Method and apparatus for structure optimization

Abstract

The present disclosure provides a computer-implemented method for optimizing a structure of an object. The method comprises: receiving a structure quantity through a hard constrained neural network (NN) framework, the structure quantity being used to describe a boundary of the object, the hard constrained NN framework comprising at least one primary NN, at least one secondary NN and a combination unit, and a partial differential equation (PDE) being formulated to characterize a physical system associated with the object with a physical quantity of the object; generating, by the at least one primary NN, a first prediction of a solution of the reformulated PDE; generating a second prediction of a solution to a boundary condition (BC) of the reformulated PDE by using at least the at least one secondary NN; obtaining a final prediction of the solution of the reformulated PDE by combining the first prediction of the solution and the second prediction of the solution by the combining unit; and updating a structure quantity representing a boundary of the object on the basis of the finally predicted physical quantity.

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, while the pressure and velocity affect the performance of the physical system of the wing. Physical systems can be described by partial differential equations (PDEs). Physical quantities of the physical system (such as pressure and velocity) can be obtained by solving PDEs based on structural quantities (such as structural quantities 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 called structural optimization. Structural optimization is applicable to 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 together with control PDEs, which need to be solved at each iteration of the optimization to determine the state of the structure. The efficiency and accuracy of solving PDEs, especially for objects with complex structures, will be critical to improving the performance of structural optimization of the object to be manufactured. Summary of the invention

[0005] In order to improve the performance of structural optimization, the present disclosure proposes a novel framework for structural optimization and solving PDEs, by which the time consumption and computational requirements can be reduced, and the accuracy of the solution of PDEs can be improved, and thus the performance of structural optimization can be improved.

[0006] According to an embodiment, a computer-implemented method for optimizing the structure of an object is provided. The method includes: receiving a structural quantity through a hard-constrained neural network (NN) framework, wherein the structural quantity is used to describe the boundary of the object, wherein the hard-constrained NN framework includes at least one main NN, at least one auxiliary NN, and a combination unit, wherein a partial differential equation (PDE) is formulated to characterize a physical system related to the object using physical quantities of the object; generating a first prediction of a solution of a reformulated PDE through the at least one main NN, wherein the reformulated PDE is obtained by replacing the gradient of the physical quantity in the PDE with an additional quantity, wherein the first prediction of the solution of the reformulated PDE includes the physical quantity of the object and the additional quantity representing the gradient of the physical quantity of the object prediction; generating a second prediction of a solution of a boundary condition (BC) of the reformulated PDE by at least using the at least one auxiliary NN, wherein the BC is formulated to characterize the physical system associated with the boundary of the object, wherein the second prediction of the solution of the BC of the reformulated PDE includes a prediction of the physical quantity of the object; obtaining a final prediction of a solution of the reformulated PDE by combining the first prediction of the solution and the second prediction of the solution by the combination unit, wherein the final prediction of the solution of the reformulated PDE represents a prediction of the physical quantity of the object; and updating the structural quantity representing the boundary of the object based on the physical quantity of the final prediction.

[0007] According to an embodiment, a computer-implemented method for predicting a physical quantity of an object is provided. The method includes: receiving a structural quantity through a hard-constrained neural network (NN) framework, wherein the structural quantity is used to describe the boundary of the object, wherein the hard-constrained NN framework includes at least one main NN, at least one auxiliary NN, and a combination unit, wherein a partial differential equation (PDE) is formulated to characterize a physical system related to the object using the physical quantity of the object; generating a first prediction of a solution of a reformulated PDE through the at least one main NN, wherein the reformulated PDE is obtained by replacing the gradient of the physical quantity in the PDE with an additional quantity, wherein the first prediction of the solution of the reformulated PDE includes the physical quantity of the object and a representation of the a prediction of the additional quantity of the gradient of the physical quantity of the object; generating a second prediction of the solution of the boundary condition (BC) of the reformulated PDE by at least using the at least one auxiliary NN, wherein the BC is formulated to characterize the physical system associated with the boundary of the object, wherein the second prediction of the solution of the BC of the reformulated PDE includes the prediction of the physical quantity of the object; obtaining a final prediction of the solution of the reformulated PDE by combining the first prediction of the solution and the second prediction of the solution by the combination unit, wherein the final prediction of the solution of the reformulated PDE represents the prediction of the physical quantity of the object.

[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 described in 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 described 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 as well as to perform the operations described in the method according to aspects of the present disclosure.

[0011] By using the HCNN framework in the structural optimization framework, the efficiency of structural optimization can be improved while solving the bottleneck of solving PDE. In addition, by using the HCNN framework, the most commonly used BCs (i.e., Dirichlet, Neumann, and Robin BCs) can be automatically satisfied during the training of the NN, so the model can be trained without losing these BCs, which alleviates the unbalanced competition between the loss terms of PDE and BC, and significantly improves the performance of solving geometrically complex PDEs, and thus significantly improves the performance of the structure of the object. Other advantages and enhancements are explained in the description below. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] The disclosed aspects will be described below 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 2 Three types of the most commonly used BCs in accordance with aspects of the present disclosure are shown.

[0015] Figure 3 Exemplary transformations of coordinate systems in accordance with aspects of the present disclosure are shown.

[0016] Figure 4 Exemplary structural optimizations of 2D batteries according to aspects of the present disclosure are shown.

[0017] Figure 5 An exemplary architecture of an HCNN according to aspects of the present disclosure is shown.

[0018] Figure 6 An exemplary structural optimization of a wing according to aspects of the present disclosure is shown.

[0019] Figure 7 An exemplary architecture of an HCNN according to aspects of the present disclosure is shown.

[0020] Figure 8 An exemplary architecture of an HCNN according to aspects of the present disclosure is shown.

[0021] Fig. 9 An exemplary process for optimizing the structure of an object according to aspects of the present disclosure is shown.

[0022] Fig.10 An exemplary process for predicting a physical quantity of an object according to aspects of the present disclosure is shown.

[0023] Fig.11An exemplary computing system in accordance with aspects of the present disclosure is shown. DETAILED DESCRIPTION

[0024] 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.

[0025] 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.

[0026] Figure 1 An exemplary framework for optimizing the structure of an object according to aspects of the present disclosure is shown.

[0027] exist Figure 1 In the example shown, the object to be structurally optimized is an aircraft 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 quantity θ. In this example, the structural quantity 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 label Ω represents the problem domain, or in particular, the domain of the physical system associated with the wing.

[0028] 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 physical quantities of the physical system related to the wing. In this example, the physical quantities can be the velocity u=u(x) and the pressure p=p(x). The label x represents the spatial coordinate in the domain Ω.

