WO2023206204A1

WO2023206204A1 - Method and apparatus for structure optimizition

Info

Publication number: WO2023206204A1
Application number: PCT/CN2022/089786
Authority: WO
Inventors: Jun Zhu; Ze CHENG; Songming LIU; Hang SU; Zhongkai HAO; Chengyang YING
Original assignee: Robert Bosch Gmbh; Tsinghua University
Priority date: 2022-04-28
Filing date: 2022-04-28
Publication date: 2023-11-02

Abstract

A computer implemented method for optimizing structure of an object. The method comprises receiving structural parameters by a neural boundary operator (NBO), wherein the structural parameters are used to describe boundary shape of the object, and the NBO is trained to map structural parameters of boundary shapes of the object to approximations of solutions of partial differential equations (PDEs) formulated to characterize a physical system related to the object; generating state variables by the NBO from the structural parameters, wherein the state variables correspond to the solutions of the PDEs and are used to describe physical states of the physical system related to the object; calculating an optimization objective value based on the state variables; and updating the structural parameters based on the optimization objective value.

Description

METHOD AND APPARATUS FOR STRUCTURE OPTIMIZITION

FIELD

Aspects of the present disclosure relate generally to artificial intelligence (AI) , and more particularly, to optimizing structure of an object.

BACKGROUND

The structure of an object, such as its shape, size, or distribution of materials, influences the performance of the object. Examples of the object may be the wing of an airplane, the truss of a house, the pipes in a reactor, or the like. For example, the shape of an airfoil influences the pressure and velocity of the airflow with respect to the airfoil while the pressure and velocity influencing the performance of the physical system of the airfoil. The physical system can be described by partial differential equations (PDEs) . The state variables of the physical system such as the pressure and velocity can be obtained by solving the PDEs based on the structural parameters such as those describing the shape of the object.

It is desirable to design the structure of an object so as to improve its performance, this procedure may be referred to as structural optimization. The structural optimization is applicable in many areas including scientific area, engineering area, industrial area or the like. For example, the shape of chemical catalyst pellets, the shape of auto parts or the like may be optimized through the structural optimization procedure before manufacturing.

Since most physical systems are described by partial differential equations (PDEs) , structural optimization is typically carried out with the governing PDEs. This can be time-consuming because of the need to solve PDEs at each iteration of the optimization.

SUMMARY

In order to improve the efficiency of the structural optimization, the disclosure proposes a novel framework for structural optimization, by which the time consumption and computation requirement may be reduced while optimization performance may be guaranteed.

According to an embodiment, there provides a computer implemented method for optimizing structure of an object. The method comprises: receiving structural parameters by a neural boundary operator (NBO) , wherein the structural parameters are used to describe boundary shape of the object, and the NBO is trained to map structural parameters of boundary shapes of the object to approximations of solutions of partial differential equations (PDEs) formulated to characterize a physical system related to the object; generating state variables by the NBO from the structural parameters with a forward pass, wherein the state variables correspond to the solutions of the PDEs and are used to describe physical states of the physical system related to the object; calculating an optimization objective value based on the state variables; and updating the structural parameters of the object based on the optimization objective value.

According to an embodiment, there provides 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 boundary shapes of an object to approximations of solutions of partial differential equations (PDEs) formulated to characterize a physical system related to the object, wherein the solutions of PDEs correspond to state variables used to describe physical states of the physical system related to the object. The method comprises: sampling a first number of sets of structural parameters corresponding to the 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 uniform distribution; generating the first number of sets of state variables by using a numerical solver to solve the PDEs of the object based on the first number of sets of structural parameters; sampling a second number of sets of structural parameters corresponding to the 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 the second number of sets of state variables by using the numerical solver to solve the PDEs of the object based on the second number of sets of structural parameters, 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.

According to an embodiment, there provides a computer system, which comprises one or more processors and one or more storage devices storing computer-executable instructions that, when executed, cause the one or more processors to perform the operations of the method as mentioned above as well as to perform the operations of the method according to aspects of the disclosure.

According to an embodiment, there provides 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 as mentioned above as well as to perform the operations of the method according to aspects of the disclosure.

According to an embodiment, there provides a computer program product comprising computer-executable instructions that, when executed, cause one or more processors to perform the operations of the method as mentioned above as well as to perform the operations of the method according to aspects of the disclosure.

By using the NBO, which learns mappings between the structural parameters of variant boundary shapes of an object and the solutions of the PDEs, in the optimization framework, the efficiency of structural optimization can be improved dramatically while the bottleneck of solving PDEs being resolved. Moreover, by using the adaptive sampling framework to collect training data set for training the NBO, the training cost of NBO is reduced by taking advantage of a tradeoff between exploiting the priors of the structure of the object and exploring unknown structures. Other advantages and enhancements are explained in the description hereafter.

BRIEF DESCRIPTION OF THE DRAWINGS

The disclosed aspects will hereinafter be described in connection with the appended drawings that are provided to illustrate and not to limit the disclosed aspects.

