METHOD AND EQUIPMENT FOR STRUCTURAL OPTIMIZATION

Abstract

The present disclosure provides a computer-implemented method for optimizing the structure of an object. The method comprises: receiving structural quantities by a framework for a constraint-bounded neural network (NN), wherein the structural quantities are used to describe boundaries of the object, the constraint-bounded NN framework comprising at least one primary NN, at least one secondary NN, and an assembly unit, wherein partial differential equations (PDEs) are formulated to characterize a physical system with respect to the object using physical quantities of the object; generating a first prediction of solutions of reformulated PDEs by the at least one primary NN; generating a second prediction of solutions of boundary conditions (BCs) of the reformulated PDEs by using at least one secondary NN.Obtaining a final prediction of solutions of the reformulated PDEs by joining the first prediction of a solution and the second prediction of a solution using the joining unit and updating the structural quantities representing the boundaries of the object, based on the physical quantities of the final prediction.

Description

AREA

[0001] The aspects of this disclosure generally relate to artificial intelligence (AI) and, in particular, to the optimization of the structure of an object. BACKGROUND

[0002] The structure of an object, such as its shape, size, or material distribution, influences its performance. Examples of such objects include the wing of an airplane, the framework of a house, the pipes in a reactor, and so on. For instance, the shape of an airfoil affects the pressure and velocity of the airflow with respect to the blade, while the pressure and velocity affect the performance of the physical system of the airfoil. This physical system can be described by partial differential equations (PDEs). The physical quantities of the system, such as pressure and velocity, can be obtained by solving the PDEs based on the structural quantities, such as those describing the object's shape.

[0003] It is desirable to design the structure of an object in such a way as to improve its performance; this procedure can be called structural optimization. Structural optimization is applicable in many fields, including science, engineering, industry, and the like. For example, the shape of chemical catalyst pellets, the shape of automotive parts, or similar items can be optimized before manufacturing using structural optimization procedures.

[0004] Since most physical systems are described by partial differential equations (PDEs), structural optimization is typically performed using the relevant PDEs, which must be solved to determine the state of the structure at each optimization iteration. The efficiency and accuracy of solving PDEs, especially for an object with a complex structure, would be crucial for improving the performance of the structural optimization of the object being manufactured. SUMMARY

[0005] To improve the performance of structural optimization, the disclosure proposes a novel framework for structural optimization and the solution of PDEs, which can reduce the time consumption and computational requirements and improve the accuracy of the solution of PDEs and, consequently, the performance of structural optimization.

[0006] According to one embodiment, a computer-implemented method for optimizing the structure of an object is provided. The method comprises: receiving structural quantities through a constraint-boundary neural network (NN) framework, wherein the structural quantities are used to describe the boundaries of the object, wherein the constraint-boundary NN framework comprises at least one primary NN, at least one secondary NN, and an assembly unit, wherein partial differential equations (PDEs) are formulated to characterize a physical system with respect to the object, using physical quantities of the object;Generating a first prediction of solutions of reformulated PDEs by the at least one primary neural network, wherein the reformulated PDEs are obtained by replacing gradients of the physical quantities in the PDEs with additional quantities, the first prediction of the solutions of the reformulated PDEs comprising a prediction of the physical quantities of the object and the additional quantities that represent gradients of the physical quantities of the object; Generating a second prediction of solutions of boundary conditions (BCs) of the reformulated PDEs by at least the use of the at least one secondary neural network, wherein the BCs are formulated to characterize the physical system with respect to the boundaries of the object, the second prediction of the solutions of the BCs of the reformulated PDEs comprising a prediction of the physical quantities of the object;Obtaining a final prediction of solutions of the reformulated PDEs by joining the first prediction of a solution and the second prediction of a solution by the joining unit, wherein the final prediction of solutions of the reformulated PDEs represents a prediction of the physical quantities of the object; and updating the structural quantities representing the boundaries of the object based on the physical quantities of the final prediction.

