Method and apparatus for structure optimization

Abstract

The present disclosure provides a computer-implemented method for optimizing a structure of an object. The method includes receiving structural parameters by a neural boundary operator (NBO), where the structural parameters are used to describe a boundary shape of the object, and the NBO is trained to map the structural parameters of the boundary shape of the object to an approximation of a solution of a partial differential equation (PDE) formulated to characterize a physical system associated with the object; generating, by the NBO, a state variable from the structural parameter, where the state variable corresponds to a solution of the PDE and is used to describe a physical state of a physical system related to the object; calculating an optimization target value based on the state variable; and updating the structure parameter based on the optimization target value.

Description

Technical Field

[0001] Generally speaking, aspects of the present disclosure relate to artificial intelligence (AI), and more specifically, aspects of the present disclosure relate to optimizing the structure of an object. Background Art

[0002] The structure of an object (such as its shape, size or material distribution) affects the performance of the object. Examples of objects can be the wing of an airplane, the trusses of a house, the pipes in a reactor, etc. For example, the shape of the wing affects the pressure and velocity of the airflow relative to the wing, and the pressure and velocity affect the performance of the physical system of the wing. Physical systems can be described by partial differential equations (PDEs). The state variables of the physical system (such as pressure and velocity) can be obtained by solving the PDE based on structural parameters (such as structural parameters describing the shape of the object).

[0003] It is desirable to design the structure of an object in order to improve its performance, and this process can be referred to as structural optimization. Structural optimization can be applied in many fields, including science, engineering, industry, etc. For example, the shape of chemical catalyst particles, the shape of automobile parts, etc. can be optimized through the structural optimization process before manufacturing.

[0004] Since most physical systems are described by partial differential equations (PDEs), structural optimization is usually performed using governing PDEs. This can be time consuming since the PDEs need to be solved at each iteration of the optimization. Summary of the invention

[0005] In order to improve the efficiency of structural optimization, the present disclosure proposes a novel framework for structural optimization, through which the time consumption and computational requirements can be reduced while ensuring the optimization performance.

[0006] According to an embodiment, a computer-implemented method for optimizing the structure of an object is provided. The method includes: receiving structural parameters by a neural boundary operator (NBO), wherein the structural parameters are used to describe the boundary shape of the object, and the NBO is trained to map the structural parameters of the boundary shape of the object to an approximation of a solution of a partial differential equation (PDE), the PDE being formulated to characterize a physical system associated with the object; generating state variables from the structural parameters by the NBO using forward transfer, wherein the state variables correspond to the solution of the PDE and are used to describe the physical state of the physical system associated with the object; calculating an optimization target value based on the state variables; and updating the structural parameters of the object based on the optimization target value.

[0007] According to an embodiment, a method for collecting a training data set for training a neural boundary operator (NBO) is provided, wherein the NBO is trained to map structural parameters of a boundary shape of an object to an approximation of a solution of a partial differential equation (PDE), the PDE being formulated to characterize a physical system associated with the object, wherein the solution of the PDE corresponds to state variables used to describe a physical state of the physical system associated with the object. The method includes: sampling a first number of sets of structural parameters corresponding to a first number of boundary shapes of the object according to a first distribution of the structural parameters of the object, wherein the first distribution is a uniform distribution; solving the PDE of the object based on the first number of sets of structural parameters by using a numerical solver to generate a first number of sets of state variables; sampling a second number of sets of structural parameters corresponding to a second number of boundary shapes of the object according to a second distribution of the structural parameters of the object, wherein the second distribution is determined based on the first number of sets of structural parameters; and solving the PDE of the object based on the second number of sets of structural parameters by using the numerical solver to generate a second number of sets of state variables, wherein the first number of sets of structural parameters, the corresponding first number of sets of state variables, the second number of sets of structural parameters, and the corresponding second number of sets of state variables are included in the training data set.

[0008] According to an embodiment, a computer system is provided, which includes one or more processors and one or more storage devices storing computer-executable instructions, which, when executed, cause the one or more processors to perform the operations of the method described above and to perform the operations of the method according to various aspects of the present disclosure.

[0009] According to an embodiment, one or more computer-readable storage media storing computer-executable instructions are provided, which, when executed, cause one or more processors to perform the operations of the methods described above and to perform the operations of the methods according to various aspects of the present disclosure.

[0010] According to an embodiment, there is provided a computer program product comprising computer executable instructions which, when executed, cause one or more processors to perform the operations of the method as described above and to perform the operations of the method according to aspects of the present disclosure.

[0011] By using NBO (which learns the mapping between the structural parameters of various boundary shapes of the object and the solution of the PDE), in the optimization framework, the efficiency of structural optimization can be significantly improved while solving the bottleneck of solving the PDE. In addition, by using an adaptive sampling framework to collect the training data set for training NBO, the training cost of NBO is reduced by exploiting the trade-off between the exploitation of the prior of the structure of the object and the exploration of the unknown structure. Other advantages and enhancements are explained in the description below. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] The disclosed aspects will hereinafter be described in conjunction with the accompanying drawings, which are provided to illustrate and not to limit the disclosed aspects.

[0013] Figure 1 An exemplary framework for optimizing the structure of an object according to aspects of the present disclosure is shown.

[0014] Figure 2A The architecture of an NBO according to aspects of the present disclosure is shown.

[0015] Figure 2B The architecture of an NBO according to aspects of the present disclosure is shown.

[0016] Figure 3 Exemplary structural optimizations of flow baffles according to aspects of the present disclosure are shown.