[0029] The velocity u=u(x) and the pressure p=p(x) can be characterized by the nonlinear PDE of the Navier-Stokes (NS) equations, where the constraint on the shape of the wing is represented by the boundary 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 formulated according to the following: Figure 1The PDEs are established by the relevant physical laws shown in the label 140 of . Since the PDEs are used to control the optimization of the structure of the wing, they can be called control PDEs. It should be understood that the formulation of the control PDEs can be implemented using known physical knowledge, and other types of PDEs used to describe the physical system of an object can also be used as control PDEs in various aspects of the present disclosure.

[0030] like Figure 1 As shown in label 130, the purpose or goal 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 The objective function J can be formulated as shown in label 130, 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 quantity θ can be iteratively optimized to minimize the objective function J so as to achieve a desired pressure distribution p on the wing surface. ref .

[0031] In each cycle of structural optimization, PDE 140 needs to be solved to obtain physical quantities (such as velocity u and pressure p), and then the objective function J can be evaluated based on the physical quantities (such as pressure p). Then, the structural quantity θ can be updated or optimized based on the calculated objective function J, as shown in label 160.

[0032] In the example, as shown in label 150, the NN model can be used to obtain the velocity u and pressure p by solving the PDE 140. For example, the physically-informed neural network (PINN) (Maziar Raissi, Paris Perdikaris, and George EKarniadakis, Physically-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations, Journal of Computational Physics, 378:686-707, 2019) is one of the most influential works for predicting the solution of PDEs and can be used to implement the NN model 150. The PINN is trained in a manner that uses the residuals of both the PDE and the BC as multiple terms in the loss function. However, there is an unbalanced competition between the terms of the PDE and the BC, resulting in large computational requirements for the PINN when solving PDEs with complex BCs, and the accuracy of the PINN is reduced. One way to solve this problem is to embed the BC into the ansatz so that the neural network can automatically satisfy the BC and there is no need to add the corresponding loss term. However, the challenge is that, such as Figure 2 The equation forms of the complex BCs shown for Neumann BC and Robin BC usually have no analytical solutions and are therefore difficult to embed into the anatomy.

[0033] Figure 2Three types of the most commonly used BCs according to aspects of the present disclosure are shown. In the example of heat transfer shown, T = T (x) is the temperature, k is the thermal conductivity, and h is the heat transfer coefficient. Figure 2 As shown in (a), Dirichlet BC specifies the value of the solution of the PDE at the boundary. Assume a constant temperature T at the heat source (x = x0) H .like Figure 2 As shown in (b), Neumann BC specifies the value of the derivative at the boundary. Assume that the right wall (x = x1) is adiabatic and force the derivative to be zero. Figure 2 As shown in (c), Robin BC is a combination of Dirichlet BC and Neumann BC. And it is used to describe the relationship between the heat source and the environment (T ∞ ) between the heat convection.

[0034] In an embodiment, Figure 1 As shown, a hard-constrained neural network (HCNN) framework 150 is employed in the structural optimization framework 100 to predict solutions to PDE 140 with various boundary shapes represented by the structural quantity θ. HCNN 150 is a unified hard-constrained framework for all three most commonly used BCs (i.e., Dirichlet BC, Neumann BC, and Robin BC). Using HCNN, the assumptions can be constructed to automatically satisfy the three types of BCs. Therefore, the model can be trained without these BC losses, which alleviates the unbalanced competition and significantly improves the accuracy of solving geometrically complex PDEs, and thus significantly improves the efficiency and performance of structural optimization of objects (such as the wing shown).

[0035] PINN is briefly introduced to better understand HCNN according to various aspects of the present disclosure. The following Laplace equation can be considered as an example, Δu(x1,x2)=0,x1∈(0,1],x2∈[0,1](1a) u(x1,x2)=g(x2),x1=0,x2∈[0,1](1b) Where equation (1a) gives the form of the PDE, and equation (1b) is the Dirichlet boundary condition (BC). The solution to the above problem is the solution to equation (1a) that also satisfies equation (1b).

[0036] PINN uses a neural network NN(x1,x2;w) to approximate or predict the solution of PDE, that is, Where w represents the trainable parameters of the neural network. And the parameters w are learned by minimizing the following loss function: The first loss is configured to solve the prediction The constraint is to satisfy the PDE (Equation (1a)), and the second loss term is configured to solve the prediction Restricted to satisfy BC (Equation (1b)), is from the domain [0,1] 2 A set of N samples f configuration points, and is a set of N samples from the boundary x1=0, x2∈[0,1] b Configuration points.

[0037] PINN has a wide range of applications, including thermal, flow, and atmospheric. However, PINN has some problems in terms of performance. Previous analysis has shown that the first loss term The convergence of can be significantly faster than the second loss term This anomaly may lead to nonphysical solutions that do not satisfy the BCs or initial conditions (ICs). Moreover, for geometrically complex PDEs where the number of BCs is large, this problem is exacerbated and can severely affect the accuracy.

[0038] One potential approach to overcome this anomaly is to embed the BC into the anomaly in such a way that any instance from the anomaly can automatically satisfy the BC. In this way, the loss term corresponding to the BC is no longer needed, and thus the anomaly mentioned above is mitigated. Such methods are called hard constraint methods, and they share a similar formulation of the anomaly, as follows: where x is the coordinate, Ω is the domain of interest, It's the border The general solution at is an extended distance function that satisfies the following formula: In equation (1) (where x = (x1, x2), Ω = [0, 1] 2 ), the general solution is exactly g(x2), and the following assumption can be used which automatically satisfies BC in equation (1b).

[0039] However, it is difficult to extend this approach directly to the more general case of Robin BC (see Eq. (7)), since it is difficult to obtain the general solution analytically. Existing attempts are either grid-dependent methods (which are time-consuming for high-dimensional and geometrically complex PDEs) or self-organizing methods for specific (geometrically simple) physical systems. For geometrically complex PDEs and the most commonly used Dirichlet BC, Neumann BC, and Robin BC, there is still a lack of a unified hard constraint framework.

[0040] Consider a geometrically complex domain The following PDE defined on controls the physical system. in It includes N PDE operators that map u to x, functions of u and their derivatives. Here, u(x)=(u1(x),…,u n (x)) is the solution of the PDE, which represents the physical quantity of interest. For each u j , j = 1, ..., n, the appropriate boundary condition (BC) can be proposed as in It's the border The closure of a subset of is disjoint, a i (x) and b i (x) is satisfied right Two functions that hold true, and n(x)=(n1(x),…,n d (x)) is the corresponding boundary γ at point x i The (outward) cell normal of . Note that if a i (x)≡1,b i (x)≡0, then equation (7) represents Dirichlet BC, if a i (x)≡0,b i (x)≡1, then equation (7) represents Neumann BC, otherwise equation (7) represents Robin BC.

