following equation (15): J(W)=12∫γ1,right|u1−u1¯|2ds+β[k2∫Ω‖∇uT+Δu‖F2dx] where β represents the equilibrium coefficient, for example β = 0.01, u = (u1, u2) stands for the velocity, the viscosity is and the goal of structural optimization is to minimize J(W).

[0111] In step 1210, the structural parameter set is determined, and this structural parameter set serves to describe the boundaries of the object. With regard to the in Fig. The flow deflectors shown in Figure 11, serving as objects, can include the structural parameters circle center and radius of the circle edges γ2 to γ5, for example the structural features W = (x0, r0, ..., x3, r3).

[0112] In step 1220, the SNI method is applied according to the embodiments of the present disclosure (for example, in combination with the one described in Fig. 6A process) predicted the physical state of the flow deflector, namely velocity u = (u1, u2).

[0113] In step 1130, based on the predicted physical state u = (u1, u2) and equation (15) J(W) and based on J(W) the loss l(W) is determined, for example J(W) can serve as loss l(W).

[0114] The procedure then returns to step 1210 in step 1200 and updates the structural features W = (x0, r0, ..., x3, r3) based on the loss l(W). The next iteration round is then executed from steps 1120 to 1130 to 1110 based on these updated structural features.

[0115] It is understood that any suitable method for determining the loss l(W) in step 1130 can be used and that any suitable method for updating the structural features in step 1110 can be used.

[0116] In the exemplary embodiment from Fig. 11 and Fig. 12. Using a flow deflector as an example, the process of generating training data, the process of training the local neural operator, the SNI inference process, and the process of structural optimization of an object according to the embodiments of this disclosure are described. It is understood that the aforementioned individual processes of the embodiments of this disclosure can be applied to any type of object and any type of PDE; for example, the object may be a bipolar plate of a fuel cell, a motor vehicle part, an aircraft part, a part of a building, a reactor tube, a flow deflector, etc., and the physical state corresponding to the object may include velocity, pressure, temperature, electric field, magnetic field, etc.By using the method according to the embodiments of the present disclosure, sufficient training data for the NN model can be generated so that the NN model performs ideally, and the physical state of an object with any shape can be determined based on such a trained NN model, which increases the training and usage efficiency of the NN model.It is understood that if the object has a complex shape, the method according to the embodiments of the present disclosure has even more significant advantages over traditional methods in which a NN model is trained to solve the entire shape; for example, it can more easily obtain the training data for the basic shape, which gives the trained local neural operator a better generalization capability, so that it can effectively process objects with complex shapes and also effectively process objects with relatively large shape changes.

[0117] Fig. Figure 13 shows a flowchart of the computer-implemented method 1300 for generating training data according to an exemplary embodiment.

[0118] In step 1310, a basic shape is generated.

[0119] In step 1320, boundary conditions relating to a first partial differential equation (PDE) are assigned to the boundary of the basic form.

[0120] In step 1330, based on the basic shape and the boundary conditions, the solution of the first PDE for the basic shape is determined, where the solution expresses the physical state of the object with the basic shape.

[0121] According to one embodiment, the training data is obtained based on the basic shape, the boundary conditions and the solution.

[0122] According to one embodiment, the basic shape is bounded by a defined area. According to another embodiment, the basic shape is a planar polygon without self-intersections and without holes. According to another embodiment, the planar polygon comprises n vertices and 3 ≤ n ≤ 12.

[0123] According to one embodiment, the boundary conditions comprise several types of boundary conditions. According to one embodiment, step 1320 comprises: subdividing the boundary of the basic shape into several parts; and assigning one type of boundary condition from the several types of boundary conditions to each part of the several parts. According to one embodiment, the boundary conditions comprise first-type boundary conditions and second-type boundary conditions, and step 1320 comprises: subdividing the boundary of the basic shape into a first part and a second part; and assigning a first-type boundary condition to the first part and assigning a second-type boundary condition to the second part. According to one embodiment, the first-type boundary condition is a Dirichlet boundary condition and the second-type boundary condition is a Neumann boundary condition.

[0124] According to one embodiment, the boundary conditions are random values ​​from a defined range.