Fig. 1 illustrates an exemplary framework for optimizing structure of an object according to aspects of the disclosure.

Fig. 2A illustrates an architecture of NBO according to aspects of the disclosure.

Fig. 2B illustrates an architecture of NBO according to aspects of the disclosure.

Fig. 3 illustrates an exemplary structural optimization of flow baffles according to aspects of the disclosure.

Fig. 4 illustrates an exemplary structural optimization of conductive-media interfaces according to aspects of the disclosure.

Fig. 5 illustrates an exemplary process for optimizing structure of an object according to aspects of the disclosure.

Fig. 6 illustrates an exemplary process for collecting a training data set for training NBO according to aspects of the disclosure.

Fig. 7 illustrates an exemplary computing system according to aspects of the disclosure.

DETAILED DESCRIPTION

The present disclosure will now be discussed with reference to several example implementations. It is to be understood that these implementations are discussed only for enabling those skilled in the art to better understand and thus implement the embodiments of the present disclosure, rather than suggesting any limitations on the scope of the present disclosure.

Various embodiments will be described in detail with reference to the accompanying drawings. Wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts. References made to particular examples and embodiments are for illustrative purposes, and are not intended to limit the scope of the disclosure.

The object to be structurally optimized in the example shown in Fig. 1 is an airfoil. As shown in block 120 of Fig. 1, the shape of the airfoil is presented by a boundary γ, which is parameterized by structural parameters θ. In this example, the structural parameters θ may be a set of control points which consist of splines representing the shape of the airfoil. In other words, the structure to be optimized is the shape of the airfoil represented by splines with a set of control points θ. The label Ω denotes the problem domain or particularly denotes the domain of the physical system related to the airfoil.

As shown in block 110 of Fig. 1, the arrowed lines denote airflows on the airfoil. The physical state of the airflow on the airfoil may be represented by state variables of the physical system related to the airfoil. In this example, the state variables may be the velocity u=u (x) and the pressure p=p (x) . The label x denotes spatial coordinates in the domain Ω.

The velocity u=u (x) and the pressure p=p (x) may be characterized by nonlinear PDEs of Navier–Stokes (NS) equations, with the constraint of the shape of the airfoil represented by the boundary γ. It is appreciated that NS equations are well known physical equations that can be used to describe the three-dimensional motion of viscous fluid substances. The NS equations are second-order nonlinear PDEs, and may be used to model the weather, ocean currents, heat conducting, air flow around an airfoil, water flow in a pipe or in a reactor and many other applications. The PDEs of the NS equations for a physical system of an object may be formulated according to physical laws related to the physical system of the object. In the illustrated example, the PDEs of the physical system of the airfoil may be established according to the related physical law as shown by label 140 of Fig. 1. As the PDEs are used to govern the optimization of the structure of the airfoil, they may be referred to as governing PDEs. It is appreciated that the formulation of the governing PDEs may be implemented with well-known physical knowledge, and other kinds of PDEs used to describe a physical system of an object may also be used as the governing PDEs in aspects of the disclosure.

As shown by label 130 of Fig. 1, the objective or goal of the structural optimization of the airfoil is to reach the desired pressure distribution p _ref on the airfoil surface by changing the structural parameter θ. The objective function J may be formulated as shown by label 130, and the aim of the structural optimization is to minimize the objective function J, which is a functional of the pressure p. The shape of the airfoil corresponding to the boundary γ parameterized by structural parameters θ may be iteratively optimized to minimize the objective function J, so as to reach the desired pressure distribution p _ref on the airfoil surface.

In each loop of the structural optimization, the PDEs 140 need to solved to obtain the physical state variables such as the velocity u and pressure p, and then the objective function J may be evaluated based on the physical state variables such as the pressure p. Then the structural parameters θ may be updated or optimized based on the calculated objective function J as shown by label 160.

In an example, existing numerical methods such as finite element methods (FEM) and finite volume methods (FVM) may be used to obtain the velocity u and pressure p by solving the PDEs 140. For example, FEM are frequently used to model the behavior of the structure, since they are accurate and stable to a wide range of PDEs including some complicated nonlinear PDEs (e.g., NS equations) . Despite the accuracy and stability, the FEM solvers are frustrated by the large computational overhead. For each instance of the PDEs, the FEM solvers need to generate a mesh of the structure of the object and solve a large system of equations from scratch, which is exceedingly slow for high dimensional problems. For example, even for a medium-sized system, for example, a system represented with a 50 × 50 two-dimensional (2D) mesh, the numerical methods often require a few seconds to obtain the solutions of the PDEs. Since the solutions of PDEs are extremely sensitive to boundary conditions and such methods lack the ability to generalize to various boundary shapes, they require independently solving PDEs millions of times during the optimization process, incurring unbearable computational overhead and corresponding time consuming. Using the existing numerical PDE solvers such as FEM solvers and FVM solvers in the structural optimization frame 100 would lead to inefficiency of the structural optimization of the object.