[0017] Figure 4 Exemplary structural optimization of a conductor-dielectric interface according to aspects of the present disclosure is shown.

[0018] Figure 5 An exemplary process for optimizing the structure of an object according to aspects of the present disclosure is shown.

[0019] Figure 6 An exemplary process for collecting a training data set for training an NBO in accordance with aspects of the present disclosure is shown.

[0020] Figure 7 An exemplary computing system in accordance with aspects of the present disclosure is shown. DETAILED DESCRIPTION

[0021] The present disclosure will now be discussed with reference to several example implementations. It will be understood that these implementations are discussed only to enable those skilled in the art to better understand and thus implement embodiments of the present disclosure, and do not imply any limitation on the scope of the present disclosure.

[0022] Various embodiments will be described in detail with reference to the accompanying drawings. Wherever possible, the same reference numerals will be used throughout the drawings to refer to the same or similar parts. Reference to specific examples and embodiments is for illustrative purposes and is not intended to limit the scope of the present disclosure.

[0023] Figure 1 An exemplary framework for optimizing the structure of an object according to aspects of the present disclosure is shown.

[0024] exist Figure 1 In the example shown, the object to be structurally optimized is the wing. Figure 1 As shown in block 120 of , the shape of the wing is given by a boundary γ, which is parameterized by a structural parameter θ. In this example, the structural parameter θ may be a set of control points consisting of a spline representing the shape of the wing. In other words, the structure to be optimized is the shape of the wing represented by a spline with a set of control points θ. The letter Ω represents the problem domain, or in particular, the domain of the physical system associated with the wing.

[0025] like Figure 1 As shown in the box 110 of FIG. 1 , the arrow line represents the airflow over the wing. The physical state of the airflow over the wing can be represented by the state variables of the physical system associated with the wing. In this example, the state variables can be the velocity u=u(x) and the pressure p=p(x). The letter x represents the spatial coordinate in the domain Ω.

[0026] The velocity u=u(x) and the pressure p=p(x) can be characterized by nonlinear PDEs of the Navier–Stokes (NS) equations, where the constraints on the shape of the wing are represented by the boundaries γ. It should be understood that the NS equations are well-known physical equations that can be used to describe the three-dimensional motion of viscous fluid materials. The NS equations are second-order nonlinear PDEs and can be used to model weather, ocean currents, heat conduction, airflow around wings, water flow in pipes or reactors, and many other applications. The PDEs of the NS equations for the physical system of an object can be formulated according to the physical laws associated with the physical system of the object. In the example shown, the PDEs for the physical system of the wing can be established according to the relevant physical laws, such as Figure 1 140. Since the PDEs are used to manage the optimization of the structure of the wing, they can be referred to as management PDEs. It should be understood that the formulation of the management PDEs can be implemented using well-known physical knowledge, and in various aspects of the present disclosure, other types of PDEs used to describe the physical system of an object can also be used as management PDEs.

[0027] like Figure 1 As shown in the reference numeral 130, the goal or purpose of the structural optimization of the wing is to achieve a desired pressure distribution p on the wing surface by changing the structural parameter θ.ref As indicated by reference numeral 130, an objective function J may be formulated, and the goal of the structural optimization is to minimize the objective function J, which is a function of the pressure p. The shape of the wing corresponding to the boundary γ parameterized by the structural parameter θ may be iteratively optimized to minimize the objective function J so as to achieve a desired pressure distribution p on the wing surface. ref .

[0028] In each cycle of structural optimization, the PDE 140 needs to be solved to obtain physical state variables such as velocity u and pressure p, and then the objective function J can be evaluated based on the physical state variables such as pressure p. Then, the structural parameter θ can be updated or optimized based on the calculated objective function J, as shown by reference numeral 160.

[0029] In the example, existing numerical methods such as the finite element method (FEM) and the finite volume method (FVM) can be used to obtain the velocity u and the pressure p by solving the PDE 140. For example, FEM is often used to model the behavior of the structure because they are accurate and stable for a wide range of PDEs including some complex nonlinear PDEs (e.g., NS equations). Despite its accuracy and stability, FEM solvers are hampered by large computational overhead. For each instance of the PDE, the FEM solver needs to generate a mesh of the structure of the object and solve a large set of equations from scratch, which is very slow for high-dimensional problems. For example, even for a medium-sized system, such as a system represented by a 50×50 two-dimensional (2D) grid, numerical methods typically require several seconds to obtain a solution to the PDE. Since the solution of the PDE is extremely sensitive to boundary conditions, and such methods lack the ability to generalize to various boundary shapes, they need to independently solve the PDE millions of times during the optimization process, resulting in unbearable computational overhead and corresponding time consumption. Using existing numerical PDE solvers such as FEM solvers and FVM solvers in the structural optimization framework 100 will result in inefficiency in the structural optimization of the object.

[0030] In an embodiment, Figure 1 As shown, a neural boundary operator (NBO) 150 is employed in the structural optimization framework 100 to solve the PDE 140 with various boundary shapes represented by the structural parameters θ. The NBO can be implemented using a deep learning architecture that is used to learn operators from the structural parameters of the various boundary shapes to the solutions of the PDE. In the example shown, the NBO 150 has learned the latent operator It can be generalized to the space of structural parameters θ, in other words, NBO has learned the mapping between the structural parameters of the boundary shape of the object and the solution of the PDE. Since NBO can be effectively generalized across variable boundary shapes, the overall structural optimization is an iterative process, where during each iteration, the PDE can be efficiently solved via the forward pass of NBO instead of calling a costly FEM solver. Therefore, by using NBO 150, the solution process of the PDE can be accelerated by orders of magnitude compared to existing PDE solvers, making the structural optimization framework 100 much more efficient.