[0125] According to one embodiment, method 1300 further comprises: assigning a PDE parameter function related to the first PDE to the basic form, wherein the PDE parameter function comprises coefficients and / or source terms related to the first PDE, and wherein determining the solution of the first PDE for the basic form comprises: determining the solution of the first PDE for the basic form based on the basic form, the boundary conditions, and the PDE parameter function. According to one embodiment, the value of the PDE parameter function is a random value from a defined range.According to one embodiment, the following is further included: performing data augmentation based on at least one of the elements basic shape, boundary condition values, PDE parameter function and solution, in order to obtain at least one of the elements data-augmented basic shape, data-augmented boundary condition values, data-augmented PDE parameter function and data-augmented solution; and wherein the training data is based on at least one of the elements data-augmented basic shape, data-augmented boundary condition values, data-augmented PDE parameter function and data-augmented solution, as well as on the basic shape, the boundary conditions, the PDE parameter function and the solution.According to one embodiment, performing data augmentation based on at least one of the elements basic shape, boundary condition values, PDE parameter function and solution includes performing at least one of the following operations: performing a spatial translation, a spatial scaling and / or a spatial rotation of the basic shape; performing a value offset and / or a value scaling for at least one of the elements boundary condition values, PDE parameter function and solution.

[0126] According to one embodiment, the solution expresses the physical state of the object with the basic shape over time.

[0127] Fig. Figure 14 shows a flowchart of the computer-implemented method 1400 for training a NN model to determine the physical state of an object with a shape according to an exemplary embodiment.

[0128] In step 1410, a solution for a basic shape in the training data is predicted using the NN model based on the training data generated using the method according to the embodiments of the present disclosure, wherein the solution expresses the physical state of the object with the basic shape.

[0129] In step 1420, the loss is determined based on the solution predicted by the NN model and the solution in the training data.

[0130] In step 1430, the learnable parameters of the NN model are updated based on the loss.

[0131] Fig. Figure 15 shows a flowchart of the computer-implemented method 1500 for determining the physical state of an object with a shape according to an exemplary embodiment.

[0132] In step 1510, the shape of the object is divided into several sub-shapes, the totality of which corresponds to the several sub-areas covering the entire area of ​​this shape.

[0133] In step 1520, based on the global boundary conditions for this shape and these multiple sub-shapes, a local solution for each sub-shape of the multiple sub-shapes is determined using the neural network (NN) model, where the local solution for each sub-shape expresses the physical state of the part of the object that has this sub-shape.

[0134] In step 1530, based on the local solutions for each sub-shape of the multiple sub-shapes, the global solution for that shape is determined, where the global solution for that shape expresses the physical state of the object.

[0135] According to one embodiment, step 1510 comprises: subdividing this mold into several non-overlapping sub-molds to form several of these non-overlapping sub-molds; or subdividing this mold into several overlapping sub-molds to form several of these overlapping sub-molds. According to one embodiment, forming several of these overlapping sub-molds comprises: subdividing this mold into several non-overlapping sub-molds to form several non-overlapping intermediate sub-molds; and extending a portion or all of the intermediate sub-molds of the several non-overlapping intermediate sub-molds to form several of these overlapping sub-molds.

[0136] According to one embodiment, step 1520 further comprises, based on the PDE parameter function of the partial differential equation (PDE) corresponding to the NN model, determining a local solution for each subform of these several subforms using the NN model, wherein the PDE parameter function includes coefficients and / or source terms.

[0137] According to one embodiment, step 1520 comprises: obtaining, based on these subforms, subform preprocessing models and subform postprocessing models, and applying the subform preprocessing models to the subforms to obtain preprocessed subforms, and / or, based on the PDE parameter function, obtaining a PDE parameter function preprocessing model and a PDE parameter function postprocessing model, and applying the PDE parameter function preprocessing model to the PDE parameter function to obtain a preprocessed PDE parameter function; and, using the NN model, determining a local solution for each subform of the multiple subforms based on the preprocessed subforms and / or the preprocessed PDE parameter function, and based on the subform postprocessing model and / or the PDE parameter function postprocessing model.