In an embodiment, as illustrated in Fig. 1, a Neural Boundary Operator (NBO) 150 is employed in the structural optimization frame 100 to solve the PDEs 140 with various boundary shapes represented by structural parameters θ. The NBO may be implemented with a deep learning architecture for learning operators from the structural parameters of the various boundary shapes to the solutions of PDEs. In the illustrated example, the NBO 150 has learned the latent operator

which can generalize to the space of the structural parameter θ, in other words, the NBO has learned the mappings between the structural parameters of boundary shapes of the object and the solutions of the PDEs. As the NBO can effectively generalize across variant boundary shapes, the overall structure optimization is an iterative process, of which during each iteration, the PDEs can be solved efficiently via a forward pass of NBO, instead of calling on an expensive FEM solver. Therefore, by using the NBO 150, the solution process of the PDEs may be accelerated with orders of magnitude compared with existing PDE solvers, leading the structural optimization framework 100 to be much more efficient.

Fig. 2A and Fig. 2B each illustrates an architecture of NBO according to aspects of the disclosure.

The structural optimization problem may be represented by the following equation (1) :

where θ denotes the structural parameters of the structure of the object to be optimized, which are also the parameters of the boundary shapes, Θ denotes the space of θ. The structural parameters θ is changing during the optimization of the structure but is within the design space Θ. J is the objective function whose value measures how good a given structure is, W stands for the constraints on both the structural parameters θ and the state variable s = s (x) which denotes the state of the physical system, for example, velocity, pressure or the like, where x denotes spatial coordinates. And the symbol “≤°” is an elementwise comparison.

Since the state variables s are associated with the physical system of the object to be structurally optimized, it is characterized by the PDEs as shown in the following equation (2) :

where F denotes a vector-valued operator to represent the PDEs, B _i denotes a vector-valued operator to represent the equation of the ith boundary condition, γ _i denotes a domain on which the ith boundary condition is defined, Ω denotes the domain of the physical system, the shape of the boundary γ _i (θ) is parameterized by structural parameters θ and all the boundaries γ _i (θ) together correspond to the structure to be optimized. For any given structural parameters θ, it may assume the PDEs have a unique solution s. Hence, an operator

can defined as G (θ) (x) = s (x) .

As an example, the problem of drag minimization of 2D fluid as shown in the following equation (3) may be considered:

where v is the viscosity of the fluid, ∥·∥ _F is the Frobenius norm. In this example, the physical system includes a 2D channel Ω with fluid and an elliptical baffle. Moreover, the structural parameters θ = (θ ₁, θ ₂) ∈ Θ = [0, 1] ², where θ ₁ and θ ₂ are respectively corresponding to the width and height of the elliptical baffle. The state variable is s = (p, u) , where p = p (x) and u = u (x) are the pressure and the velocity of the fluid respectively. The structural optimization goal is to minimize the energy dissipated by the fluid by changing the structural parameters θ (i.e., the shape of the elliptical baffle) , with the constraint that the area of the elliptical baffle is 1. And the physical state variable s = (p, u) is characterized by the nonlinear PDEs of NS equations (4) :

where Ω is the problem domain (i.e., the 2D channel) , γ ₁ and γ ₂ are the inlet and outlet of the channel, respectively, and γ ₃ is the geometry of the elliptical baffle. It is appreciated that the PDEs (4) describing the physical system may be formulated according to the physical law of the physical system.

In the equation (4) , the boundary γ ₃ (θ ₁, θ ₂) is the structure to be optimized. Since the structure is changing during the optimization process, the equation (4) corresponds to PDEs with various boundary shapes. Furthermore, in equation (3) , the objective function J depends on the physical state variable u, which is defined by equation (4) . The physical state variable u may be obtained by solving the PDEs (4) . In the structural optimization process, the PDEs need to be solved simultaneously with the optimization of the structure. However, previous methods are too time-consuming for applications, since they cannot generalize to PDEs with various boundary shapes and have to solve the PDEs from scratch repetitively.

NBO may be used to address the above challenges by generalizing to PDEs with various boundary shapes and mapping the structural parameters θ to the solution of the PDEs. The NBO is a neural network

to approximate the latent operator

NBO takes the coordinate x and the parameters θ as inputs. The output is an approximation of the solution of the PDEs at x, i.e.,

In the embodiment shown in Fig. 2A, the NBO 200A comprises a branch subnetwork 210A and a trunk subnetwork 220A. The branch subnetwork 210A takes the structural parameter θ as its input and outputs n p-dimensional vectors

and the trunk subnetwork 220A takes the coordinate x as its input and outputs n p-dimensional vectors

where n equals to the dimension of the state variable s, p is a hyperparameter. As shown at 230A, the respective dot products between the n vectors

and n vectors

are taken as the prediction of the solution of the PDEs

The branch subnetwork 210A and the trunk subnetwork 220A may be implemented as fully-connected networks (FCNs) .