[0007] According to one embodiment, a computer-implemented method for predicting the physical quantities of an object is provided. The method comprises: receiving structural quantities through a constraint-boundary neural network (NN) framework, wherein the structural quantities are used to describe the boundaries of the object, the constraint-boundary NN framework comprising at least one primary NN, at least one secondary NN, and an assembly unit, and wherein partial differential equations (PDEs) are formulated to characterize a physical system with respect to the object, using the object's physical quantities.Generating a first prediction of solutions of reformulated PDEs by the at least one primary neural network, wherein the reformulated PDEs are obtained by replacing gradients of the physical quantities in the PDEs with additional quantities, the first prediction of the solutions of the reformulated PDEs comprising a prediction of the physical quantities of the object and the additional quantities that represent gradients of the physical quantities of the object; Generating a second prediction of solutions of boundary conditions (BCs) of the reformulated PDEs by at least the use of the at least one secondary neural network, wherein the BCs are formulated to characterize the physical system with respect to the boundaries of the object, the second prediction of the solutions of the BCs of the reformulated PDEs comprising a prediction of the physical quantities of the object;Obtaining a final prediction of solutions of the reformulated PDEs by combining the first prediction of a solution and the second prediction of a solution by the combining unit, wherein the final prediction of solutions of the reformulated PDEs represents a prediction of the physical quantities of the object.

[0008] According to one embodiment, a computer system is provided comprising one or more processors and one or more storage devices that store computer-executable instructions which, when executed, cause the one or more processors to perform the operations of the method as mentioned above, as well as to carry out the operations of the method according to the aspects of the disclosure.

[0009] According to one embodiment, one or more computer-readable storage media are provided which store computer-executable instructions which, when executed, cause one or more processors to perform the operations of the method as mentioned above, as well as to carry out the operations of the provided method in accordance with the aspects of the disclosure.

[0010] According to one embodiment, a computer program product is provided that stores computer-executable instructions which, when executed, cause one or more processors to perform the operations of the method as mentioned above, as well as to carry out the operations of the provided method in accordance with the disclosure.

[0011] By using the HCNN framework within the structure optimization framework, the efficiency of structure optimization can be improved while overcoming the bottleneck in solving PDEs. Furthermore, the most commonly used BCs (i.e., Dirichlet, Neumann, and Robin BCs) can be automatically satisfied during NN training by using the HCNN framework, allowing the model to be trained without loss of these BCs. This mitigates the unbalanced competition between the lossy concepts of PDEs and BCs, significantly improving performance in solving geometrically complex PDEs and, consequently, the overall performance of an object's structure. Further advantages and improvements are explained in the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] The disclosed aspects are described below in conjunction with the accompanying drawings, which are provided to illustrate, not limit, the disclosed aspects. Fig. Figure 1 illustrates an exemplary framework for optimizing the structure of an object according to the principles of revelation. Fig. 2 illustrates three types of the most commonly used BCs according to the points of revelation. Fig. Figure 3 illustrates an exemplary transformation of coordinate systems according to the principles of revelation. Fig. Figure 4 illustrates an exemplary structural optimization of a 2D battery pack according to the aspects of the disclosure. Fig. Figure 5 illustrates an exemplary architecture of an HCNN according to the principles of revelation. Fig. Figure 6 illustrates an exemplary structural optimization of a paddle blade according to the aspects of the revelation. Fig. Figure 7 illustrates an exemplary architecture of an HCNN according to the principles of revelation. Fig. Figure 8 illustrates an exemplary architecture of an HCNN according to the principles of revelation. Fig. Figure 9 illustrates an exemplary process for optimizing the structure of an object according to the principles of revelation. Fig. 10 illustrates an exemplary process for predicting the physical quantities of an object according to the principles of revelation. Fig. 11 illustrates an exemplary calculation system according to the principles of revelation. DETAILED DESCRIPTION

[0013] The present disclosure will now be discussed with reference to several exemplary implementations. It is understood that these implementations are discussed only to enable the person skilled in the art to better understand and thus implement the embodiments of the present disclosure, and not to suggest any limitations to the scope of protection afforded by the present disclosure.

[0014] Various embodiments are described in detail with reference to the accompanying drawings. Wherever possible, the same reference numerals are used in the drawings to indicate identical or similar parts. References made to specific examples and embodiments are for illustrative purposes only and are not intended to limit the scope of protection of the disclosure.

[0015] Fig. Figure 1 illustrates an exemplary framework for optimizing the structure of an object according to the principles of revelation.

[0016] The object that is in the Fig. The first example shown, which is to be structurally optimized, is a paddle blade. As shown in Block 120 of Fig. As shown in Figure 1, the shape of the airfoil is represented by a boundary γ, which is parameterized by structural quantities θ. In this example, the structural quantities θ can be a set of control points consisting of splines that represent 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 notation Ω denotes the problem domain or, more specifically, the domain of the physical system with respect to the airfoil.