[0138] According to one embodiment, the local solutions for each sub-shape and the global solution for that shape are determined through several iteration rounds, and step 1520 includes: setting the local boundary conditions for these sub-shapes based on at least one of the elements global boundary conditions and global solution of the previous round for that shape; and, based on these sub-shapes and the local boundary conditions for these sub-shapes, determining the local solutions for these sub-shapes using the NN model.According to one embodiment, step 1530 includes: obtaining value preprocessing models and value postprocessing models for these subshapes based on the local boundary conditions for these subshapes; applying the value preprocessing models to the local boundary conditions to obtain preprocessed local boundary conditions; determining intermediate values ​​for the local solutions of the subshapes using the NN model based on these subshapes and the preprocessed local boundary conditions; and applying value postprocessing operators to the intermediate values ​​of the local solutions to determine the local solutions for the subshapes.

[0139] According to one embodiment, the local solutions for each sub-shape and the global solution for that shape are determined by several iteration rounds, and step 1530 comprises: obtaining a boundary model and an extension model based on each of the multiple sub-shapes; determining an intermediate value for the global solution of the shape based on the local solution for each sub-shape and the extension model for each sub-shape, or determining an intermediate value for the global solution of the shape based on the local solution for each sub-shape, the extension model and the boundary model for each sub-shape, and the global solution of the previous round for the shape; and determining the global solution for that shape based on the intermediate value of the global solution for that shape and the global solution of the previous round for that shape.

[0140] According to one embodiment, the solution for the shape expresses the physical state of the object with this shape over time.

[0141] According to one embodiment, the object is at least one of the following: bipolar plate of a fuel cell, motor vehicle part, aircraft part, part of a building, reactor tube and flow deflector, wherein the physical state comprises at least one of the elements velocity, pressure, temperature, electric field and magnetic field.

[0142] According to one embodiment, the shape of the object in the structural data is expressed by at least one of the following elements: circle center, radius, width, height, length, grid cell or geometric shape.

[0143] According to one embodiment, the NN model is trained according to the computer-implemented method described in this text.

[0144] Fig. Figure 16 shows a flowchart of the computer-implemented method 1600 for performing a structural optimization of an object according to an exemplary embodiment.

[0145] In step 1610, the physical state of the object is predicted using the method described according to the embodiments of the present disclosure.

[0146] In step 1620, the structural features of the object are updated based on the predicted physical state of the object.

[0147] According to one embodiment, step 1620 includes: updating the shape and / or material of the object based on the predicted physical state of the object.

[0148] Fig. Figure 17 shows a block diagram of a device 1700 for generating training data according to an embodiment.

[0149] The device 1700 comprises a shape generation module 1710, a boundary condition module 1720, and a solution module 1730. The shape generation module 1710 generates the basic shape S 115. The boundary condition module 1720 assigns boundary conditions B 125 to the boundary of the basic shape, based on a first partial differential equation (PDE). The solution module 1730 determines the solution u 135 of the first PDE for the basic shape based on the basic shape S 115 and the boundary conditions B 125, wherein the solution expresses the physical state of the object with the basic shape. According to one embodiment, the training data is obtained based on the aforementioned basic shape, the boundary conditions, and the solution.

[0150] According to one embodiment, the basic shape is bounded by a defined area. According to another embodiment, the basic shape is a planar polygon without self-intersections and without holes. According to another embodiment, the planar polygon comprises n vertices and 3 ≤ n ≤ 12.

[0151] According to one embodiment, the boundary conditions comprise several types of boundary conditions. According to one embodiment, the boundary condition module 1720 assigns boundary conditions related to a first PDE to the boundary of the basic shape, using the following modules: a subdivision module that divides the boundary of the basic shape into several parts; and a boundary condition type assignment module that assigns one type of boundary condition from the several types of boundary conditions to each part of the several parts. According to one embodiment, the boundary conditions comprise boundary conditions of the first type and boundary conditions of the second type, and the subdivision module divides the boundary of the basic shape into a first part and a second part; the boundary condition type assignment module assigns boundary conditions of the first type to the first part and boundary conditions of the second type to the second part.According to one embodiment, the boundary condition of the first kind is a Dirichlet boundary condition and the boundary condition of the second kind is a Neumann boundary condition.

[0152] According to one embodiment, the boundary conditions are random values ​​from a defined range.