In the embodiment shown in Fig. 2B, the NBO 200B comprises a branch subnetwork 210B, a trunk subnetwork 220B and a port subnetwork 240B. The branch subnetwork 210B takes the structural parameter θ as its input and outputs n p-dimensional vectors

and the trunk subnetwork 220B takes the coordinate x as its input and outputs n p-dimensional vectors

where n equals to the dimension of the state variable s, p is a hyperparameter. The n vectors

and n vectors

are concatenated as shown at 230B, and then input into the port subnetwork 240, which outputs the prediction of the solution of the PDEs

The port subnetwork 240B is a micro neural network. Taking advantage of the potent nonlinear approximation ability of the port subnetwork 240B, the abstraction ability of the NBO 200B for high-level features may be further improved compared to the NBO 200A. The branch subnetwork 210B, the trunk subnetwork 220B and the port subnetwork 240B may be implemented as fully-connected networks (FCNs) .

In order to train the NBO which is the neural network

for approximating the latent operator

instances of the structural parameters are firstly sampled, that is, { (θ _i, x _ij) ∣1≤i≤N, 1≤j≤M} are firstly sampled according to a custom distribution

in the space Θ×Ω. One instance of structure parameter θ _i may be referred to a set of structure parameter θ _i for sake of easy description and understanding. Then a numerical solver such as the FEM solver is utilized to solve the corresponding PDEs to obtain the solutions {G (θ _i) (x _ij) } . In this way, a data set { (θ _i, x _ij, G (θ _i) (x _ij) ) } can be constructed as the training and testing samples.

The NBO may be trained with the training data set. The training objective for the NBO as shown in the following equation (5) may be employed:

where w are the neural network weights, i.e., the parameters of the NBO, and

is the space of neural network weights, G (θ) (x) is the ground truth and

is the prediction of the NBO model. Instead of approximating the solution of a specific instance of the PDEs, the training data are sampled in the structural parameter space Θ and the NBO is trained with these samples to learn the underlying operator

In this way, for a new structure, it can take advantage of the generalizability of NBO to transfer to the solution of the corresponding PDEs without solving from scratch.

Since NBO is the key component in the overall structural optimization framework, the distribution

of the training data may significantly affect the performance of the structural optimization. In an embodiment,

is the uniform distribution. However, uniform sampling is inefficient since it is not focusing 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 custom distribution so as to sample those structures with good performance. The following equation (6) shows the density function of

p (θ, x) =p (θ) p (x∣θ) (6)

Since the structural optimizers are more likely to stay in the locations where the value of the objective function J is lower, which means a better structure, it is advantageous to sample those points with lower J values. The objective J (θ, s) (abbreviated as J (θ) hereinafter) may be chosen as the prior, guiding the NBO to fit better around potential minimums of the objective function.

The exploitation (empirical prior) and the exploration (space filling) can be balanced. The distribution of p (θ) may be defined as shown in the following equation (7) in order for the balance of exploitation and exploration:

where

is the unnormalized density function,

is a balance factor, S _t is a set of θ _i already sampled so far, and ∥·∥ ₂ is the l ₂ norm.

Since the objective function J (θ) of the structural optimization does not have an analytic expression from θ to J, an estimator of the objective function J (θ) may be utilized in an embodiment. A small set of structural instances {θ _i} may be firstly sampled according to a sampling distribution, which may be a uniform distribution. Then the FEM solver may be used to obtain corresponding solutions {G (θ _i) =s _i} of PDEs, which are used to evaluate the corresponding objective function {J (θ _i, s _i) } . Then the date set {θ _i, J (θ _i, s _i) } can serve as training data of the estimator

In an example, the estimator

may be implemented as a linear model

where w denotes the parameters or weights of the estimator

It is appreciated that the estimator

may be implemented as a non-linear model in another example. Assuming that the structural parameter space Θ is bounded (i.e.,

) ,

can be defined on [-d, d] ^q and be normalized to get p (θ) as shown in the following equation (8) :

The expression of the normalized coefficient

may be given as shown in the following equation (9) :

Where

and erfi is the imaginary error function.

In an embodiment, the distribution p (θ) can be determined based on equation (7) . In another embodiment, the distribution p (θ) can be determined based on equation (8) or based on equations (8) and (9) . In another embodiment, the distribution p (θ) can be determined based on equation (7) by taking

as p (θ) . It is appreciated that there may be other ways for implementing the custom sampling of structural parameter instances θ _i as defined in equation (7) .