[0017] As in Block 110 of Fig. As shown in Figure 1, the arrow lines represent airflows on the airfoil. The physical state of the airflow on the airfoil can be represented by physical quantities of the physical system with respect to the airfoil. In this example, the physical quantities can be the velocity u = u(x) and the pressure p = p(x). The term x denotes spatial coordinates in the region Ω.

[0018] The velocity u=u(x) and the pressure p=p(x) can be characterized by nonlinear PDEs of the Navier-Stokes equations (NS equations), where the constraint on the blade shape is represented by the boundary γ. It is important to note that NS equations are well-known physical equations that can be used to describe the three-dimensional motion of viscous fluids. The NS equations are second-order nonlinear PDEs and can be used to model weather, ocean currents, heat conduction, airflow around a blade, water flow in a pipe or reactor, and in many other applications. The PDEs of the NS equations for a physical system of an object can be formulated according to the physical laws relating to that physical system.In the illustrated example, the PDEs of the physical system of the blade can be set up according to the associated physical law, as indicated by the designation 140 of . Fig. Figure 1 shows that since the PDEs are used to control the optimization of the blade structure, they can be called controlling PDEs. It should be noted that the formulation of the controlling PDEs can be implemented using known physical knowledge, and that other types of PDEs used to describe a physical system of an object can also be used as controlling PDEs under certain aspects of the disclosure.

[0019] As indicated by the designation 130 of Fig. As shown in Figure 1, the task or goal of structural optimization of the airfoil is to achieve the desired pressure distribution p. refon the blade surface by changing the structural parameter θ. The objective function J can be formulated as shown by designation 130, and the purpose of the structural optimization is to minimize the objective function J, which is a function of the pressure p. The shape of the blade, corresponding to the boundary γ parameterized by structural quantities θ, can be iteratively optimized to minimize the objective function J and thus achieve the desired pressure distribution p. ref to be achieved on the blade surface.

[0020] In each loop of the structural optimization, the PDEs 140 must be solved to obtain the physical quantities, such as the velocity u and the pressure p, and then the objective function J can be evaluated based on these physical quantities, such as the pressure p. The structural quantities θ can then be updated or optimized based on the calculated objective function J, as indicated by the notation 160.

[0021] In an example, as shown by the designation 150, a neural network (NN) model can be used to obtain the velocity u and pressure p by solving the PDEs 140. For example, Physical-informed neural network (PINN) (Maziar Raissi, Paris Perdikaris, and George E. Karniadakis. Physics-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 on predicting the solutions of the PDEs and can be used to implement the NN model 150. The PINN is trained to use the residuals of both the PDEs and the BCs as multiple terms of the loss function.However, there is an unbalanced competition between the terms of PDEs and BCs, resulting in high computational costs for PINNs and reduced accuracy for solving PDEs with complex BCs. One way to address this problem is to embed BCs in the approach so that the neural networks can automatically satisfy the BCs without having to add corresponding loss terms. The challenge, however, lies in the fact that the equation forms of complex BCs, such as Neumann BCs and Robin BCs, as in [reference missing]. Fig. 2 illustrates that, in general, they do not have analytical solutions and are therefore difficult to incorporate into the approach.

[0022] Fig. Figure 2 illustrates three types of the most commonly used BCs according to the aspects of disclosure. In the illustrated example of heat transfer, T = T(x) is the temperature, k is the thermal conductivity, and h is the heat transfer coefficient. As in Fig. As illustrated in 2(a), the Dirichlet BC gives the value of the solution of PDEs at the boundary. A constant temperature T is assumed. H assumed at the heat source (x = x0). As in Fig. As illustrated in 2(b), the Neumann BC gives the value of the derivative at the boundary. It is assumed that the right wall (x = x1) is adiabatic and forces the derivative to zero. As shown in Fig. As illustrated in 2(c), the Robin BC is a combination of the Dirichlet BC and the Neumann BC. It is used to measure heat convection between the heat source and the surroundings (Tr). ∞ ) to describe the surface (x = x2).

[0023] In one embodiment as in Fig. As illustrated in Figure 1, a framework for a coercive-constraint neural network (HCNN) 150 is used in the structural optimization framework 100 to predict the solutions of PDEs 140 with different boundary shapes, represented by structural quantities θ. The HCNN 150 is a unified coercive-constraint framework for all three most commonly used BCs, i.e., Dirichlet, Neumann, and Robin BCs. Using the HCNN, an approach can be constructed to automatically satisfy the three types of BCs. Therefore, the model can be trained without the losses of these BCs, which mitigates unbalanced competition and significantly improves the accuracy of solving geometrically complex PDEs, thus considerably improving the efficiency and performance of structural optimization for the object, such as the illustrated sheet.