[0153] According to one embodiment, the device 1700 further comprises a PDE parameter function module which assigns to the basic form a PDE parameter function related to a first PDE, wherein the PDE parameter function comprises coefficients and / or source terms related to the first PDE, wherein the solution module determines the solution of the first PDE for the basic form by means of the following process: Determining the solution of the first PDE for the basic shape based on the basic shape, the boundary conditions, and the PDE parameter function. According to one embodiment, the value of the PDE parameter function is a random value from a defined range.According to one embodiment, the device 1700 comprises a data augmentation module which performs data augmentation based on at least one of the elements basic shape, boundary condition values, PDE parameter function and solution in order to obtain at least one of the elements data-augmented basic shape, data-augmented boundary condition values, data-augmented PDE parameter function and data-augmented solution; wherein the training data is based on at least one of the elements data-augmented basic shape, data-augmented boundary condition values, data-augmented PDE parameter function and data-augmented solution as well as on the aforementioned basic shape, the boundary conditions, the PDE parameter function and the solution.According to one embodiment, the data augmentation module performs data augmentation by executing at least one of the following operations based on at least one of the elements basic shape, boundary condition values, PDE parameter function and solution: performing a spatial translation, a spatial scaling and / or a spatial rotation of the basic shape; performing a value offset and / or a value scaling for at least one of the elements boundary condition values, PDE parameter function and solution.

[0154] According to one embodiment, the solution expresses the physical state of the object with the basic shape over time.

[0155] Fig. Figure 18 shows a block diagram of a device 1800 for training a NN model to determine the physical state of an object with a shape according to an embodiment.

[0156] The device 1800 comprises a neural network model 1810, a loss module 1820, and an update module 1830. Based on the training data generated using the method according to the embodiments of this disclosure, the neural network model 1810 predicts a solution for a basic shape in the training data, wherein the solution expresses the physical state of the object with the basic shape. Based on the solution predicted by the neural network model and the solution in the training data, the loss module 1820 determines the loss. Based on the loss, the update module 1830 updates the learnable parameters of the neural network model.

[0157] Fig. Figure 19 shows a block diagram of a device 1900 for determining the physical state of an object with a shape according to an embodiment.

[0158] The device 1900 comprises a subdivision module 1910, a local solution module 1920, and a global solution module 1930. The subdivision module 1910 subdivides the shape of the object into several sub-shapes, the totality of the sub-areas corresponding to the several sub-shapes covering the entire area of ​​the shape. Based on the global boundary conditions for this shape and these several sub-shapes, the local solution module 1920 uses the neural network (NN) model to determine local solutions for each sub-shape of these several sub-shapes, the local solution for each sub-shape expressing the physical state of the part of the object that possesses that sub-shape. Based on the local solutions for each sub-shape of these several sub-shapes, the global solution module 1930 determines the global solution for this shape, the global solution for this shape expressing the physical state of this object.

[0159] According to one embodiment, the subdivision module 1910 comprises the following modules to subdivide this shape into these several sub-shapes: a non-overlapping subdivision module that subdivides this shape into several non-overlapping sub-shapes in order to form several of these non-overlapping sub-shapes; or an overlapping subdivision module that subdivides this shape into several overlapping sub-shapes in order to form several of these overlapping sub-shapes.According to one embodiment, the overlapping subdividing module comprises the following modules to form these multiple overlapping sub-forms: an intermediate subdivision module that subdivides this form into multiple non-overlapping sub-forms to form multiple non-overlapping intermediate sub-forms; and a form extension module that extends some or all of the intermediate sub-forms of the multiple non-overlapping intermediate sub-forms to form multiple of these overlapping sub-forms.

[0160] According to one embodiment, the local solution module 1920 further determines a local solution for each subform of the several subforms using the following process: furthermore, determining, using this NN model, a local solution for each subform of these several subforms based on the PDE parameter function of the partial differential equation (PDE) corresponding to this NN model, wherein this PDE parameter function includes coefficients and / or source terms.

[0161] According to one embodiment, the local solution module 1920 uses this NN module to determine a local solution for each sub-shape of these multiple sub-shapes and uses the following modules for this purpose: a sub-shape model generation module that obtains sub-shape pre-processing models and sub-shape post-processing models based on these sub-shapes, and a sub-shape pre-processing module that applies the sub-shape pre-processing models to these sub-shapes to obtain pre-processed sub-shapes, and / or a PDE parameter function model generation module that obtains PDE parameter function pre-processing models and PDE parameter function post-processing models based on the PDE parameter function, and a PDE parameter function pre-processing module that applies PDE parameter function pre-processing models to the PDE parameter functions to obtain pre-processed PDE parameter functions;and this NN model determines a local solution for each subform of the multiple subforms based on the preprocessed subforms and / or the preprocessed PDE parameter function, as well as based on the subform post-processing model and / or this PDE parameter function post-processing model.