As for p (x | θ) , it may be determined with the well-known adaptive mesh refinement (AMR) technique. The AMR uses a set of refinement criteria based on the local field, slope, or curvature of the variables to generate and actively adapt the mesh in each zone of a structure, which can improve the utilization of the nodes and reduce the computational cost in consequence. 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 as corresponding coordinate samples

Compared to the limitations of existing methods that are of high computational cost and cannot generalize to PDEs with various boundary shapes, the NBO can generalize to various boundary shapes and thus can estimate the solutions of the PDEs with a forward pass in each iteration of structural optimization, avoiding the expensive solving process of PDEs. As detailed above, the overall approach contains two stages. The first stage is collecting the dataset and training NBO. The second stage is to employ NBO to estimate the solutions of the PDEs in structural optimization. In the second stage, note that it may require NBO, which is an NN model, to make predictions on multiple points to evaluate the objective function J, especially when it has an integral term as shown in equation (3) for example, this provides a chance for GPU parallel acceleration.

Fig. 3 is a schematic diagram illustrating the structural optimization of Flow Baffles according to aspects of the disclosure.

As shown in Fig. 3, γ ₂ to γ ₅ located in corresponding grids are the flow baffles to be optimized. The goal of the structural optimization is to optimize the shape and position of the baffles γ ₂ to γ ₅ so as to obtain an even distribution of fluid flow at the outlet γ _1, _right with the little energy dissipated by the fluid.

The objective function of the structural optimization may be formulated based on fluid mechanics as shown in the following equation (10) :

where β is a balance factor, for example, β = 0.01, u= (u ₁, u ₂) is the velocity, v is the viscosity, for example v = 1.

The governing PDEs describing the physical system related to the flow baffles may be formulated according to physical laws, for example, the PDEs may be NS equations as shown in the following equation (11) :

Where x= (x ₁, x ₂) is the spatial coordinate, p is the pressure, u= (u ₁, u ₂) is the velocity, v is the viscosity, for example v = 1, γ _1, left is the left part of γ ₁, γ _{1, left, up} is the upper half of γ _1, left , γ _{1, left, down} is the lower half of γ _1, left , γ _wall =γ _1, up ∪γ _1, down ∪γ ₂∪γ ₃∪γ ₄∪γ ₅∪γ _{1, left, down}, and u ₁ (x) is

The structural parameters include the center and the radius of circular baffles γ ₂ to γ ₅, that is, θ= (x ₀, r ₀, …, x ₃, r ₃) . The constraints W (θ) can be defined to confine each of γ ₂ to γ ₅ within its grid as shown in Fig. 3. In an embodiment, in order to make the sampling process efficient, a mapping of the structural parameter can be made:

where

and minDist is the shortest distance from ith circle to the edge of the corresponding grid as shown in Fig. 3. Hence, the allowable values of each dimension of θ′ are in a closed interval and the allowable parameters can be normalized to [-3, 3] ¹², where the superscript means 12 dimensions of the structural parameters. Let the final parameters as θ″, W (θ) is defined as:

The training of the NBO model is then performed on this mapped space, which is a normalized structural space of the original structural space.

In an embodiment, the Adam algorithm may be utilized to optimize the structure of the baffles with NBO providing forward predictions of the PDEs as well as backward calculations of derivatives. In an example, the number of iterations of optimizing the structure θ is 200, the learning rate is 0.1. The loss function is defined as

where λ is a predefined coefficient such as λ=100, ∥·∥ _absSum stands for the sum of absolute values of all elements of a vector.

In an embodiment, a training data set for training the NBO model is generated. The NBO may have the structure as shown in Fig. 2A or 2B. It is appreciated that the NBO may also have other structure.

Assuming the data size of the data set is N, for example, N = 5000, which means 5000 structure instances

are included in the data set. Firstly, a small subset of structural instances

may be sampled according to uniform distribution, where N ₁＜＜N, for example, N ₁ = 100. The well-known FEM solver is used to generate the corresponding meshes for the structural instances as well as the values of the state variables {G (θ _i) (x _ij) ∣1≤i≤N ₁, 1≤j≤M} according to the PDEs (11) , where the coordinate x _ij corresponds to the node of the mesh, M is the number of nodes of the mesh, M may be a variable value related to the specific structure θ _i. And the values of objective function

are obtained according to the equation (10) . The estimator

of the objective function (10) may be trained based on

The rest structural instances

may be sampled according to the distribution defined in equation (7) with J replaced by

For example, Markov Chain Monte Carlo (MCMC) method may be used to sample the rest 4900 structural instances according to the distribution defined in equation (7) . Then, the FEM solver is used to generate the corresponding meshes for the structural instances as well as the values of the state variables {G (θ _i) (x _ij) ∣N ₁+1≤i≤N, 1≤j≤M} . And the training data set for training the NBO is obtained, 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) .

In each iteration of the structural optimization, the trained NBO is used to estimate the state variables u and optional p, from which the objective J (θ) is obtained according to equation (10) . Then the loss

as shown in equation (13) is obtained, and the derivates of the loss

with respect to θ can be obtained accordingly. Finally the structural parameter θ is updated based on the loss

or its derivates in this structural optimization iteration. In an embodiment, the Adam algorithm may be utilized to update the structure θ with the aim of minimizing J shown in equation (10) .