[0024] The PINN is briefly introduced to better understand the HCNN from a revelatory perspective. The following Laplace equation can be considered an example. Δu(x1,x2)=0,x1∈(0,1], x2∈[0,1] u(x1,x2)=g(x2),x1=0, x2∈[0,1] where Eq. (1a) gives the form of the PDE and Eq. (1b) is a Dirichlet boundary condition (BC). A solution to the above problem is a solution of Eq. (1a) that also satisfies Eq. (1b).

[0025] PINNs employ a neural network NN(x1, x2; w) to approximate or predict the solution of the PDE, i.e., û(x1, x2; w) = NN(x1, x2; w) ≈ u(x1, x2), where w denotes the trainable parameters of the neural network. These parameters w are learned by minimizing the following loss function: L(w)=LF(w)+LB(w)≜1Nƒ∑i=1Nƒ|Δu^(xƒ,1(i),xƒ,2(i);w)|2+1Nb∑i=1Nb|u^(0,xb,2(i);w)−g(xb,2(i))|2 where the first loss term LF is configured such that the prediction û of the solution is restricted to satisfy the PDE (Eq. (1 a)), while the second loss term LB is configured such that the prediction û of the solution is restricted to satisfy BC (Eq. (1 b)), wherein {xƒ(i)=(xƒ,1(i),xƒ,2(i))}i=1Nƒ a set of N f collocation points are those from the range [0,1] 2 can be extracted, and {xb(i)=(0,xb,2(i))}i=1Nb a set of N b Collocation points are those taken from the boundary x1 = 0, x2 E [0,1].

[0026] PINNs have a wide range of applications, including heat, flow, and atmosphere. However, PINNs suffer from some performance issues. Previous analyses have shown that the convergence of the first loss term LF can be significantly faster than that of the second loss term LB. This pathology can lead to non-physical solutions that do not satisfy the BCs or initial conditions (ICs). Furthermore, this problem is exacerbated in geometrically complex PDEs where the number of BCs is large, and can significantly impair accuracy.

[0027] One possible approach to overcoming this pathology is to embed the BCs in the approach in such a way that every instance of the approach can automatically satisfy the BCs. In this way, the loss terms corresponding to the BCs are no longer needed, and thus the aforementioned pathology is mitigated. Such procedures are called constrained procedures and have a similar approach formula to [example missing]. u^(x;w)=u∂Ω(x)+l∂Ω(x)NN(x;w) where x is the coordinate, Ω is the area of ​​interest, u ∂Ω (x) the general solution at the boundary ∂Ω is and l ∂Ω(x) is an extended distance function that satisfies l∂Ω(x)={0if x∈∂Ω>0otherwise

[0028] In the case of Eq. (1) (where x = (x1, x2), Ω = [0,1] 2 ) the general solution is exactly g(x2), and the following approach, which automatically satisfies the BC in Eq. (1 b), can be used. u^(x1,x2;w)=g(x2)+x1 NN(x1,x2;w)

[0029] However, it is difficult to extend this procedure directly to more general cases of Robin-BCs (see Eq. (7)), since it is difficult to find the general solution u (∂Ω)(x) to obtain analytically. Existing attempts are either mesh-dependent methods, which are time-consuming for high-dimensional and geometrically complex PDEs, or ad-hoc methods for specific (geometrically simple) physical systems. A unified framework with mandatory constraints for both geometrically complex PDEs and the most commonly used Dirichlet, Neumann, and Robin BCs is still lacking.

[0030] A physical system is considered that is defined by the following PDEs, which are on a geometrically complex domain Ω ⊂ ℝ d are defined, is controlled. F[u(x)]=0, x=(x1,...,xd)∈Ω where F=(F1,...,FN) N PDE operators are included, which map u to a function of x, u and their derivatives. Here, u(x) = (u1(x), ... , u n (x)) the solution of the PDEs, which represents physical quantities of interest. For each u jFor j = 1, ...,n, the appropriate boundary conditions (BC) can be formulated as ai(x)uj+bi(x)(n(x)⋅∇uj)=gi(x),x∈γi,∀i=1,...,mj where γ1, ..., γ mj subsets of the boundary ∂Ω whose closures are disjoint, a i (x) and b i (x) are two functions that satisfy that ai2(x)+bi2(x)≠0 for ∀x ∈ γ i This applies, and n(x) = (n1(x), ...,n d (x)) the (outward-pointing) unit normal of a corresponding boundary γ i at point x. It is pointed out that Eq. (7) is a Dirichlet BC if a i (x) ≡ 1, b i (x) ≡ 0, a Neumann BC if a i (x) ≡ 0, b i (x) ≡ 1, and otherwise represents a Robin-BC.