[0041] For such geometrically complex PDEs, if PINN is to be adopted, there will be at least The difficulty of multi-task learning with multiple loss terms is that the training convergence will be seriously affected due to the unbalanced competition between those loss terms. If BC can be embedded in the premise, each instance automatically satisfies BC. However, it is not feasible to directly follow the pipeline of hard constraint methods (see Eq. (3)) because Eq. (7) does not have a general solution in analytical form. Therefore, a new method is needed to solve this thorny problem.

[0042] According to aspects of the present disclosure, a general solution of BC is given and used to construct hard constraint anatomy. Additional quantities can be introduced to reformulate the PDE equivalently. Let Substituting into equation (6) and equation (7) to obtain the equivalent PDE: where (u(x),p1(x),…,p n(x)) is the solution to the new PDE, is the reformulated PDE operator. And for j=1,…,n, the corresponding BC is as follows:

[0043] By adding the additional quantities shown in equation (8b), equation (7) has been transformed into j ,p j ) is a linear equation, which makes it easier to derive a general solution. In the following, (u j ,p j )Depend on Represents, and they are on the boundary γ i The general solution at (i.e., )Depend on express.

[0044] According to aspects of the present disclosure, a basis B(x) of the null space can be used to obtain a general solution to Eq. (9), where the dimension of the null space is d. B(x) should be carefully chosen. Since Eq. (9) is parameterized by x, for any x∈γ i , b(x) should always be a basis of the null space, that is, its columns cannot degenerate into linearly dependent vectors, otherwise it will not be able to represent all possible solutions.

[0045] In general, for any dimension Finding a simple expression for the basis is not trivial. According to aspects of the present disclosure, it is preferred to find (d+1) vectors in the null space where d is linearly independent (i.e., ).Then, At the Border i The general solution at can be constructed as: in It has trainable parameters A neural network, and b(x) is given by:

[0046] According to aspects of the present disclosure, a simpler expression for B(x) can be found when d=1 or d=2. In an embodiment, when d=1, a basis of the null space can be found: in

[0047] In an embodiment, when d=2, the basis of the null space can be found: b(x)=[β1(x),β2(x),β3(x)], in

[0048] Using the parameterization of the neural network, equation (10) can express the boundary γ i Any function in , as long as the function satisfies BC (see equation (9)). Since the problem domain contains multiple boundaries, we can combine the corresponding i Therefore, the assumption can be constructed as follows: in is a trainable parameter w main The main neural network. is a continuous expansion distance function (see equation (4)), and α i (i=1,…,m j ) is determined by the following formula: in is a hyperparameter of “hardness” in the spatial domain. Represents except γ i In equation (11), the extended distance function is used to divide the problem domain into several parts, where ( is its learnable part) responsible for the boundary γ i The approximation of , and NN main is responsible for the approximation of the interior separated from the boundary. In addition, equation (12) ensures that The weight at the boundary γ i The nearest neighbor of Make Do not interfere with approximations on other boundaries.

[0049] According to aspects of the present disclosure, if the parameter a in equation (10) and equation (11) i , b i , n or g i Only at the border i , their definitions can be extended to the domain using interpolation or approximation via neural networks As an example, consider the wing boundary (i.e., γ af ). Assume that f(x) is only on the wing boundary γ af The task is to extend its definition to like Figure 3 As shown, two reference points (i.e., x0 and x1) are placed on the front and rear halves of the wing. It can be expressed as polar coordinates about x0 and x1 respectively. The two polar coordinates are concatenated to form a new space. The next interpolation and approximation is performed in the new space. This is because in the new space, it can better characterize the shape of the wing. It should be understood that there are many ways for coordinate transformation, not limited to the examples here. For interpolation, at the wing boundary γ af Sample several points to obtain a data set For any The corresponding expansion f(x) is generated by interpolating in the data set. The interpolation method used here depends on the proposed smoothness requirement. In addition, the number of reference points can also be changed, and it was found in experiments that only one reference point is sufficient.

[0050] Approximation via a neural network is a general approach that does not require manual design. In this case, the wing boundary γ af Sampling several points at Then train the neural network on the data set, namely NN(φ0(x (i) ),φ1(x (i) ))≈f (i) For any NN(φ0(x), φ1(x)) is considered as the corresponding expansion f(x). According to aspects of the present disclosure, the neural network can also be trained in the original space. However, experimental results show that training on the new space can achieve better results. The reason may be that complex geometric shapes become smoother and easier to learn in the new space.

[0051] It is worth noting that in addition to the cases mentioned above, the extended distance function l(x) can also be treated similarly (superscripts are omitted here and see equation (4) for its definition). Because for complex geometries, the distance function may be complex and can be replaced by a cheap alternative model. The methods are similar, including approximating the distance function with a neural network, or constructing a spline function (Hailong Sheng and Chao Yang, Pfnn: A penalty-free neural network method for solving a class of second-order boundary value problems on complex geometries, Journal of Computational Physics, 428:110085, 2021).

[0052] Finally, the HCNN model can be trained using the following loss function: in is defined in equation (11), and is a set of collocation points sampled in the domain Ω. For simplicity, the trainable parameters of the neural network are omitted here. The loss function of equation (13) measures the PDE (i.e., ) and the equilibrium equation introduced by the additional quantity (i.e., Eq. (8b)) in N f The difference at each configuration point.

[0053] According to equation (13), BC is successfully embedded into the anatomy, and it is no longer necessary to use the residual of BC as an additional term in the loss function performed by PINN. That is, the proposed anatomy strictly conforms to BC throughout the training process, thereby greatly reducing the possibility of generating non-physical solutions. However, this is at the expense of introducing (nd) additional equilibrium equations (see equation (8b)). But in many physical systems, especially those with complex geometries, the number of BCs (cnt(BC)) is much larger than (nd) (for example, for classical physical systems, i.e., heat exchangers, n=3, d=2, cnt(BC)=1260,). Therefore, various aspects of the present disclosure actually reduce the number of loss terms by one or two orders of magnitude (Δcnt(loss)=nd-cnt(BCs)<<0), thereby alleviating the unbalanced competition between loss terms.