Fig. 4 is a schematic diagram illustrating the structural optimization of conductive-media interfaces according to aspects of the disclosure.

The conductive-media interface shown in Fig. 4 may be an infinite-length conductive-media interface with thickness H = 8 and period L = 8. Ω ₁ and Ω ₂ are two conductive mediums with different thermal conductivity, governed by the PDEs as shown in the following equation (14) :

Where x= (x ₁, x ₂) is the spatial coordinate, k ₁ and k ₂ are the thermal conductivity of Ω ₁ and Ω ₂ respectively, for example, k ₁= 1 and k ₂ = 2, y _i is the temperature distribution of Ω _i, i = 1, 2, and

and

are the unit normal for the Neumann boundary condition. The structure to be optimized is the conductive-media interface γ _S represented by splines. The leftmost and rightmost anchor points of the splines are fixed (marked as circular dots in Fig. 4) , while the four middle anchor points are variable (marked as squared dots in Fig. 4) but within corresponding rectangular grids (marked as dotted lines in Fig. 4) . Given the positions of the four anchor points, the splines can be obtained by interpolation. Hence, the four anchor points are set as the structural parameters, i.e., θ= (x ₀, x ₁, x ₂, x ₃) .

The goal of the structural optimization is to maximize the heat transfer through the interface γ _S by changing the structural parameters θ. The goal of the structural optimization may be described as shown in the following equation (15) :

Then the goal may be converted to a minimization problem by taking the objective function J as 1/h _T (y) , as shown in the following equation (16) :

J (θ) = 1/h _T (y) (16)

The constraints W (θ) can be defined to confine each of anchor points within its grid as shown in Fig. 4. In an embodiment, the allowable parameters can be normalized to [-3, 3] ⁸, where the superscript means 8 dimensions of the structural parameters. Let the final parameters as θ″, W (θ) is defined as:

In an embodiment, the Adam algorithm may be utilized to optimize the structure of the interface with NBO providing forward predictions of the PDEs as well as backward calculations of derivatives. In an example, the number of iterations of optimizing the structure θ is 200, the learning rate is 0.1. The loss function is defined as

Assuming the data size of the data set is N, for example, N = 2000, which means 2000 structure instances

are included in the data set. Firstly, a small subset of structural instances

may be sampled according to uniform distribution, where N ₁＜＜N, for example, N ₁ = 100. The well-known FEM solver with AMR function is used to generate the corresponding meshes for the structural instances as well as the values of the state variables {G (θ _i) (x _ij) ∣1≤i≤N ₁, 1≤j≤M} according to the PDEs (14) , where the coordinate x _ij corresponds to the node of the mesh, M is the number of nodes of the mesh, M may be a variable value related to the specific structure θ _i. And the values of objective function

are obtained according to the equation (16) . The estimator

of the objective function (16) may be trained based on

The rest structural instances

For example, Markov Chain Monte Carlo (MCMC) method may be used to sample the rest 1900 structural instances according to the distribution defined in equation (7) . Then, the FEM solver is used to generate the corresponding meshes for the structural instances as well as the values of the state variables {G (θ _i) (x _ij) ∣N ₁+1≤i≤N, 1≤j≤M} . And the training data set for training the NBO is obtained, 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) .

In each iteration of the structural optimization, the trained NBO is used to estimate the state variables y _i, from which the objective J (θ) is obtained according to equation (16) . Then the loss

as shown in equation (18) is obtained, and the derivates of the loss

with respect to θ can be obtained accordingly. The structural parameter θ is updated based on the loss

or its derivates in this structural optimization iteration. In an embodiment, the Adam algorithm may be utilized to update the structure θ with the aim of minimizing J shown in equation (16) .

It is appreciated that the structural optimization framework proposed in the disclosure is applicable to optimize the structure of any suitable object in addition to the specific objects exampled in Figs. 3 and 4. For example, the object to be structural optimized may be a part of a vehicle such as the car hood, a part of a plane such as the wine of the plane, a part of a rocket such as the rocket shall, fuel cell bipolar plate, pines of a reactor, or any suitable part of any suitable device.

In an embodiment, the process of the structural optimization is shown in the following table 1.

table 1

Fig. 5 illustrates an exemplary process for optimizing structure of an object according to aspects of the disclosure. It is appreciated that the process may be implemented with computers or processors.

At block 510, structural parameters are received by a neural boundary operator (NBO) , wherein the structural parameters are used to describe boundary shape of the object, and the NBO is trained to map structural parameters of various boundary shapes of the object to approximations of solutions of partial differential equations (PDEs) formulated to characterize a physical system related to the object. It is appreciated that the structural parameters may also include other structural related parameters such as information related to materials or the like, in addition to those describing the boundary shape.

At block 520, state variables are generated by the NBO from the structural parameters with a forward pass, wherein the state variables correspond to the solutions of the PDEs and are used to describe physical states of the physical system related to the object.