[0162] According to one embodiment, the local solutions for each sub-shape and the global solution for that shape are determined through several iteration rounds, and the local solution module 1920 uses this NN module to determine a local solution for each of the multiple sub-shapes and comprises the following modules: a local boundary conditions module, which, based on at least one of the elements global boundary conditions and global solution of the previous round for that shape, defines the local boundary conditions for these sub-shapes; and a local solution determination module, which, based on these sub-shapes and the local boundary conditions for these sub-shapes, uses the NN model to determine the local solutions for these sub-shapes.According to one embodiment, the local solution module 1920 determines the local solutions for these subforms based on these subforms and the local boundary conditions for these subforms using the NN model and comprises the following modules: a value model generation module that obtains value preprocessing models and value postprocessing models for these subforms based on the local boundary conditions for these subforms; a value preprocessing module that applies the value preprocessing models to the local boundary conditions to obtain preprocessed local boundary conditions; wherein the NN model determines intermediate values ​​for the local solutions of these subforms based on these subforms and the preprocessed local boundary conditions; and a postprocessing module that applies the value postprocessing models to the intermediate values ​​of the local solutions to determine the local solutions for these subforms.

[0163] According to one embodiment, the local solutions for each sub-shape and the global solution for this shape are determined by several iteration rounds, wherein the global solution module 1930 determines the global solution for this shape based on the local solutions for each of these multiple sub-shapes and comprises the following modules: a boundary and extension model generation module that obtains a boundary and an extension model for each of these multiple sub-shapes based on each of these multiple sub-shapes; a linking module that determines an intermediate value for the global solution of this shape based on the local solution for each sub-shape and the extension model for each sub-shape, or that determines an intermediate value for the global solution of the shape based on the local solution for each sub-shape, the extension model and the boundary model for each sub-shape and the global solution of the previous round for the shape;as well as an update module that obtains the global solution for this shape based on the intermediate value of the global solution for this shape and the global solution of the previous round for this shape.

[0164] According to one embodiment, the solution for this shape expresses the physical state of the object with this shape over time.

[0165] According to one embodiment, the object is at least one of the following: bipolar plate of a fuel cell, motor vehicle part, aircraft part, part of a building, reactor tube and flow deflector, wherein the physical state comprises at least one of the elements velocity, pressure, temperature, electric field and magnetic field.

[0166] According to one embodiment, the shape of the object in the structural data is expressed by at least one of the following elements: circle center, radius, width, height, length, grid cell or geometric shape.

[0167] According to one embodiment, the NN model is trained according to the computer-implemented method described in this text.

[0168] Fig. Figure 20 shows a flowchart of the computer-implemented procedure 2000 for performing a structural optimization of an object according to an exemplary embodiment.

[0169] The device 2000 comprises a prediction module 2010 and an update module 2020. The prediction module 2010 uses the method according to the embodiments of this disclosure to predict the physical state of the object. The update module 2020 updates the structural features of the object based on the predicted physical state of the object.

[0170] According to one embodiment, the update module 2020 updates the shape and / or material of the object based on the predicted physical state of the object.

[0171] Fig. Figure 21 shows a block diagram of a processing device 2100 according to an exemplary embodiment.

[0172] The processing device or processing system 2100 comprises one or more control units or processing units 2110 that execute one or more machine-readable instructions stored or encoded on a machine-readable storage medium (i.e., memory 2120). In one embodiment, the processing unit 2110 is configured to execute the various commands described above in conjunction with the program instructions. Fig. to perform the processes and functions described in points 1 to 9.

[0173] Although in Fig. 1, Fig. 2C, Fig. 3C, Fig. 5, Fig. 6A, Fig. 7C to Fig. 9 and Fig. 17 to Fig. Not shown in Figure 21, it is understandable to those skilled in the field that the device described in the embodiments of the present disclosure may contain a variety of further components, such as various communication modules, bus modules and possibly user interface modules.

[0174] According to one embodiment, a program product, such as a non-volatile machine-readable medium, is provided. The non-volatile machine-readable medium can contain instructions which, when executed by the 2110 processing unit, can perform the various operations and functions described above in conjunction with Fig. 1 to Fig. 20 in various embodiments described in the present disclosure.