[0054] According to aspects of the present disclosure, the HCNN framework detailed above can be extended to the spatiotemporal domain. The following time-dependent PDE defined on controls the physical system. Where t is the time coordinate, and the other symbols are the same as those in equations (6) and (7). j ,j=1,…,n, the appropriate boundary condition (BC) can be proposed as: and initial conditions (IC) u j (x,0)=f j (x),x∈Ω (16)

[0055] Following the process described above with reference to equations (9) to (13), At the Border i The general solution at can be constructed as: in is a neural network, and For the sake of brevity, the trainable parameters of the neural network are omitted here.

[0056] Finally, the hypothesis can be constructed as follows: in is an intermediate variable that combines hard constraints in the spatial dimension, is the main neural network, (i=1,…,m j ) is the extended distance function (see equation (4)), α i (i=1,…,m j ) is determined in equation (12), and is a hyperparameter of “hardness” in the time domain.

[0057] Figure 4 is a schematic diagram illustrating structural optimization of a two-dimensional (2D) battery pack according to aspects of the present disclosure.

[0058] like Figure 4 As shown, the battery boundary γ in the domain Ω c,i and cooling tube boundary γ p,i The goal of structural optimization is to optimize the shape and position of the battery and cooling tubes in order to obtain a uniform distribution of temperature over time.

[0059] The structural optimization problem can be expressed as: Where θ represents the structural parameters of the structure of the object to be optimized, which are also parameters of the boundary shape, and Θ represents the space of θ. The structural parameters θ change during the optimization of the structure, but within the design space Θ. J is an objective function whose value measures how good a given structure is, and W represents constraints on both the structural parameters θ and the state variables s=s(x), which represent the state of the physical system, such as velocity, pressure, temperature, etc., where x represents the spatial coordinate. The symbol "≤°" is an element-by-element comparison.

[0060] Figure 4 The structural optimization problem of the 2D battery pack shown can be expressed as follows: J(θ)=∫ t ∫ Ω |T(x)-T ref (x)|dx dt (20) Where T is the temperature, T ref is the reference temperature.

[0061] The control PDE describing the physical system associated with the 2D battery pack can be formulated in terms of the laws of physics. For example, the PDE can be given by: T(x,0)=T0,x∈Ω,(21e)where x=(x1,x2) is the spatial coordinate, t is the time coordinate, T(x,t) is the temperature over time, k is the thermal conductivity and is set to k=1 in this example, h is the heat transfer coefficient and is set to h=1 in this example. T a , T c , T w They are air, battery (n c = 11 with radius r c Battery), cooling tube (n w = 6 with radius r w The temperature of the tube is set to T a =0.1, T c =5, T w =1. T0 is the initial temperature and is set to T0=0.1 in this example. Figure 4 Medium, γ ou represents the outer boundary of the battery pack, γ c,i represents the boundary of the battery, γ p,i Represents the boundaries of the cooling pipes.

[0062] The structural quantities include the center and radius of the battery and cooling tube. In order to optimize the structure of the 2D battery pack based on equation (20), the control PDE needs to be solved to obtain the physical quantity, namely the temperature T.

[0063] According to aspects of the present disclosure, the PDE of Eq. (21a) to Eq. (21e) is reformulated by adding an additional quantity according to the HCNN model. The additional quantity introduced is p(x, t) shown in Eq. (22b), and the reformulated PDE is: k(n(x)·p(x,t))=h(T a -T(x,t)),x∈γ ou ,t∈(0,1], (22c) k(n(x)·p(x,t))=h(T c -T(x,t)),x∈γ c,i ,t∈(0,1],i=1,…,n c , (22d) k(n(x)·p(x,t))=h(T w -T(x,t)),x∈γ p,i ,t∈(0,1],i=1,…,n w , (22e) T(x,0)=T0,x∈Ω, (22f)

[0064] Since the solution of a PDE or reformulated PDE is a scalar function, namely the temperature T(x,t), the solution is represented by u(x,t) = T(x,t). Let Denote (u, p), where p is the additional quantity shown in Eq. (22b). The bounds γ are derived separately. ou , γ c,i , γ p,i The general solution of .

[0065] For any point on the outer boundary, that is, for x∈γ ou , the coefficients of BC shown in equation (9) can be obtained from BC shown in equation (22c), that is, a(x) = h, b(x) = k, g(x) = hT a According to equation (10), the general solution at the outer boundary is Given by: where B(x) is given by equation (10a) or equation (10c), where

[0066] For any point on the battery boundary, that is, for x∈γ c,i , the coefficients of BC shown in equation (9) can be obtained from BC shown in equation (22d), that is, a(x) = h, b(x) = k, g(x) = hT c According to equation (10), the general solution at the battery boundary is Given by:

[0067] For any point on the cooling tube boundary, that is, for x∈γ p,i , the coefficients of BC shown in equation (9) can be obtained from BC shown in equation (22e), that is, a(x) = h, b(x) = k, g(x) = hT w According to equation (10), the general solution at the cooling pipe boundary is Given by:

[0068] The extended distance function in equation (11) It can be implemented in various ways according to various aspects of the present disclosure. In one embodiment, for the battery boundary γ c,i and cooling tube boundary γ p,i Since they are 2D circles, we can use the extended distance function and Choose to be the distance between x and the center minus the radius. For the outer boundary of the rectangle γ ou , assuming it is given by [a1, a2] × [b1, b2], then the extended distance function It can be constructed as follows: Where x = (x1, x2), SoftMin is a differentiable version of the minimum function. In one implementation, SoftMin is implemented by LogSumExp in PyTorch, i.e., SoftMin(y) = LogSumExp(-βy) / (-β), β = 4. And the expansion function is calculated by taking the SoftMin of the distance to all boundaries

[0069] Figure 5 The architecture of an HCNN 500 for predicting a PDE solution for a 2D battery pack is shown in accordance with aspects of the present disclosure.

[0070] HCNN 500 includes a main NN denoted as M-NN 510 and three sub-NNs denoted as S-NN 520-1, 520-2 and 520-3. In one implementation, the main NN is a multi-layer perceptron (MLP) of size [3]+4x

[50] +[3], which means 3 inputs, 4 hidden layers of width 50, and 3 outputs. The three inputs are spatial coordinates x=(x1, x2) and time t, and the three outputs are temperature T and its two derivatives Sub-NNs 520-1, 520-2, and 520-3 are MLPs of size [3]+3×

[20] +[3], which means 3 inputs, 3 hidden layers of width 20, and 3 outputs. The three inputs are spatial coordinates x=(x1, x2) and time t, and the three outputs are temperature T and its two derivatives The main NN 510 is responsible for predicting the PDE solution in the out-of-bounds region of the battery pack, and the child NNs 520-1, 520-2, and 520-3 are responsible for predicting the PDE solution at BC corresponding to equation (22c), equation (22d), and equation (22e), respectively. In one implementation, the hyperparameter of "hardness" is β s =5 and β t =10.