[0031] Figure 2A and Figure 2B Each shows the architecture of an NBO according to aspects of the present disclosure.

[0032] The structural optimization problem can be expressed by the following equation (1): Wherein, θ represents the structural parameter of the structure of the object to be optimized, which is also the parameter of the boundary shape, and θ represents the space of θ. The structural parameter θ changes during the optimization of the structure, but within the design space θ. J is the objective function, whose value is used to measure how good a given structure is, and W represents the constraints on both the structural parameter θ and the state variable s=s(x), which represents the state of the physical system, such as velocity, pressure, etc., where x represents the spatial coordinate. And the symbol "≤°" is an element-by-element comparison.

[0033] Since the state variable s is associated with the physical system of the object to be structurally optimized, it is characterized by a PDE as shown in the following equation (2): Where F represents the vector-valued operator used to represent PDE, B i represents the vector-valued operator used to represent the equation for the ith boundary condition, γ i represents the domain on which the i-th boundary condition is defined, Ω represents the domain of the physical system, and the boundary γ i The shape of (θ) is parameterized by the structural parameter θ, and all boundaries γ i (θ) together correspond to the structure to be optimized. For any given structural parameter θ, it can be assumed that the PDE has a unique solution s. Therefore, the operator It can be defined as G(θ)(x)=s(x).

[0034] As an example, we can consider the problem of minimizing the drag of a 2D fluid, as shown in equation (3): where v is the viscosity of the fluid, ||·|| Fis the Frobenius norm. In this example, the physical system consists of a 2D channel Ω with a fluid and an elliptical baffle. In addition, the structural parameters θ=(θ1,θ2)∈Θ=[0,1] 2 , where θ1 and θ2 correspond to the width and height of the elliptical baffle, respectively. The state variables s = (p, u), where p = p(x) and u = u(x) are the pressure and velocity of the fluid, respectively. The structural optimization goal is to minimize the energy dissipated by the fluid by changing the structural parameter θ (i.e., the shape of the elliptical baffle), where the constraint is that the area of ​​the elliptical baffle is 1. And the physical state variables s = (p, u) are characterized by the nonlinear PDE of the NS equation (4): Where Ω is the problem domain (ie, a 2D channel), γ1 and γ2 are the inlet and outlet of the channel, respectively, and γ3 is the geometry of the elliptical baffle. It should be understood that the PDE (4) describing the physical system can be formulated according to the physical laws of the physical system.

[0035] In equation (4), the boundary γ3(θ1,θ2) is the structure to be optimized. Since the structure changes during the optimization process, equation (4) corresponds to a PDE with various boundary shapes. In addition, in equation (3), the objective function J depends on the physical state variable u defined by equation (4). The physical state variable u can be obtained by solving PDE (4). In the structural optimization process, the PDE needs to be solved simultaneously with the optimization of the structure. However, previous methods are too time-consuming for application because they cannot be generalized to PDEs with various boundary shapes and the PDE must be solved repeatedly from scratch.

[0036] NBO can be used to address the above challenges by generalizing to PDEs with various boundary shapes and mapping the structural parameters θ to the solutions of the PDEs. Approximate Neural Network NBO takes as input the coordinates x and the parameters θ. The output is an approximation to the solution of the PDE at x, i.e.,

[0037] exist Figure 2A In the illustrated embodiment, the NBO 200A includes a branch sub-network 210A and a trunk sub-network 220A. The branch sub-network 210A takes the structural parameter θ as its input and outputs n p-dimensional vectors And the backbone subnetwork 220A takes the coordinate x as its input and outputs n p-dimensional vectors Where n is equal to the dimension of the state variable s, and p is a hyperparameter. As shown at 230A, n vectors are and n vectors The corresponding dot product between is the prediction of the solution of the PDE The branch sub-network 210A and the trunk sub-network 220A may be implemented as a fully connected network (FCN).

[0038] exist Figure 2B In the illustrated embodiment, NBO 200B includes a branch subnetwork 210B, a trunk subnetwork 220B, and a port subnetwork 240B. Branch subnetwork 210B takes structural parameters θ as its input and outputs n p-dimensional vectors And the backbone subnetwork 220B takes the coordinate x as its input and outputs n p-dimensional vectors Where n is equal to the dimension of the state variable s, and p is a hyperparameter. As shown at 230B, n vectors are and n vectors concatenated, and then input into the port subnetwork 240, which outputs the prediction of the solution of the PDE The port subnetwork 240B is a micro-neural network. By utilizing the effective nonlinear approximation capability of the port subnetwork 240B, the abstraction capability of the NBO 200B for high-level features can be further improved compared to the NBO 200A. The branch subnetwork 210B, the trunk subnetwork 220B, and the port subnetwork 240B can be implemented as a fully connected network (FCN).

[0039] In order to train the latent operator Approximate Neural Network The NBO of the structure parameter is first sampled, that is, first according to the customized distribution in the space Θ×Ω To {(θ i ,x ij )|1≤i≤N,1≤j≤M}. For ease of description and understanding, an example of structural parameters θ i can be called the set of structural parameters θ i The corresponding PDE is then solved using a numerical solver such as a FEM solver to obtain the solution {G(θ i )(x ij )}. In this way, the dataset {(θ i ,x ij ,G(θ i )(x ij ))} can be constructed as training and testing samples.