[0175] According to one embodiment, a computer program product is provided. The computer program product comprises instructions, the instructions being that, when executed by the 2110 processing unit, can perform the various operations and functions described above in connection with Fig. 1 to Fig. 20 in various embodiments described in the present disclosure.

[0176] The specific embodiments described above in conjunction with the figures represent exemplary embodiments and do not constitute all embodiments that can be realized or that fall within the scope of the claims. The term "exemplary," used throughout this description, means "serving as an example or for illustration" and does not mean "preferred" or "advantageous" over other embodiments. To facilitate an understanding of the described technologies, the specific embodiments include specific details. However, these technologies can be implemented without these specific details. In some cases, known structures and devices are shown in the form of block diagrams to avoid making the concept of the described embodiments difficult to understand.

[0177] The foregoing description of the present application enables persons skilled in the art to implement or apply the content of the present application. The various amendments to the present application are obvious to persons skilled in the art, and the general principles defined herein can also be applied to other variants without deviating from the scope of the content of the present application. Therefore, the present application is not limited to the examples and embodiments described herein, but is consistent with the broadest scope of the principles and novel features disclosed herein.

Claims

[1] Computer-implemented method for determining the physical state of an object with a shape, comprising: Dividing the shape of the object into several sub-shapes, whereby the totality of the several sub-areas corresponding to the several sub-shapes covers the entire area of ​​the shape; Determine, using the neural network (NN) model, local solutions for each subshape of the multiple subshapes based on the global boundary conditions for the shape and the multiple subshapes, wherein the local solution for each subshape expresses the physical state of the part of the object that possesses that subshape; and Determining the global solution for the shape based on the local solutions for each sub-shape of the multiple sub-shapes, where the global solution for the shape expresses the physical state of the object. [2] Computer-implemented method according to claim 1, wherein the mold is divided into several sub-molds, comprising: Dividing the shape into several non-overlapping sub-shapes in order to create several non-overlapping sub-shapes; or Dividing the shape into several overlapping sub-shapes in order to create several overlapping sub-shapes. [3] Computer-implemented method according to claim 2, wherein several overlapping sub-forms are formed, comprising: Dividing the shape into several non-overlapping sub-shapes in order to create several non-overlapping intermediate sub-shapes; Extending a part or the entirety of the intermediate forms of several non-overlapping intermediate forms in order to create several overlapping intermediate forms. [4] Computer-implemented method according to claim 1, wherein the local solutions for each sub-shape of the multiple sub-shapes are determined using the NN model, further comprising: based on the PDE parameter function of the partial differential equation (PDE) corresponding to the NN model, using the NN model to determine a local solution for each subform of these several subforms, where the PDE parameter function includes coefficients and / or source terms. [5] Computer-implemented method according to claim 4, wherein a local solution for each sub-shape of these multiple sub-shapes is determined using the NN model, comprising: Obtaining the sub-shape pre-processing models and the sub-shape post-processing models based on these sub-shapes and applying the sub-shape pre-processing models to these sub-shapes to obtain pre-processed sub-shapes, and / or Obtaining PDE parameter function preprocessing models and PDE parameter function postprocessing models based on the PDE parameter function and applying the PDE parameter function preprocessing models to the PDE parameter functions to obtain preprocessed PDE parameter functions; and Determining a local solution for each sub-shape of the multiple sub-shapes using the NN model based on the pre-processed sub-shapes and / or the pre-processed PDE parameter function, as well as based on the sub-shape post-processing model and / or this PDE parameter function post-processing model. [6] Computer-implemented method according to claim 1, wherein the local solutions for each sub-shape and the global solution for this shape are determined by several iteration rounds, and wherein the local solutions for each sub-shape of the multiple sub-shapes are determined using the NN model, comprising performing the following processes for each sub-shape of the multiple sub-shapes: Defining the local boundary conditions for this sub-shade based on at least one of the elements from the global boundary conditions and the global solution of the previous round for the shape; and Determining the local solution for this sub-shape based on this sub-shape and the local boundary conditions for this sub-shape using the NN model. [7] Computer-implemented method according to claim 6, wherein, based on this subform and the local boundary conditions for this subform, the local solution for this subform is determined using the NN model, further comprising: Obtaining the value preprocessing model and the value postprocessing model for this subform based on the local boundary conditions for this subform; Applying the value preprocessing model to the local boundary conditions to obtain preprocessed local boundary conditions; Determining the