[0071] HCNN 500 also includes a combination unit denoted as AS 530. The combination unit 530 includes a main combination unit denoted as M-AS 530-4 and three sub-combination units denoted as S-AS 530-1, 530-2 and 530-3. S-AS 530-1, 530-2 and 530-3 take the outputs of sub-NNs 520-1, 520-2 and 520-3 as inputs, respectively, and obtain general solutions at the corresponding boundaries of the outer boundary, the battery boundary and the cooling pipe boundary. In one implementation, S-AS 530-1, 530-2 and 530-3 output the general solutions at the three boundaries according to equations (23), (24) and (25), respectively. The main combination unit 530-4 takes the outputs of M-NN 510, S-AS 530-1, 530-2 and 530-3 as input and obtains the solution of the PDE according to equation (11). The solution of the PDE of HCNN 500 is the temperature T and its two derivatives It should be understood that the output of HCNN 500 may be the temperature T only.

[0072] To train HCNN 500, a loss may be obtained based on equation (13), based on which the parameters or weights of the NN of HCNN 500 may be updated. In one implementation, the NN model is trained for, for example, 5000 Adam iterations, followed by L-BFGS optimization until convergence. In one implementation, mean square error (MSE) is used for the loss function, and tanh is used for the activation function.

[0073] After training the HCNN 500, the output of the trained HCNN 500 (i.e., temperature T) is used to optimize the structure of the 2D battery pack. For example, the structure of the 2D battery pack can be optimized based on equation (20). In one implementation, in each iteration of the structural optimization of an object (such as a 2D battery pack), the HCNN 500 is trained based on the current structural quantities (such as the position and radius of the battery and cooling pipe). Therefore, the improvement in the efficiency and accuracy of the HCNN significantly enhances the performance of the structural design process of an object (such as a 2D battery pack, etc.).

[0074] Figure 6 is a schematic diagram illustrating structural optimization of a wing according to aspects of the present disclosure.

[0075] like Figure 6 As shown, the wing boundary γ is located in the domain Ω af is the structure to be optimized. The goal of structural optimization is to achieve the desired pressure distribution p on the wing surface by changing the structural parameters θ ref .

[0076] Figure 6The structural optimization problem of the wing shown can be expressed as follows: Where p is the pressure, p ref is the reference pressure.

[0077] The control PDE describing the physical system associated with the wing can be formulated in terms of the laws of physics. For example, the PDE can be the NS equations and can be given by: u(x)=u0(x),x∈γ il ∪γ tp ∪γ bt , (29c) p(x)=1,x∈γ ol , (29d) n(x)·u(x)=0,x∈γ af , (29e)where x=(x1,x2) is the spatial coordinate, u(x)=(u1(x),u2(x)) is the velocity of the fluid, p(x) is the pressure of the fluid, and v is the viscosity of the fluid and is set to 1 / 50 in this example. u0(x)=(1,0). is the Jacobian matrix, exist Figure 6 Medium, γ il represents the entry boundary of the domain, γ tp represents the top border, γ bt represents the bottom boundary, γ ol represents the exit boundary, γ af Represents the wing boundary.

[0078] The structural quantity includes the boundary γ af In order to optimize the structure of the wing based on equation (28), it is necessary to solve the control PDE to obtain the physical quantities, namely, the velocity u and the pressure p.

[0079] According to aspects of the present disclosure, the PDEs of Eqs. (29a) to (29e) are reformulated by adding additional quantities according to the HCNN model. The additional quantities introduced are shown in Eqs. (30c) and (30d). And the reformulated PDE is: u(x)=u0(x),x∈γ il ∪γ tp ∪γ bt , (30e) p(x)=1,x∈γ ol , (30f) n(x)·u(x)=0,x∈γ af , (30g)

[0080] The solutions to the PDE or reformulated PDE are velocity and pressure (u(x), p(x)). For pressure p(x), the outlet boundary γ ol The general solution in is The assumption for the pressure p is given by: Where [3] means taking NN main The third element of the output of (x).

[0081] For velocity u(x), the inlet boundary, top boundary, and bottom boundary γ*=γ il ∪γ tp ∪γ bt The general solution in is exactly u γ* (x) = u0(x). Wing boundary γ af The general solution at is given by: Where, according to equation (10b), B(x) = [n2(x), -n1(x)] T , and NN γ* The output of (x) is a scalar.

[0082] According to equation (11), we can combine u γ* and The velocity u can be obtained by the following formula: Where [1:2] means taking NN main (x), and as well as is defined similarly to equation (12).

[0083] The extended distance function in equation (11) This can be implemented in various ways according to various aspects of the present disclosure. In one implementation, for the wing boundary, obtain The direct way is to calculate x and wing γ af However, due to the wing boundary γ af is complex, so this may be time consuming. In another implementation, the true distance can be approximated using an MLP. For example, Figure 6A NN with 3 hidden layers of width 30 is trained using 1024×6 points sampled from the domain Ω shown. A portion of these points are sampled in the bounding box of the wing, and the rest are sampled in Ω along with their true distances, which can be calculated using the formula for the distance to the polygon.

[0084] For the export boundary γ ol , which is a vertical line, for example, the exit boundary is x1 = a. The expansion distance function can be calculated as Where x=(x1,x2).

[0085] For the set of entrance boundary, top boundary and bottom boundary γ * =γ il ∪γ tp ∪γ bt , boundary γ * is an open rectangle, so the extended distance function can be constructed similarly to the case of a rectangle as shown in equation (26) The right side is ignored. In addition, the distance to all boundaries is calculated by taking the SoftMin As shown in equation (27).

[0086] Figure 7 The architecture of an HCNN 700 for predicting a PDE solution for an airfoil in accordance with aspects of the present disclosure is shown.