[0040] The NBO can be trained using the training data set. The training objective for the NBO can be adopted as shown in the following equation (5): where w is the neural network weight, i.e., the parameters of the NBO, and is the space of neural network weights, G(θ)(x) is the true value and is the prediction of the NBO model. Instead of approximating the solution to a particular instance of the PDE, training data is sampled in the structural parameter space Θ and the NBO is trained with these samples to learn the underlying operator In this way, for new structures, the generalization power of NBO can be exploited to transfer to the solution of the corresponding PDE without solving it from scratch.

[0041] Since NBO is a key part in the overall structure optimization framework, the distribution of training data Can significantly affect the performance of structural optimization. In an embodiment, is a uniform distribution. However, uniform sampling is inefficient because it does not focus on those structures with good performance. Therefore, in another embodiment, the instances of the structural parameters of the training data are sampled according to a customized distribution so as to sample those structures with good performance. The following equation (6) shows the density function p(θ,x)=p(θ)p(x|θ) (6)

[0042] Since the structure optimizer is more likely to stay at locations with lower values ​​of the objective function J (which means better structures), it is advantageous to sample those points with lower values ​​of J. The target J(θ,s) (abbreviated as J(θ) below) can be selected as a prior to guide the NBO to fit better around potential minima of the objective function.

[0043] It is possible to balance exploitation (empirical prior) and exploration (space filling). In order to achieve a balance between exploitation and exploration, the distribution p(θ) can be defined as shown in the following equation (7): in is the non-normalized density function, is the balance factor, S t is the θ that has been sampled so far i is a set of , and ||·||2 is the l2 norm.

[0044] Since the objective function J(θ) of structural optimization does not have an analytical expression from θ to J, an estimator of the objective function J(θ) can be used in an embodiment. First, a small set of structural instances {θ i} is sampled, and the sampling distribution can be a uniform distribution. Then, the FEM solver can be used to obtain the corresponding solution of the PDE {G(θ i )=s i}, which is used to evaluate the corresponding objective function {J(θi ,s i )}. Then, the dataset {θ i ,J(θ i ,s i )} can be used as an estimator In this example, the estimator Can be implemented as a linear model Where w represents the estimator It should be understood that in another example, the estimator can be implemented as a nonlinear model. Assume that the structural parameter space Θ is bounded (i.e., ), can be in [-d,d] q Definition And normalize it to obtain p(θ), as shown in the following equation (8):

[0045] The normalization coefficient can be given The expression of is shown in the following equation (9): in And erfi is the imaginary error function.

[0046] In an embodiment, the distribution p(θ) may be determined based on equation (7). In another embodiment, the distribution p(θ) may be determined based on equation (8) or based on equations (8) and (9). In another embodiment, the distribution p(θ) may be determined based on equation (7) by taking The distribution p(θ) is determined as p(θ). It should be understood that there may be instances of structural parameters θ for achieving the structure defined in equation (7) i Other ways to customize sampling.

[0047] For p(x|θ), it can be determined using the well-known adaptive mesh refinement (AMR) technique. AMR uses a set of refinement criteria to generate and actively adapt the mesh in each region of the structure based on the local field, slope, or curvature of the variable, which can improve node utilization and thus reduce computational cost. In an embodiment, given a structure instance θ i , the AMR technique is used to generate the mesh of the structure, and the nodes of the mesh are set to the corresponding coordinate samples

[0048] Compared to the limitations of existing methods that have high computational cost and cannot be generalized to PDEs with various boundary shapes, NBO can be generalized to various boundary shapes, and therefore the forward pass can be used to estimate the solution of the PDE in each iteration of the structural optimization, thereby avoiding the expensive solution process of the PDE. As described above, the overall method includes two stages. The first stage is to collect the data set and train the NBO. The second stage is to use the NBO in the structural optimization to estimate the solution of the PDE. In the second stage, it should be noted that the NBO (which is a NN model) may be required to predict multiple points to evaluate the objective function J, especially when it has an integral term as shown in, for example, equation (3), which provides an opportunity for GPU parallel acceleration.

[0049] Figure 3 is a schematic diagram illustrating structural optimization of a flow baffle according to aspects of the present disclosure.

[0050] like Figure 3 As shown in FIG. 1 , γ2 to γ5 located in the corresponding grid are the flow baffles to be optimized. The purpose of structural optimization is to optimize the shape and position of the baffles γ2 to γ5 in order to obtain the flow baffles at the outlet γ 1,右 A uniform distribution of fluid flow at a location where the energy dissipated by the fluid is small.

[0051] The objective function of structural optimization can be formulated based on fluid mechanics as shown in the following equation (10): Where β is the balance factor, for example, β=0.01, u=(u1,u2) is the velocity, and v is the viscosity, for example, v=1.

[0052] The governing PDE describing the physical system associated with the flow baffle can be formulated according to the laws of physics. For example, the PDE can be a NS equation as shown in the following equation (11): Where x = (x1, x2) is the spatial coordinate, p is the pressure, u = (u1, u2) is the velocity, and v is the viscosity. For example, v = 1, γ 1,左 is the left part of γ1, γ 1,左上 Yes 1,左 The upper part, γ 1,左下 Yes 1,左 The lower part, γ 壁 =γ 1,上 ∪γ 1,下 ∪γ2∪γ3∪γ4∪γ5∪γ 1,左下 , and u1(x) is

[0053] The structural parameters include the centers and radii of the circular baffles γ2 to γ5, i.e., θ = (x0, r0, ..., x3, r3). ​​Constraints W(θ) can be defined to restrict each of γ2 to γ5 within its mesh, such as Figure 3 In an embodiment, in order to make the sampling process efficient, a mapping of structural parameters may be performed: in And minDist is the shortest distance from the i-th circle to the edge of the corresponding grid, such as Figure 3 Therefore, the allowed values ​​of each dimension of θ′ are in a closed interval, and the allowed parameters can be normalized to [3,3] 12 , where the superscripts refer to the 12 dimensions of the structural parameters. Let the final parameter be θ″, W(θ) is defined as:

[0054] The training of the NBO model is then performed on this mapped space, which is a normalized structure space of the original structure space.