[0031] If PINNs were to be used for such geometrically complex PDEs, this would be a difficult multitasking learning process with at least (∑j=1nmj+N) Terms in the loss function. This would severely impair the training convergence due to the unbalanced competition between these loss terms. If the BCs can be embedded in the approach, each instance will automatically satisfy the BCs. However, it is not possible to directly follow the pipeline of procedures with constraints (see Eq. (3)) because Eq. (7) does not have a general solution of analytical form. Therefore, a new approach is needed to tackle this unsolvable problem.

[0032] From a disclosure perspective, the general solutions of the BCs are presented and used to construct the approach with constraints. Additional fields can be introduced to reformulate the PDEs equivalently. Let p j (x) = (p j1 (x), ...,p jd (x)) = ∇u j ,j = 1, ...,n, which are substituted in Eq. (6) and Eq. (7) to obtain the equivalent PDEs F˜[u(x),p1(x),…,pn(x)]=0,x∈Ω pj(x)=∇uj,x∈Ω∪∂Ω, ∀j=1,…,n where (u(x), p1(x), ... , p n (x)) the solution of the new PDEs is, F˜=(F˜1,…,F˜N) The PDE operators after the reformulation are as follows. And for j = 1, ..., n, the corresponding BCs are as follows. ai(x)uj+bi(x)(n(x)⋅pj(x))=gi(x),x∈γi,∀i=1,…,mj

[0033] By adding the additional fields shown in Eq. (8b), Eq. (7) was transformed into linear equations with respect to (u j , p j ) converted, for which general solutions can be derived much more easily. In the following, (u j , p j ) with p̃ j denoted, and their general solutions at the boundary γ i (i.e. (ujγi,pjγi) ) are with p˜jγi designated.

[0034] From the perspective of revelation, a basis B(x) of the null space can be used to obtain the general solution of 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 every x ∈ γ i B(x) must always be a basis of the zero space, i.e., its columns must not degenerate into linearly dependent vectors, otherwise it cannot represent all possible solutions.

[0035] In general, for any dimension d ∈ ℝ, it is not trivial to find a simple expression for the basis. From the perspective of revelation, it is preferable to find (d + 1) vectors in the null space of which d are linearly independent (thereby B(x) ∈ ℝ). (d+1)×(d+1) Then the general solution of p˜jγi at the edge γ i be constructed as p˜jγi(x;wjγi)=B(x)NNjγi(x;wjγi)+n˜(x)g˜i(x) where n˜=(ai,bin) / ai2+bi2, g˜i= / ai2+bi2, NNjγi:ℝd→ℝd+1 a neural network with trainable parameters wjγi is and B(x) is given by B(x)=ld+1−n˜(x)n˜(x)T

[0036] According to the disclosure, in the case of d = 1 or d = 2, a simpler expression for B(x) can be found. In one embodiment, in the case of d = 1, the basis of the null space can be found as B(x)=[n˜2(x),−n˜1(x)]T where n˜=(n˜1,n˜2)=(ai,bin) / ai2+bi2.

[0037] In one embodiment, in the case of d = 2, the basis of the zero space can be found as B(x)=[β1(x),β2(x),β3(x)],β1(x)=[0,n˜3(x),−n˜2(x)]Tβ2(x)=[−n˜3(x),0,n˜1(x)]Tβ3(x)=[n˜2(x),−n˜1(x),0]T where n˜=(n˜1,n˜2,n˜3)=(ai,bin) / ai2+bi2.