At block 530, an optimization objective value is calculated based on the state variables. It is appreciated that the optimization objective value may be calculated based on the state variables and the structural parameters.

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

In an embodiment, the processes at blocks 510 to 540 are repeated by taking the updated structural parameters obtained in last loop at block 540 as the structural parameters in current loop at block 510.

In an embodiment, the NBO comprises a branch network and a trunk network, the branch network takes the structural parameters as its input and outputs a first set of vectors, the trunk network takes coordinates of the object as its input and outputs a second set of vectors, wherein respective dot products between the first set of vectors and the second set of vectors are taken as the state variables. In an embodiment, the branch network and the trunk network are full connection networks.

In an embodiment, the NBO comprises a branch network, a trunk network and a port network, the branch network takes the structural parameters as its input and outputs a first set of vectors, the trunk network takes coordinates of the object as its input and outputs a second set of vectors, and the port network takes the first set of vectors and the second set of vectors as input and outputs the state variables. In an embodiment, the branch network, the trunk network and the port network are full connection networks.

In an embodiment, the process 500 further comprises collecting a training data set for training the NBO, wherein the training data set comprises 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 is appreciated that the first number of sets of structural parameters correspond to the structural instances

as detailed above, where one set of structural parameters correspond to one structural instance θ _i. Accordingly, the first number of sets of state variables correspond to the state variables {G (θ _i) (x _ij) ∣1≤i≤N, 1≤j≤M} of the structural instances

as detailed above, where one set of state variables correspond to the state variables {G (θ _i) (x _ij) ∣1≤j≤M} of one structural instance θ _i.

In an embodiment, the collecting the 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 uniform distribution; generating a second number of sets of state variables by using a numerical solver to solve the PDEs of the object based on the second number of sets of structural parameters; 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 PDEs of the object based on the third number of sets of structural parameters, wherein the first number of sets of structural parameters consist of the second number of sets of structural parameters and the third number of sets of structural parameters, the first number of sets of state variables consist 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 methods (FEM) solver.

In an embodiment, the second distribution is determined based further on the optimization objective value of unsampled structural parameters of the object. In an embodiment, the optimization objective value of the unsampled structural parameters of the object is estimated by an estimator, wherein the estimator is trained based on the second number of sets of structural parameters and a second number of corresponding optimization objective values, wherein the second number of corresponding optimization objective values are obtained based on the second number of sets of state variables.

In an embodiment, the second distribution p (θ) is determined by means of the following equation:

wherein J (θ) denotes the optimization objective function of the unsampled structural parameters θ,

denotes a balance factor, S _t denotes a set of θ _i sampled so far, ∥θ _i-θ∥ ₂ denotes l ₂ norm.

In an embodiment, the object is one of a fuel cell bipolar plate, a part of a car, a part of a plane, a part of a building, pipes of a reactor, flow baffles. In an embodiment, the structural parameters comprise at least one of position, radius, width, height, length, anchor points. In an embodiment, the state variables comprise at least one of velocity, pressure, temperature.

Fig. 6 illustrates an exemplary process for collecting a training data set for training a neural boundary operator (NBO) . The NBO is trained to map structural parameters of boundary shapes of an object to approximations of solutions of partial differential equations (PDEs) formulated to characterize a physical system related to the object, wherein the solutions of PDEs correspond to state variables used to describe physical states of the physical system related to the object.

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

At block 620, a first number of sets of state variables are generated by using a numerical solver to solve the PDEs of the object based on the first number of sets of structural parameters.

At block 630, a second number of sets of structural parameters corresponding to a second number of boundary shapes of the object are sampled 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.

At block 640, a second number of sets of state variables are generated by using the numerical solver to solve the PDEs of the object 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.

In an embodiment, the second distribution is determined based further on the optimization objective value of unsampled structural parameters of the object. In an embodiment, the optimization objective value of the unsampled structural parameters of the object is estimated by an estimator, wherein the estimator is trained based on the first number of sets of structural parameters and a first number of corresponding optimization objective values, wherein the first number of corresponding optimization objective values are obtained based on the first number of sets of state variables.

Fig. 7 illustrates an exemplary computing system according to aspects of the disclosure. The computing system 700 may comprise at least one processor 710. The computing system 700 may further comprise at least one storage device 720. The storage device 720 may store computer-executable instructions that, when executed, cause the processor 710 to perform any operations according to the embodiments of the present disclosure as described in connection with Figs. 1-6.

The embodiments of the present disclosure may be embodied in a computer-readable medium such as non-transitory computer-readable medium. The non-transitory computer-readable medium may comprise instructions that, when executed, cause one or more processors to perform any operations according to the embodiments of the present disclosure as described in connection with Figs. 1-6.

The 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 any operations according to the embodiments of the present disclosure as described in connection with Figs. 1-6.