[0055] In an embodiment, the Adam algorithm can be used to optimize the structure of the baffle and the reverse calculation of the derivative, where the NBO provides the forward prediction of the PDE. In the example, the number of iterations for optimizing the structure θ is 200 and the learning rate is 0.1. The loss function is defined as where λ is a predefined coefficient, such as λ=100, ||·|| absSum Represents the sum of the absolute values ​​of all elements of a vector.

[0056] In an embodiment, a training data set is generated for training the NBO model. NBO may have Figure 2A or Figure 2B It should be understood that NBO may also have other structures.

[0057] Assume that the data size of the dataset is N, for example, N = 5000, which means that the dataset includes 5000 structure instances. First, a small subset of structure instances can be distributed according to a uniform distribution. Sampling is performed, where N1<<N, for example, N1=100. A well-known FEM solver is used to generate the corresponding mesh of the structure instance and the values ​​of the state variables {G(θ i )(x ij )|1≤i≤N1,1≤j≤M}, where the coordinate x ij corresponds to the nodes of the grid, M is the number of nodes of the grid, and M can be related to the specific structure θ i The relevant variable values. And according to equation (10) to obtain the value of the objective function Can be based on To train the estimator of the objective function (10) It can be obtained from the distribution defined in equation (7) (where J is replaced by ) for the remaining structure instances For example, the Markov Chain Monte Carlo (MCMC) method can be used to sample the remaining 4900 structural instances according to the distribution defined in equation (7). Then, a FEM solver is used to generate the corresponding meshes of the structural instances and the values ​​of the state variables {G(θ i )(x ij )|N1+1≤i≤N,1≤j≤M}. And obtain the training data set for training NBO, which is {(θ i ,x ij ,G(θ i )(x ij ))|1≤i≤N,1≤j≤M}. In an embodiment, the NBO may be trained based on the training objective function (5).

[0058] At each iteration of the structural optimization, the trained NBO is used to estimate the state variables u and optionally p, from which the target J(θ) is obtained according to Equation (10). Then, the loss is obtained as shown in Equation (13): And the loss can be obtained accordingly The derivative of θ. Finally, the loss in the iteration is optimized based on this structure. Or its derivative to update the structure parameter θ. In an embodiment, the Adam algorithm can be used to update the structure θ, aiming to minimize J shown in equation (10).

[0059] Figure 4 is a schematic diagram illustrating structural optimization of a conductor-dielectric interface according to aspects of the present disclosure.

[0060] Figure 4 The conductor-dielectric interface shown can be an infinitely long conductor-dielectric interface with a thickness of H = 8 and a period of L = 8. Ω1 and Ω2 are two conductor media with different thermal conductivities, governed by the PDE shown in the following equation (14): Where x = (x1, x2) is the spatial coordinate, k1 and k2 are the thermal conductivities of Ω1 and Ω2, respectively, for example, k1 = 1 and k2 = 2, y i is Ω i The temperature distribution of , i = 1, 2, and and is the unit normal of the Neumann boundary condition. The structure to be optimized is the conductor-dielectric interface represented by the spline γ SThe leftmost and rightmost anchor points of the spline are fixed (in Figure 4 The four middle anchor points are variable (marked as dots in Figure 4 ), but in the corresponding rectangular grid (in Figure 4 Given the positions of the four anchor points, the spline can be obtained by interpolation. Therefore, the four anchor points are set as structural parameters, i.e., θ = (x0, x1, x2, x3).

[0061] The goal of structural optimization is to change the structural parameter θ so that S The heat transfer is maximized. The goal of structural optimization can be described as shown in the following equation (15): Then, we can take the objective function J as 1 / h T (y) to transform the objective into a minimization problem, as shown in the following equation (16): J(θ)=1 / h T (t) (16)

[0062] A constraint W(θ) can be defined to restrict each anchor point to its grid, such as Figure 4 In an embodiment, the allowed parameter can be normalized to [-3, 3] 8 , where the superscripts refer to the 8 dimensions of the structural parameters. Let the final parameter be θ″, W(θ) is defined as:

[0063] Then, the training of the NBO model is performed on this mapped space, which is the normalized structure space of the original structure space.

[0064] In an embodiment, the Adam algorithm can be used to optimize the structure of the interface and the reverse calculation of the derivative, where NBO provides the forward prediction of the PDE. In the example, the number of iterations for optimizing the structure θ is 200 and the learning rate is 0.1. The loss function is defined as l(θ)=J(θ)+λ||relu(W(θ))|| absSum (18) where λ is a predefined coefficient, such as λ=100, ||·|| absSum Represents the sum of the absolute values ​​of all elements of a vector.

[0065] In an embodiment, a training data set is generated for training the NBO model. NBO may have Figure 2A or Figure 2B It should be understood that NBO may also have other structures.