[0038] By parameterizing a neural network, Eq. (10) can be used to define any function in the boundary γ.i represent as long as the function satisfies BC (see Eq. (9)). Since the problem domain contains multiple boundaries, the general solutions that satisfy each boundary γ can be i These values ​​can be combined to achieve an overall approximation or prediction. The approach can therefore be constructed as follows: (u^j,p^j)=l∂Ω(x)NNmain(x;wmain)+∑i=1mjexp[−αilγi(x)]p˜jγi(x;wjγi),∀j=1,…,n where NN main : ℝ d → ℝ d+1 the higher-level neural network with trainable parameters w main is. ∂Ω , l γi , i = 1, ... , m j are continuous extended distance functions (see Eq. (4)), and α i (i = 1, ..., m j ) are determined by αi=βsminx∈∂Ω\γilγi(x) where β s ∈ ℝ is a hyperparameter of "hardness" in the spatial domain. ∂Ω \ γ i stands for the edges except γ iIn Eq. (11), extended distance functions are used to divide the problem domain into several parts, where p˜jγi(NNjγi (is their learnable part) for the approximation at the edges γ i is responsible, while NN main is responsible for the approach of the interior except at the edges. Furthermore, Eq. (12) ensures that the weight of p˜jγi on e -βs at the nearest neighbor of the edge γ i falls, so that p˜jγi does not interfere with the approach at other edges.

[0039] From the perspective of disclosure, if the parameters a i , b i , n or g i in Eq. (10) and Eq. (11) only at the edge γ i are defined, their definition can be extended to the region Ω u ∂Ω using interpolation or approximation via neural networks. The blade edge (i.e., γ) af) is considered as an example. Assume that f(x) is only in the blade edge γ. af If a definition is given, then the task is to extend its definition to Ω u ∂Ω. As in Fig. As shown in Figure 3, two reference points (i.e., x0 and x1) are placed on the leading and trailing halves of the airfoil. For any x ∈ Ω u ∂Ω, this can be expressed as polar coordinates with respect to x0 and x1, respectively. The two polar coordinates are concatenated to form a new space. The subsequent interpolation and approximation are performed in this new space. This is because the shape of the airfoil can be better characterized in the new space. It should be noted that there are many possibilities for coordinate transformations, not limited to this example. Regarding the interpolation, several points on the airfoil edge γ are used. af scanned to obtain the data set {((θ0(i),θ1(i)),f(i))}i=1N To obtain the desired result, for each x ∈ Ω u ∂Ω, the corresponding augmented f(x) is generated by interpolation within the dataset. The interpolation method used here depends on the smoothness requirements of the approach. Additionally, the number of reference points can also be changed, and experiments have shown that a single reference point is sufficient.

[0040] The neural network approach is a general method that requires no manual design. In this case, several points on the blade edge γ can be used. af be scanned to obtain the data set {((ϕ0(x(i)),ϕ1(x(i))),f(i))}i=1N to construct, followed by training a neural network on the dataset, i.e. NN(ϕ0(x) (i) ),ϕ1(x (i) )) ≈ f (i)For each x ∈ Ω ∪ ∂Ω, NN(ϕ0 (x), ϕ1 (x)) is used as the corresponding augmented f(x). From a revelation standpoint, the neural network can also be trained in the original space. However, experimental results show that training in the new space can achieve better results. This could be because the complex geometry in the new space becomes smoother and easier to learn.

[0041] It is worth noting that, in addition to the cases mentioned above, the extended distance functions I(x) (where the superscript is omitted and see Eq. (4) for its definition) can also be treated similarly. For complex geometry, the distance function can be complex and replaced by a suitable substitute 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).

[0042] Finally, the HCNN model can be trained using the following loss function. L=1NfN∑k=1Nf∑j=1N|F¯j[u^(x(k)),p^1(x(k)),…,p^n(x(k))]|2+1Nfn∑k=1Nf∑j=1n‖p^j(x(k))−∇u^j(x(k))‖22 where u^=(u^1,…,u^n),{u^j,p^j}j=1N is defined in Eq. (11) and {x(k)}k=1Nf a set of collocation points that were sampled in the region Ω. For clarity, the trainable parameters of neural networks are omitted here. The loss function of Eq. (13) measures the discrepancy of both the PDEs (i.e., F˜1,…F˜N as well as the equilibrium equations introduced by the additional fields (i.e. Eq. (8b)) at N f Collocation points.

[0043] According to Eq. (13), BCs are successfully embedded in the approach, and it is no longer necessary to include the residuals of BCs as additional terms in the loss function, as is done by PINN. That is, the proposed approach strictly corresponds to BCs throughout the training process, thus significantly reducing the possibility of generating non-physical solutions. However, this comes at the cost of introducing (nd) additional equilibrium equations (see Eq. (8b)). In many physical systems, especially those with complex geometries, the number of BCs (cnt(BCs)) is far greater than (nd) (e.g., n = 3, d = 2, cnt(BCs) = 1260 for a classical physical system, a heat exchanger). Aspects of revelation thus actually reduce the number of loss terms by an order of magnitude or two (Δcnt(losses) = nd - cnt(BCs) « 0), thereby mitigating the unbalanced competition between loss terms.