It should be appreciated that all the operations in the methods described above are merely exemplary, and the present disclosure is not limited to any operations in the methods or sequence orders of these operations, and should cover all other equivalents under the same or similar concepts.

It should also be appreciated that all the modules in the apparatuses described above may be implemented in various approaches. These modules may be implemented as hardware, software, or a combination thereof. Moreover, any of these modules may be further functionally divided into sub-modules or combined together.

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 readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other aspects. Thus, the claims are not intended to be limited to the aspects shown herein. All structural and functional equivalents to the elements of the various aspects described throughout the present disclosure that are known or later come to be known to those of ordinary skill in the art are expressly incorporated herein by reference and are intended to be encompassed by the claims.

Claims

A computer implemented method for optimizing structure of an object, comprising:

receiving structural parameters by a neural boundary operator (NBO) , wherein the structural parameters are used to describe boundary shape of the object, and the NBO is trained to map structural parameters of boundary shapes of the object to approximations of solutions of partial differential equations (PDEs) formulated to characterize a physical system related to the object;

generating state variables by the NBO from the structural parameters with a forward pass, wherein the state variables correspond to the solutions of the PDEs and are used to describe physical states of the physical system related to the object;

calculating an optimization objective value based on the state variables; and

updating the structural parameters of the object based on the optimization objective value.
The method of claim 1, further comprising:

repeating the receiving step, the generating step, the calculating step and the updating step by taking the updated structural parameters in the updating step as the structural parameters in the receiving step.
The method of claim 1, wherein the NBO comprises a branch network and a trunk network, the branch network takes the structural parameters as its input and outputs a first set of vectors, the trunk network takes coordinates of the object as its input and outputs a second set of vectors, wherein respective dot products between the first set of vectors and the second set of vectors are taken as the state variables.
The method of claim 1, wherein the NBO comprises a branch network, a trunk network and a port network, the branch network takes the structural parameters as its input and outputs a first set of vectors, the trunk network takes coordinates of the object as its input and outputs a second set of vectors, and the port network takes the first set of vectors and the second set of vectors as input and outputs the state variables.
The method of claim 4, wherein the branch network, the trunk network and the port network are full connection networks.
The method of claim 1, further comprising:

collecting a training data set for training the NBO, wherein the training data set comprises 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

training the NBO based on the training data set.
The method of claim 6, wherein the collecting the training data set further comprising:

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 uniform distribution;

generating a second number of sets of state variables by using a numerical solver to solve the PDEs of the object based on the second number of sets of structural parameters;

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 PDEs of the object based on the third number of sets of structural parameters,

wherein the first number of sets of structural parameters consist of the second number of sets of structural parameters and the third number of sets of structural parameters, the first number of sets of state variables consist of the second number of sets of state variables and the third number of sets of state variables.
The method of claim 7, wherein the numerical solver is a finite element methods (FEM) solver.
The method of claim 7, wherein the second distribution is determined based further on the optimization objective value of unsampled structural parameters of the object.
The method of claim 9, wherein the optimization objective value of the unsampled structural parameters of the object is estimated by an estimator, wherein the estimator is trained based on the second number of sets of structural parameters and a second number of corresponding optimization objective values, wherein the second number of corresponding optimization objective values are obtained based on the second number of sets of state variables.
The method of claim 9, wherein the second distribution p (θ) is determined by means of the following equation:

wherein J (θ) denotes the optimization objective function of the unsampled structural parameters θ,
denotes a balance factor, S _t denotes a set of θ _i sampled so far, ||·|| ₂ denotes l ₂ norm.
The method of claim 1, wherein the object is one of a fuel cell bipolar plate, a part of a car, a part of a plane, a part of a building, pipes of a reactor, flow baffles.
The method of claim 1, wherein the structural parameters comprise at least one of position, radius, width, height, length, anchor points, and the state variables comprise at least one of velocity, pressure, temperature.
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 boundary shapes of an object to approximations of solutions of partial differential equations (PDEs) formulated to characterize a physical system related to the object, wherein the solutions of PDEs correspond to state variables used to describe physical states of the physical system related to 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 uniform distribution;

generating a first number of sets of state variables by using a numerical solver to solve the PDEs of the object 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 PDEs of the object based on the second number of sets of structural parameters,

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.
The method of claim 14, wherein the second distribution is determined based further on the optimization objective value of unsampled structural parameters of the object.
The method of claim 15, wherein the optimization objective value of the unsampled structural parameters of the object is estimated by an estimator, wherein the estimator is trained based on the first number of sets of structural parameters and a first number of corresponding optimization objective values, wherein the first number of corresponding optimization objective values are obtained based on the first number of sets of state variables.
The method of claim 15, wherein the second distribution p (θ) is determined by means of the following equation:

wherein J (θ) denotes the optimization objective function of the unsampled structural parameters θ,
denotes a balance factor, S _t denotes a set of θ _i sampled so far, ||·|| ₂ denotes l ₂ norm.
A computer system, comprising:

one or more processors; and

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