[0066] Assume that the data size of the data set is N, for example, N = 2000, which means that the data set includes 2000 structure instances. First, a small subset of structure instances can be distributed according to a uniform distribution. Sampling is performed, where N1<<N, for example, N1=100. A well-known FEM solver with AMR function is used to generate the corresponding mesh of the structure instance and the values ​​of the state variables {G(θ i )(x ij )|1≤i≤N1,1≤j≤M}, where the coordinate x ij corresponds to the nodes of the grid, M is the number of nodes of the grid, and M can be related to the specific structure θ i The relevant variable values. And according to equation (16) to obtain the value of the objective function Can be based on To train the estimator of the objective function (16) It can be obtained from the distribution defined in equation (7) (where J is replaced by ) for the remaining structure instances For example, the Markov Chain Monte Carlo (MCMC) method can be used to sample the remaining 1900 structural instances according to the distribution defined in equation (7). Then, a FEM solver is used to generate the corresponding mesh of the structural instance and the values ​​of the state variables {G(θ i )(x ij )|N1+1≤i≤N,1≤j≤M}. And obtain the training data set for training NBO, which is {(θ i ,x ij ,G(θ i )(x ij ))|1≤i≤N,1≤j≤M}. In an embodiment, the NBO may be trained based on the training objective function (5).

[0067] In each iteration of structural optimization, the trained NBO is used to estimate the state variable y i , from which the target J(θ) is obtained according to equation (16). Then, the loss is obtained as shown in equation (18) And the loss can be obtained accordingly The derivative with respect to θ. Based on this structure, the loss in the optimization iteration is Or its derivative to update the structure parameter θ. In an embodiment, the Adam algorithm can be used to update the structure θ, aiming to minimize J shown in equation (16).

[0068] It should be understood that except for Figure 3 and Figure 4In addition to the specific objects exemplified in the examples, the structural optimization framework proposed in the present disclosure is also applicable to optimizing the structure of any suitable object. For example, the object to be structurally optimized can be a part of a vehicle (such as a car hood), a part of an aircraft (such as an aircraft wing), a part of a rocket (such as a rocket shell), a fuel cell bipolar plate, a pump of a reactor, or any suitable part of any suitable device.

[0069] In the embodiment, the process of structural optimization is shown in Table 1 below. Table 1

[0070] Figure 5 An exemplary process for optimizing the structure of an object according to various aspects of the present disclosure is shown. It should be understood that the process can be implemented using a computer or a processor.

[0071] At block 510, a neural boundary operator (NBO) receives structural parameters, wherein the structural parameters are used to describe the boundary shape of the object, and the NBO is trained to map the structural parameters of various boundary shapes of the object to an approximation of a solution of a partial differential equation (PDE), the PDE being formulated to characterize a physical system associated with the object. It should be understood that in addition to the structural parameters describing the boundary shape, the structural parameters may also include other structure-related parameters, such as information related to materials, etc.

[0072] At block 520 , state variables are generated from the structural parameters by the NBO using a forward pass, wherein the state variables correspond to solutions to the PDE and are used to describe a physical state of a physical system associated with the object.

[0073] At block 530, an optimization target value is calculated based on the state variables. It should be understood that the optimization target value may be calculated based on the state variables and the structural parameters.

[0074] At block 540 , structural parameters of the object are updated based on the optimization objective value.

[0075] In an embodiment, the process at blocks 510 to 540 is repeated by taking the updated structure parameters obtained at block 540 in the previous cycle as the structure parameters in the current cycle at block 510 .

[0076] In an embodiment, NBO includes a branch network and a trunk network, the branch network takes the structural parameters as its input and outputs a first vector set, the trunk network takes the coordinates of the object as its input and outputs a second vector set, wherein the corresponding dot product between the first vector set and the second vector set is taken as the state variable. In an embodiment, the branch network and the trunk network are fully connected networks.

[0077] In an embodiment, the NBO includes a branch network, a trunk network, and a port network, the branch network takes a structural parameter as its input and outputs a first vector set, the trunk network takes the coordinates of the object as its input and outputs a second vector set, and the port network takes the first vector set and the second vector set as input and outputs a state variable. In an embodiment, the branch network, the trunk network, and the port network are fully connected networks.

[0078] In an embodiment, process 500 further includes: collecting a training data set for training the NBO, wherein the training data set includes a first number of sets of structural parameters corresponding to a first number of boundary shapes of the object and a first number of sets of state variables corresponding to the first number of sets of structural parameters, wherein the first number is the size of the training data set. The NBO is trained based on the training data set. It should be understood that the first number of sets of structural parameters corresponds to the structural instances as detailed above. Among them, a set of structural parameters corresponds to a structural instance θ i Thus, the first number of sets of state variables corresponds to the structure instance as detailed above The state variable {G(θ i )(x ij )|1≤i≤N,1≤j≤M}, where a set of state variables corresponds to a structure instance θ i The state variable {G(θ i )(x ij )|1≤j≤M}.

[0079] In an embodiment, collecting a training data set further comprises: initially, sampling a second number of sets of structural parameters corresponding to a second number of boundary shapes of the object according to a first distribution of the structural parameters of the object, wherein the first distribution is a uniform distribution; solving the PDE of the object based on the second number of sets of structural parameters by using a numerical solver to generate a second number of sets of state variables; sampling a third number of sets of structural parameters corresponding to a third number of boundary shapes of the object according to the second distribution of the structural parameters of the object, wherein the second distribution is determined based on the second number of sets of structural parameters; and solving the PDE of the object based on the third number of sets of structural parameters by using a numerical solver to generate a third number of sets of state variables, wherein the first number of sets of structural parameters consists of the second number of sets of structural parameters and the third number of sets of structural parameters, and the first number of sets of state variables consists of the second number of sets of state variables and the third number of sets of state variables. In an embodiment, the numerical solver is a finite element method (FEM) solver.