[0044] From the perspective of the disclosure, the HCNN framework described above can be extended to the spatiotemporal domain. Consider a physical system governed by the following time-dependent PDEs, which operate on a geometrically complex domain Ω ⊂ ℝ. d are defined, is controlled. F[u(x,t)]=0,x=(x1,…,xd)∈Ω,t∈(0,T] where t is the time coordinate and the other notations are the same as in Eq. (6) and Eq. (7). For each u j For j = 1, ... , n, the appropriate boundary conditions (BC) can be formulated as ai(x,t)uj+bi(x,t)(n(x)⋅∇uj)=gi(x,t),x∈γi,t∈(0,T]∀i=1,…,mj and an initial condition (IC) uj(x,0)=fi(x),x∈Ω

[0045] According to the pipeline described above with respect to Eqs. (9) to (13), the general solution of p˜jγi at the edge γ i be constructed as p˜jγi=B(x,t)NNjγi(x,t)+n˜(x,t)g˜i(x,t) where n˜=(ai,bin) / ai2+bi2, g˜i=gi / ai2+bi2, NNiγi:ℝd+1→ℝd+1 a neural network and B(x, t) = I d+1 - ñ(x, t)ñ(x, t) T The trainable parameters of neural networks are omitted for the sake of clarity.

[0046] Finally, the approach can be constructed as (uj†,p^j)=l∂Ω(x)NNmain(x,t)+∑i=1mjexp[−αilγi(x)]p˜jγi(x,t),∀j=1,…,nu^j=uj†(x,t)(1−exp[−βtt])+fj(x)exp[−βtt], ∀j=1,…,n where uj† an intermediate variable that incorporates constraints in spatial dimensions, NN main : ℝ d+1 → ℝ d+1 the overarching neural network is, l ∂Ω , l γi , i = 1, ..., m j extended distance functions (see Eq. (4)) are, α i (i = 1, ..., m j) is determined in Eq. (12) and β s ∈ ℝ is a hyperparameter of “hardness” in the time domain.

[0047] Fig. Figure 4 is a schematic diagram illustrating the structural optimization of a 2-dimensional (2D) battery pack according to the aspects of the disclosure.

[0048] As in Fig. As shown in 4, the cell borders are γ c,i and cooling pipe edges γ p,i The structure to be optimized is located within this area. The goal of structural optimization is to optimize the shape and position of the cells and cooling tubes to achieve a uniform temperature distribution over time.

[0049] The problem of structural optimization can be represented by: minimizeθ∈Θ J(θ,s)subject to W(θ,s)≤°0 where θ denotes the structural parameters of the structure of the object to be optimized, which are also the parameters of the boundary forms, and Θ denotes the space of θ. The structural parameters θ change during the optimization of the structure but lie within the design space Θ. J is the objective function, whose value measures how well a given structure performs. W represents the constraints of both the structural parameters θ and the state variable s = s(x), which denotes the state of the physical system, for example, velocity, pressure, temperature, or the like, where x denotes spatial coordinates. And the symbol “≤°” is an element-wise comparison.

[0050] The problem of structural optimization for the in Fig. The 4 shown 2D battery packs can be represented by J(θ)=∫t∫Ω|T(x)−Tref(x)|dx dt where T is the temperature, T ref The reference temperature is...

[0051] The controlling PDEs that describe the physical system with respect to the 2D battery pack can be formulated according to physical laws; for example, the PDEs can be given by: ∂T∂t=kΔT(x,t),x∈Ω,t∈(0,1], k(n(x)⋅∇T(x,t))=h(Ta−T(x,t)),x∈γou,t∈(0,1], k(n(x)⋅∇T(x,t))=h(Tc−T(x,t)),x∈γc,i,t∈(0,1],i=1,…,nc, k(n(x)⋅∇T(x,t))=h(Tw−T(x,t)),x∈γp,i,t∈(0,1],i=1,…,nw, T(x,0)=T0,x∈Ω, where x = (x1, x2) is the spatial coordinate, t is the temporal coordinate. T(x, t) is the temperature over time, k is the thermal conductivity and is set to k = 1 in this example. ΔT=∂2T / ∂x12+∂2T / ∂x22, h is the heat transfer coefficient and h is set to 1 in this example. VT = (∂T / ∂x1, ∂T / ∂x2), T a , T c , T w are the temperature of the air, of the cells (n) c = 11 cells with radius r c ), the cooling pipes (nw = 6 pipes with radius r c ) and are on 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. Fig. 4 stands for γ ou for the outer edges of the battery pack, γ c,i stands for the edges of the cells, γ p,i stands for the edges of the cooling pipes.