[0087] HCNN 700 includes a main NN denoted as M-NN 710 and a sub-NN denoted as S-NN 720. In one implementation, the main NN is a multi-layer perceptron (MLP) of size [2]+6x

[50] +[7], which means 2 inputs, 6 hidden layers of width 50, and 7 outputs. The two inputs are the spatial coordinates x=(x1, x2), and the seven outputs are velocity u=(u1, u2), pressure p, and two derivatives of u1 and two derivatives of u2. Sub-NN 720 is an MLP of size [2]+4x

[40] +[1], which means 2 inputs, 4 hidden layers of width 40, and 1 output. As shown in equations (30e), (30f), and (30g), BC does not include the additional quantities p1(x) and p2(x), so sub-NN 720 does not need to output the corresponding terms. The BCs of equations (30e) and (30f) provide constant velocity and pressure at the corresponding boundaries, so the sub-NN 720 does not need to provide outputs for these BCs. For the BCs of equation (30g), by using As a basis for the solution, the sub-NN 720 only needs to provide a scalar, as shown in equation (32). The main NN 710 is responsible for predicting the PDE solution in the out-of-bounds region of the wing, and the sub-NN 720 is responsible for predicting the PDE solution at BC corresponding to equation (30g). And the PDE solution at BC corresponding to equation (30e) and equation (30f) can be obtained analytically. In one implementation, the hyperparameter of "hardness" is β s =5.

[0088] HCNN 700 also includes a combining unit denoted as AS 730. Combining unit 730 includes a main combining unit denoted as M-AS 730-2 and a sub-combining unit denoted as S-AS 730-1. S-AS 730-1 takes the output of sub-NN 720 as input and obtains the corresponding wing boundary γ af The general solution at the boundary is output by S-AS 730-1 according to equation (30g) or equation (32). The main combination unit 730-2 takes the outputs of the M-NN 710 and the S-AS 730-1 as inputs and obtains the solution of the PDE according to equation (11) or according to equations (31) and (33). The solution of the PDE of the HCNN 700 is the velocity u = (u1, u2), the pressure p and the derivative of the velocity It should be understood that the output of HCNN 700 may be only velocity u=(u1, u2) and pressure p.

[0089] To train HCNN 700, a loss may be obtained based on equation (13), based on which the parameters or weights of the NN of HCNN 700 may be updated. In one implementation, the NN model is trained for, for example, 5000 Adam iterations, followed by L-BFGS optimization until convergence. In one implementation, mean square error (MSE) is used for the loss function, and tanh is used for the activation function.

[0090] After training the HCNN 700, the output of the trained HCNN 700 (i.e., temperature T) is used to optimize the structure of the wing. For example, the shape of the wing can be optimized based on equation (28). In one implementation, at each iteration of structural optimization of an object (such as a wing), the HCNN 700 is trained based on the current structural quantity (such as a set of control points consisting of a spline curve representing the shape of the wing). Therefore, the improvement in the efficiency and accuracy of the HCNN significantly enhances the performance of the structural design process for objects such as wings.

[0091] Figure 8 The architecture of an HCNN 800 for predicting a PDE solution for an object to be designed or manufactured is shown in accordance with aspects of the present disclosure.

[0092] HCNN 800 includes at least one main NN denoted as M-NN 810-1 to M-NN 810-m, and at least one sub-NN denoted as S-NN 820-1 to S-NN 820-n. It should be understood that although one main NN can be used to predict all solutions u=(u1,…,u n ), but it is feasible to use multiple main NNs to predict the corresponding parts of all solutions. One or more main NNs 810-1 to 810-m are responsible for predicting the PDE solutions in the area outside the boundary of the object, and one or more sub-NNs 820-1 to 820-n are respectively responsible for predicting the PDE solutions at the corresponding boundaries according to the corresponding BC.

[0093] HCNN 800 also includes a combination unit denoted as AS 830. The combination unit 830 includes a main combination unit denoted as M-AS 830-k and at least one sub-combination unit denoted as S-AS 830-1 to S-AS 830-n. One or more S-AS 830-1 to 830-n respectively take the output of sub-NN 820-1 to 820-n as input and obtain a general solution at the corresponding boundary of the object. The main combination unit 830-k takes the output of one or more main NNs 810-1 to 810-m and one or more sub-AS 830-1 to S-AS 830-n as input and obtains the solution of the PDE according to equation (11).

[0094] To train HCNN 800, a loss may be obtained based on equation (13), based on which the parameters or weights of the NN of HCNN 800 may be updated. In one implementation, the NN model is trained, for example, for a defined number of Adam iterations, followed by L-BFGS optimization, until convergence.

[0095] After training HCNN 800, the output of trained HCNN 800 (i.e., the solution u = (u1, ..., u n ) to optimize the structure of the object. Therefore, the improved efficiency and accuracy of HCNN significantly enhances the performance of the structural design process of the object to be manufactured.

[0096] Fig. 9 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.

[0097] At block 910, a structural quantity is received by the HCNN framework, wherein the structural quantity is used to describe the boundary of an object. The HCNN includes at least one main NN, at least one auxiliary NN, and a combination unit, wherein a PDE is formulated to characterize a physical system associated with the object using the physical quantity of the object.

[0098] At block 920, a first prediction of a solution to the reformulated PDE is generated by at least one master NN. The reformulated PDE is obtained by replacing the gradient of a physical quantity in the PDE with an additional quantity, wherein the first prediction of the solution to the reformulated PDE includes the physical quantity of the object and a prediction of the additional quantity representing the gradient of the physical quantity of the object.

[0099] At block 930, a second prediction of a solution to the BC of the reformulated PDE is generated by using at least the at least one auxiliary NN. The BC is formulated to characterize a physical system associated with a boundary of the object, wherein the second prediction of the solution to the BC of the reformulated PDE includes a prediction of a physical quantity of the object.

[0100] At block 940, a final prediction of the solution of the reformulated PDE is obtained by combining the first prediction of the solution and the second prediction of the solution by a combining unit. The final prediction of the solution of the reformulated PDE represents a prediction of a physical quantity of the object.

[0101] At block 950 , a structural quantity representing a boundary of the object is updated based on the final predicted physical quantity.

[0102] According to an embodiment, the BC of the boundary of the object includes one or more of Dirichlet BC, Neumann BC, and Robin BC.

[0103] According to an embodiment, the primary NN corresponds to an out-of-boundary region of the object, and at least one secondary NN corresponds to at least one of the boundaries of the object.

[0104] According to an embodiment, the second prediction of the solution of the BC of the reformulated PDE also includes a prediction of an additional quantity representing the gradient of the physical quantity of the object.

[0105] According to an embodiment, generating a second prediction of a solution of the BC of the reformulated PDE at box 930 also includes: for each of at least one boundary corresponding to at least one auxiliary NN, generating a corresponding part of the second prediction based on the constraints of the BC of the boundary and the output of the corresponding auxiliary NN.