[0080] In an embodiment, the second distribution is also determined based on the optimization target values ​​of the unsampled structural parameters of the object. In an embodiment, the optimization target values ​​of the unsampled structural parameters of the object are estimated by an estimator, wherein the estimator is trained based on a second number of sets of structural parameters and a second number of corresponding optimization target values, wherein the second number of corresponding optimization target values ​​are obtained based on a second number of sets of state variables.

[0081] In an embodiment, the second distribution p(θ) is determined by the following equation: Where J(θ) represents the optimization objective function of the unsampled structural parameter θ, represents the balance factor, S t represents the θ sampled so far i The set of ||θ i -θ||2 represents the l2 norm.

[0082] In an embodiment, the object is one of a fuel cell bipolar plate, a part of a car, a part of an airplane, a part of a building, a pipe of a reactor, a flow baffle. In an embodiment, the structural parameters include at least one of position, radius, width, height, length, anchor point. In an embodiment, the state variable includes at least one of velocity, pressure, temperature.

[0083] Figure 6 An exemplary process for collecting a training data set for training a neural boundary operator (NBO) is shown. The NBO is trained to map structural parameters of an object's boundary shape to an approximation of a solution of a partial differential equation (PDE), the PDE being formulated to characterize a physical system associated with the object, wherein the solution of the PDE corresponds to state variables describing a physical state of the physical system associated with the object.

[0084] At block 610 , initially, a first number of sets of structural parameters corresponding to a first number of bounding shapes of an object are sampled according to a first distribution of structural parameters of the object, wherein the first distribution is a uniform distribution.

[0085] At block 620 , a first number of sets of state variables are generated by using a numerical solver to solve a PDE of the plant based on the first number of sets of structural parameters.

[0086] At block 630 , a second number of sets of structural parameters corresponding to a second number of bounding shapes of the object are sampled according to a second distribution of structural parameters of the object, wherein the second distribution is determined based on the first number of sets of structural parameters.

[0087] At block 640, a second number of sets of state variables are generated by using a numerical solver to solve the PDE of the plant based on the second number of sets of structural parameters. The first number of sets of structural parameters, the corresponding first number of sets of state variables, the second number of sets of structural parameters, and the corresponding second number of sets of state variables are included in the training data set.

[0088] In an embodiment, the second distribution is also determined based on the optimization target values ​​of the unsampled structural parameters of the object. In an embodiment, the optimization target values ​​of the unsampled structural parameters of the object are estimated by an estimator, wherein the estimator is trained based on a first number of sets of structural parameters and a first number of corresponding optimization target values, wherein the first number of corresponding optimization target values ​​are obtained based on a first number of sets of state variables.

[0089] In an embodiment, the second distribution p(θ) is determined by the following equation: Where J(θ) represents the optimization objective function of the unsampled structural parameter θ, represents the balance factor, S t represents the θ sampled so far i The set of ||θ i -θ||2 represents the l2 norm.

[0090] Figure 7 1 shows an exemplary computing system according to aspects of the present disclosure. The computing system 700 may include at least one processor 710. The computing system 700 may also include at least one storage device 720. The storage device 720 may store computer executable instructions that, when executed, cause the processor 710 to perform operations according to the embodiment of the present disclosure. Figure 1-6 Any operation of an embodiment of the present disclosure is described.

[0091] Embodiments of the present disclosure may be embodied in a computer-readable medium such as a non-transitory computer-readable medium. The non-transitory computer-readable medium may include instructions that, when executed, cause one or more processors to perform operations according to the embodiment of the present disclosure. Figure 1-6 Any operation of an embodiment of the present disclosure is described.

[0092] Embodiments of the present disclosure may be embodied in a computer program product comprising computer executable instructions that, when executed, cause one or more processors to perform operations according to the present disclosure. Figure 1-6 Any operation of an embodiment of the present disclosure is described.

[0093] It should be understood that all operations in the above methods are merely exemplary operations, and the present disclosure is not limited to any operations in these methods or the order of these operations, and should encompass all other equivalents under the same or similar concept.

[0094] It should also be understood that all modules in the above-mentioned device can be implemented in various ways. These modules can be implemented as hardware, software or a combination thereof. In addition, any module in these modules can be further divided into submodules or combined together in function.

[0095] The previous description is provided to enable any person skilled in the art to practice the various aspects described herein. Various modifications to these aspects will be apparent to those skilled in the art, and the general principles defined herein may be applied to other aspects. Therefore, the claims are not intended to be limited to the aspects shown herein. All structural and functional equivalents of the elements of the various aspects described throughout this disclosure are expressly incorporated herein by reference and are intended to be covered by the claims, and these structural and functional equivalents are known or will be known later to those of ordinary skill in the art.

Claims

1. A computer-implemented method for optimizing a structure of an object, comprising: receiving, by a neural boundary operator (NBO), structure parameters describing a boundary shape of the object and trained to map the structure parameters of the boundary shape of the object to an approximation of a solution of a partial differential equation (PDE) formulated to characterize a physical system associated with the object; generating state variables from the structural parameters using forward transfer by the NBO, wherein the state variables correspond to the solutions of the PDE and are used to describe the physical state of the physical system associated with the object; calculating an optimization target value based on the state variable; and The structural parameters of the object are updated based on the optimization target value.