intermediate value for the local solution of this subform based on this subform and the preprocessed local boundary conditions using the NN model; and Applying the value post-processing operator to the intermediate value of the local solution to determine the local solution for the subform. [8] Computer-implemented method according to claim 1, wherein the local solutions for each sub-shape and the global solution for this shape are determined by several iteration rounds, and wherein the global solution for the shape is determined based on the local solutions for each sub-shape of the several sub-shapes, comprising: Obtaining an extension model for each subform based on each subform of the multiple subforms; Determining an intermediate value for the global solution of this shape based on the local solution for each sub-shape and the extension model for each sub-shape; and Obtaining the global solution for this shape based on the intermediate value of the global solution for this shape and the global solution of the previous round for this shape. [9] Computer-implemented method according to claim 1, wherein the local solutions for each sub-shape and the global solution for this shape are determined by several iteration rounds, and wherein, based on the local solutions for each sub-shape of the several sub-shapes, the global solution for the shape is determined, comprising: Obtaining a boundary model and an extension model for each subform based on each subform of the multiple subforms; Determining an intermediate value for the global solution of this shape based on the local solution for each sub-shape, the extension model and the boundary model for each sub-shape, and the global solution of the previous round for the shape; and Obtaining the global solution for this shape based on the intermediate value of the global solution for this shape and the global solution of the previous round for this shape. [10] Computer-implemented method according to claim 1, wherein the global solution for this shape expresses the physical state of the object with this shape over time. [11] Computer-implemented method according to claim 1, wherein the object is at least one of the objects bipolar plate of a fuel cell, motor vehicle part, aircraft part, part of a building, reactor tube and flow deflector, and wherein the physical state comprises at least one of the elements velocity, pressure, temperature, electric field and magnetic field. [12] Computer-implemented method according to claim 1, wherein the shape of the object is expressed by at least one of the following elements: Circle center, radius, width, height, length, grid cell or geometric shape. [13] Computer-implemented method according to claim 1, wherein the NN model is trained by the following steps: Predictions using the NN model of a solution for a basic shape in the training data, where the solution expresses the physical state of the object with the basic shape; Determining the loss based on the solution predicted by the NN model and the solution in the training data; and Updating the learnable parameters of the NN model based on the loss. [14] Computer-implemented method according to claim 13, wherein the training data are generated by the following steps: Generating a basic form; Assigning boundary conditions related to a first partial differential equation (PDE) to the boundary of the basic form; Determining the solution of the first PDE for the basic form based on the basic form and the boundary conditions, where the solution expresses the physical state of the object with the basic form; and The training data is obtained based at least partially on the basic form, the boundary conditions and the solution. [15] Computer-implemented method for performing a structural optimization of an object, comprising: Predictions of the physical state of the object using one of the methods described in claims 1 to 14; Updating the structural features of the object based on the predicted physical state of the object. [16] Device for determining the physical state of an object with a shape, comprising: a subdivision module that divides the shape of the object into several sub-shapes, wherein the totality of the several sub-areas corresponding to the several sub-shapes covers the complete area of ​​the shape; a local solution module that, based on the global boundary conditions for the shape and the multiple sub-shapes, uses the neural network (NN) model to determine a local solution for each sub-shape of the multiple sub-shapes, wherein the local solution for each sub-shape expresses the physical state of the part of the object that possesses the sub-shape; and a global solution module that, based on the local solutions for each sub-shape of the multiple sub-shapes, obtains the global solution for the shape, where the global solution for the shape expresses the physical state of the object. [17] Device for performing a structural optimization of an object, comprising: a prediction module that predicts the physical state of the object using one of the methods described in claims 1 to 14; an update module that updates the structural features of the object based on the predicted physical state of the object. [18] Processing device comprising the following: one or more processors; and one or more memories, wherein computer-executable instructions are stored in the memory and the instructions, when executed by the one or more processors, perform an operation according to one of claims 1 to 15. [19] Machine-readable storage medium on which executable instructions are stored, wherein the instructions, when executed, cause one or more processors to execute a method according to any one of claims 1 to 15. [20] Computer program product comprising executable instructions, wherein the instructions, when executed, cause one or more processors to execute a method according to any one of claims 1 to 15.

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