[0106] According to an embodiment, generating a second prediction of a solution to the BC of the reformulated PDE at box 930 also includes: for each of at least one boundary corresponding to at least one auxiliary NN, further generating a corresponding part of the second prediction based on a basis matrix or vector of the null space of the boundary, wherein the basis matrix or vector is obtained based on the unit normal of the boundary.

[0107] According to an embodiment, obtaining a final prediction of a solution to the reformulated PDE at block 940 further includes combining the first prediction of the solution and the second prediction of the solution based on distances of the sample points of the object to a boundary of the object.

[0108] According to an embodiment, combining the first prediction of the solution and the second prediction of the solution at box 940 also includes: combining corresponding portions of the second prediction corresponding to the boundaries of the object by ensuring that the portions of the second prediction corresponding to any one boundary are attenuated at other boundaries to an extent that has no significant effect on the other boundaries.

[0109] According to an embodiment, process 900 also includes: obtaining a first difference of the reformulated PDE based on a final prediction of a solution including a physical quantity of the object and a gradient of the physical quantity; obtaining a second difference between the physical quantity and the gradient of the physical quantity in the final prediction of the solution; obtaining a loss based on the first difference and the second difference; and updating at least one main NN and at least one auxiliary NN based on the loss.

[0110] According to an embodiment, the steps of generating a first prediction, generating a second prediction, obtaining a final prediction, obtaining a loss, and updating at least one main NN and at least one auxiliary NN are repeated until the training requirements are met.

[0111] According to an embodiment, at least one primary NN and at least one secondary NN are MLP NNs.

[0112] According to an embodiment, the object is one of a fuel cell bipolar plate to be manufactured or designed, a component of an automobile, a component of an aircraft, a part of a building, a pipe of a reactor, a flow baffle, etc.

[0113] According to an embodiment, the structural quantity includes at least one of position, radius, width, height, length, anchor point, etc. The physical quantity includes at least one of speed, pressure, temperature, etc.

[0114] Fig.10 An exemplary process for predicting a physical quantity 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.

[0115] At block 1010, a structural quantity is received by the HCNN framework, wherein the structural quantity is used to describe the boundary of an object. The HCNN includes at least one main NN, at least one auxiliary NN, and a combination unit, wherein a PDE is formulated to characterize a physical system associated with the object using the physical quantity of the object.

[0116] At block 1020, a first prediction of a solution to the reformulated PDE is generated by at least one master NN. The reformulated PDE is obtained by replacing the gradient of a physical quantity in the PDE with an additional quantity, wherein the first prediction of the solution to the reformulated PDE includes the physical quantity of the object and a prediction of the additional quantity representing the gradient of the physical quantity of the object.

[0117] At block 1030, a second prediction of a solution to the BC of the reformulated PDE is generated by using at least the at least one auxiliary NN. The BC is formulated to characterize a physical system associated with a boundary of the object, wherein the second prediction of the solution to the BC of the reformulated PDE includes a prediction of a physical quantity of the object.

[0118] At block 1040, a final prediction of the solution of the reformulated PDE is obtained by combining the first prediction of the solution and the second prediction of the solution by a combining unit. The final prediction of the solution of the reformulated PDE represents a prediction of a physical quantity of the object.

[0119] According to an embodiment, the BC of the boundary of the object includes one or more of Dirichlet BC, Neumann BC, and Robin BC.

[0120] According to an embodiment, the primary NN corresponds to an out-of-boundary region of the object, and at least one secondary NN corresponds to at least one of the boundaries of the object.

[0121] According to an embodiment, the second prediction of the solution of the BC of the reformulated PDE also includes a prediction of an additional quantity representing the gradient of the physical quantity of the object.

[0122] According to an embodiment, generating a second prediction of a solution of the BC of the reformulated PDE at box 1030 also includes: for each boundary of at least one boundary corresponding to at least one auxiliary NN, generating a corresponding part of the second prediction based on the constraints of the BC of the boundary and the output of the corresponding auxiliary NN.

[0123] According to an embodiment, generating a second prediction of a solution to the BC of the reformulated PDE at box 1030 also includes: for each of at least one boundary corresponding to at least one auxiliary NN, further generating a corresponding part of the second prediction based on a basis matrix or vector of the null space of the boundary, wherein the basis matrix or vector is obtained based on the unit normal of the boundary.

[0124] According to an embodiment, obtaining a final prediction of a solution to the reformulated PDE at block 1040 further includes combining the first prediction of the solution and the second prediction of the solution based on distances of the sample points of the object to a boundary of the object.

[0125] According to an embodiment, combining the first prediction of the solution and the second prediction of the solution at box 1040 also includes: combining corresponding portions of the second prediction corresponding to the boundaries of the object by ensuring that the portions of the second prediction corresponding to any one boundary are attenuated at other boundaries to an extent that has no significant impact on the other boundaries.

[0126] According to an embodiment, process 1000 also includes: obtaining a first difference of the reformulated PDE based on a final prediction of a solution including a physical quantity of the object and a gradient of the physical quantity; obtaining a second difference between the physical quantity and the gradient of the physical quantity in the final prediction of the solution; obtaining a loss based on the first difference and the second difference; and updating at least one main NN and at least one auxiliary NN based on the loss.

[0127] According to an embodiment, the steps of generating a first prediction, generating a second prediction, obtaining a final prediction, obtaining a loss, and updating at least one main NN and at least one auxiliary NN are repeated until the training requirements are met.

[0128] According to an embodiment, at least one primary NN and at least one secondary NN are MLP NNs.

[0129] According to an embodiment, the object is one of a fuel cell bipolar plate to be manufactured or designed, a component of an automobile, a component of an aircraft, a part of a building, a pipe of a reactor, a flow baffle, etc.

[0130] According to an embodiment, the structural quantity includes at least one of position, radius, width, height, length, anchor point, etc. The physical quantity includes at least one of speed, pressure, temperature, etc.

[0131] Fig.11 1 shows an exemplary computing system according to various aspects of the present disclosure. The computing system 1100 may include at least one processor 1110. The computing system 1100 may also include at least one storage device 1120. The storage device 1120 may store computer executable instructions that, when executed, cause the processor 1110 to perform operations according to the embodiment of the present disclosure. Figure 1-10 Any operation of the embodiments of the present disclosure described.

[0132] 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 1-10 Any operation of the embodiments of the present disclosure described.

[0133] Embodiments of the present disclosure may be embodied in a computer program product including computer executable instructions that, when executed, cause one or more processors to perform operations according to the embodiment of the present disclosure. Figure 1-10 Any operation of the embodiments of the present disclosure described.