2. The method according to claim 1, further comprising: The receiving step, the generating step, the calculating step and the updating step are repeated by taking the updated structural parameters in the updating step as the structural parameters in the receiving step.

3. The method according to claim 1, wherein: The NBO includes a branch network and a trunk network, wherein the branch network takes the structural parameters as its input and outputs a first set of vectors, and the trunk network takes the coordinates of the object as its input and outputs a second set of vectors, wherein the corresponding dot products between the first set of vectors and the second set of vectors are taken as the state variables.

4. The method according to claim 1, wherein: The NBO includes a branch network, a trunk network and a port network, the branch network takes the structural parameters as its input and outputs a first vector set, the trunk network takes the coordinates of the object as its input and outputs a second vector set, and the port network takes the first vector set and the second vector set as input and outputs the state variables.

5. The method according to claim 4, wherein: The branch network, the backbone network and the port network are fully connected networks.

6. The method according to claim 1, further comprising: collecting a training data set for training the NBO, wherein the training data set includes a first number of sets of structural parameters corresponding to a first number of boundary shapes of the object and a first number of sets of state variables corresponding to the first number of sets of structural parameters; and The NBO is trained based on the training data set.

7. The method according to claim 6, wherein: The collecting of the training data set further comprises: sampling a second number of sets of structural parameters corresponding to a second number of boundary shapes of the object according to a first distribution of the structural parameters of the object, wherein the first distribution is a uniform distribution; generating a second number of sets of state variables by solving the PDE of the plant based on the second number of sets of structural parameters using a numerical solver; sampling a third number of sets of structural parameters corresponding to a third number of boundary shapes of the object according to a second distribution of the structural parameters of the object, wherein the second distribution is determined based on the second number of sets of structural parameters; and generating a third number of sets of state variables by using the numerical solver to solve the PDE of the plant based on the third number of sets of structural parameters, The first number of structure parameter sets is composed of the second number of structure parameter sets and the third number of structure parameter sets, and the first number of state variable sets is composed of the second number of state variable sets and the third number of state variable sets.

8. The method according to claim 7, wherein: The numerical solver is a finite element method (FEM) solver.

9. The method according to claim 7, wherein: The second distribution is also determined based on the optimization target value of the unsampled structural parameter of the object.

10. The method according to claim 9, wherein: The optimization target value of the unsampled structural parameter of the object is estimated by an estimator, wherein the estimator is trained based on the second number of structural parameter sets and the second number of corresponding optimization target values, wherein the second number of corresponding optimization target values ​​are obtained based on the second number of state variable sets.

11. The method according to claim 9, wherein: The second distribution p(θ) is determined by the following equation: Wherein, J(θ) represents the optimization objective function of the unsampled structural parameter θ, represents the balance factor, S t represents the θ sampled so far i , ||·||2 represents the l2 norm.

12. The method according to claim 1, wherein: The object is one of a fuel cell bipolar plate, a part of a car, a part of an aircraft, a part of a building, a pipe of a reactor, a flow baffle.

13. The method according to claim 1, wherein: The structural parameters include at least one of position, radius, width, height, length, and anchor point, and the state variables include at least one of velocity, pressure, and temperature.

14. A method for collecting a training data set for training a neural boundary operator (NBO), wherein: The NBO is trained to map structural parameters of a boundary shape of an object to an approximation of a solution of a partial differential equation (PDE), the PDE being formulated to characterize a physical system associated with the object, wherein the solution of the PDE corresponds to state variables describing a physical state of the physical system associated with the object, the method comprising: sampling a first number of sets of structural parameters corresponding to a first number of boundary shapes of the object according to a first distribution of the structural parameters of the object, wherein the first distribution is a uniform distribution; generating a first number of sets of state variables by using a numerical solver to solve the PDE of the plant based on the first number of sets of structural parameters; sampling a second number of sets of structural parameters corresponding to a second number of boundary shapes of the object according to a second distribution of the structural parameters of the object, wherein the second distribution is determined based on the first number of sets of structural parameters; and generating a second number of sets of state variables by using the numerical solver to solve the PDE of the plant based on the second number of sets of structural parameters, The first number of sets of structural parameters, the corresponding first number of sets of state variables, the second number of sets of structural parameters, and the corresponding second number of sets of state variables are included in the training data set.

15. The method according to claim 14, wherein: The second distribution is also determined based on the optimization target value of the unsampled structural parameter of the object.

16. The method according to claim 15, wherein: The optimization target values ​​of the unsampled structural parameters of the object are estimated by an estimator, wherein the estimator is trained based on the first number of structural parameter sets and the first number of corresponding optimization target values, wherein the first number of corresponding optimization target values ​​are obtained based on the first number of state variable sets.

17. The method according to claim 15, wherein: The second distribution p(θ) is determined by the following equation: Wherein, J(θ) represents the optimization objective function of the unsampled structural parameter θ, represents the balance factor, S t represents the θ sampled so far i , ||·||2 represents the l2 norm.

18. A computer system comprising: one or more processors; as well as One or more storage devices storing computer-executable instructions which, when executed, cause the one or more processors to perform the operations of the method of one of claims 1-17.

19. One or more computer-readable storage media storing computer-executable instructions that, when executed, cause one or more processors to perform the operations of the method of one of claims 1-17.

20. A computer program product comprising computer executable instructions which, when executed, cause one or more processors to perform the operations of the method of one of claims 1-17.

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