[0134] It should be understood that all operations in the above methods are exemplary only, 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.

[0135] 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.

[0136] 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 in the future to those of ordinary skill in the art.

Claims

1. A computer-implemented method for optimizing a structure of an object, comprising: Receiving a structural quantity through a hard-constrained neural network (NN) framework, wherein the structural quantity is used to describe the boundary of the object, wherein the hard-constrained NN framework includes at least one main NN, at least one auxiliary NN, and a combination unit, wherein a partial differential equation (PDE) is formulated to characterize a physical system associated with the object using physical quantities of the object; generating, by the at least one master NN, a first prediction of a solution to a reformulated PDE obtained by replacing a gradient of the physical quantity in the PDE with an additional quantity, wherein the first prediction of the solution to the reformulated PDE includes a prediction of the physical quantity of the object and the additional quantity representing a gradient of the physical quantity of the object; generating a second prediction of a solution to a boundary condition (BC) of the reformulated PDE by using at least the at least one auxiliary NN, wherein the BC is formulated to characterize the physical system associated with the boundary of the object, wherein the second prediction of the solution to the BC of the reformulated PDE comprises a prediction of the physical quantity of the object; obtaining a final prediction of a solution to the reformulated PDE by combining the first prediction of a solution and the second prediction of a solution by the combining unit, wherein the final prediction of the solution to the reformulated PDE represents a prediction of the physical quantity of the object; and The structural quantity representing the boundary of the object is updated based on the final predicted physical quantity.

2. The method according to claim 1, wherein: The BC of the boundary of the object includes one or more of Dirichlet BC, Neumann BC, and Robin BC.

3. The method according to claim 1, wherein: The primary NN corresponds to an out-of-bounds region of the object, wherein the at least one secondary NN corresponds to at least one of the boundaries of the object.

4. The method according to claim 1, wherein: The second prediction of the solution to the BC of the reformulated PDE also includes a prediction of the additional quantity representing the gradient of the physical quantity of the object.

5. The method according to claim 2, wherein: Generating a second prediction of a solution to the BC of the reformulated PDE further comprises: For each of at least one boundary corresponding to the at least one auxiliary NN, a corresponding portion of the second prediction is generated based on the constraint of the BC of the boundary and an output of the corresponding auxiliary NN.

6. The method according to claim 5, wherein: Generating a second prediction of a solution of the BC of the reformulated PDE further comprises: For each of the at least one boundary corresponding to the at least one auxiliary NN, the corresponding part of the second prediction is further generated based on a basis matrix or vector of a null space of the boundary, wherein the basis matrix or vector is obtained based on a unit normal of the boundary.

7. The method according to claim 2, wherein: Obtaining a final prediction of a solution of the reformulated PDE by combining the first prediction of a solution and the second prediction of a solution by the combining unit further comprises: The first prediction of a solution and the second prediction of a solution are combined based on distances of sampling points of an object to the boundary of the object.

8. The method according to claim 7, wherein: Combining the first prediction of the solution and the second prediction of the solution also includes: combining corresponding portions of the second prediction corresponding to the boundaries of the object by ensuring that portions of the second prediction corresponding to any one boundary are attenuated at other boundaries to an extent that has no significant effect on the other boundaries.

9. The method according to claim 2, further comprising: obtaining a first difference of the reformulated PDE based on the final prediction of a solution including the physical quantity of the object and the gradient of the physical quantity; obtaining a second difference between the physical quantity in the final prediction of a solution and the gradient of the physical quantity; obtaining a loss based on the first difference and the second difference; as well as The at least one primary NN and the at least one secondary NN are updated based on the loss.

10. The method according to claim 9, further comprising: Repeat the steps of generating a first prediction, generating a second prediction, obtaining a final prediction, obtaining a loss, and updating the at least one main NN and the at least one auxiliary NN until the training requirements are met.

11. The method according to claim 1, wherein: The at least one primary NN and the at least one secondary NN are multi-layer perceptron (MLP) NNs.

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

13. The method according to claim 1, wherein: The structural quantity includes at least one of position, radius, width, height, length, and anchor point, and the physical quantity includes at least one of speed, pressure, and temperature.

14. A computer-implemented method for predicting a physical quantity of an object, comprising: Receiving a structural quantity through a hard-constrained neural network (NN) framework, wherein the structural quantity is used to describe the boundary of the object, wherein the hard-constrained NN framework includes at least one main NN, at least one auxiliary NN, and a combination unit, wherein a partial differential equation (PDE) is formulated to characterize a physical system associated with the object using physical quantities of the object; generating, by the at least one master NN, a first prediction of a solution to a reformulated PDE obtained by replacing a gradient of the physical quantity in the PDE with an additional quantity, wherein the first prediction of the solution to the reformulated PDE includes a prediction of the physical quantity of the object and the additional quantity representing a gradient of the physical quantity of the object; generating a second prediction of a solution to a boundary condition (BC) of the reformulated PDE by using at least the at least one auxiliary NN, wherein the BC is formulated to characterize the physical system associated with the boundary of the object, wherein the second prediction of the solution to the BC of the reformulated PDE comprises a prediction of the physical quantity of the object; A final prediction of a solution to the reformulated PDE is obtained by combining the first prediction of a solution and the second prediction of a solution by the combining unit, wherein the final prediction of the solution to the reformulated PDE represents a prediction of the physical quantity of the object.

15. The method according to claim 14, wherein: The BC of the boundary of the object includes one or more of Dirichlet BC, Neumann BC, and Robin BC.

16. The method according to claim 15, wherein: Generating a second prediction of a solution to the BC of the reformulated PDE further comprises: For each of the at least one boundary corresponding to the at least one auxiliary NN, a corresponding portion of the second prediction is generated based on the constraint of the BC of the boundary and an output of the corresponding auxiliary NN.

17. The method according to claim 15, wherein: Obtaining a final prediction of a solution of the reformulated PDE by combining the first prediction of a solution and the second prediction of a solution by the combining unit further comprises: The first prediction of a solution and the second prediction of a solution are combined based on distances of sampling points of an object to the boundary of the object.

18. The method according to claim 13, further comprising: obtaining a first difference of the reformulated PDE based on the final prediction of a solution including the physical quantity of the object and the gradient of the physical quantity; obtaining a second difference between the physical quantity in the final prediction of a solution and the gradient of the physical quantity; obtaining a loss based on the first difference and the second difference; as well as The at least one primary NN and the at least one secondary NN are updated based on the loss.

19. 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 according to any one of claims 1-18.

20. 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 any one of claims 1-18.

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