[0052] The structural parameters include the center and radius of the cells and the cooling tubes. To optimize the structure of the 2D battery pack based on Eq. (20), the controlling PDEs must be solved to obtain the physical quantity, i.e., the temperature T.

[0053] From a revelatory point of view, the PDEs from Eq. (21a) to Eq. (21e) are reformulated by adding additional fields according to the HCNN model. The introduced additional fields are p(x, t), shown in Eq. (22b), and the reformulated PDEs are: ∂T∂t=k(∇⋅p(x,t)),x∈Ω,t∈(0,1], p(x,t)=∇T,x∈Ω∪∂Ω,t∈(0,1], k(n(x)⋅p(x,t))=h(Ta−T(x,t)),x∈γou,t∈(0,1], k(n(x)⋅p(x,t))=h(Tc−T(x,t)),x∈γc,i,t∈(0,1],i=1,…,nc, k(n(x)⋅p(x,t))=h(Tw−T(x,t)),x∈γp,i,t∈(0,1],i=1,…,nw, T(x,0)=T0, x∈Ω,

[0054] Since the solution of the PDEs or the reformulated PDEs is a scalar function, that is, the temperature T(x, t), the solution is denoted by u(x, t) = T(x, t). Assume p̃ denotes (u, p), where p is the additional field shown in Eq. (22b). The general solutions at the boundaries γ ou , γ c,i , γ p,i are derived in each case.

[0055] For every point on the outer boundary, i.e., for x ∈ γ ou , the coefficients of the BCs shown in Eq. (9) can be obtained from the BC shown in Eq. (22c), i.e. a(x) = h, b(x) = k, g(x) = hT aAccording to Eq. (10), the general solution p̃ γou given at the outer edge by p˜γou=B(x)NNγou(x,t)+(h,kn)h2+k2hTah2+k2 where B(x) is given by Eq. (10a) or Eq. (10c), with n˜=(n˜1,n˜2,n˜3)=(h,kn) / h2+k2.

[0056] For every point at the cell boundary, i.e., for x ∈ γ c,i , the coefficients of the BCs shown in Eq. (9) can be obtained from the BC shown in Eq. (22d), that is, a(x) = h, b(x) = k, g(x) = hT c According to Eq. (10), the general solution p̃ γc,i at the cell edge given by p˜γc,i=B(x)NNγc,i(x,t)+(h,kn)h2+k2hTCh2+k2

[0057] For every point on the edge of the cooling tube, i.e. for x ∈ γ p,i , the coefficients of the BCs shown in Eq. (9) can be obtained from the BC shown in Eq. (22e), that is, a(x) = h, b(x) = k, g(x) = hT wAccording to Eq. (10), the general solution p̃ γp,i at the edge of the cooling tube given by p˜γp,i=B(x)NNγp,i(x,t)+(h,kn)h2+k2hTwh2+k2

[0058] The extended distance function l ∂Ω Eq. (11) can be implemented in various ways according to the aspects of the disclosure. In one embodiment, the cell boundary γ can be c,i and the cooling tube edge γ p,i , since they are 2D circles, the extended distance functions l γc,i (x) and l γp,i (x) is chosen as the distance between x and the center minus the radius. For the rectangular outer boundary γ ou Can the extended distance function l γou (x) assuming that it is given by [a1, a2] × [b1, b2], can be constructed as follows lγou(x)=SoftMin(x1−a1,a2−x1,x2−b1,b2−x2) where x = (x1, x2), SoftMin is a differentiable version of the Min function. In one implementation, SoftMin is implemented by LogSumExp in PyTorch, i.e., SoftMin(y) = LogSumExp(-βy) / (-β), β = 4. And the extended function l ∂Ω (x) is calculated by using SoftMin on the distances to all edges. l∂Ω(x)=SoftMin(lγou(x),lγc,1(x),...,lγc,nc(x), lγp,1(x),...,lγp,nw(x))

[0059] Fig. Figure 5 illustrates an architecture of HCNN 500 for predicting PDE solutions of the 2D battery pack according to the aspects of the revelation.

[0060] The HCNN 500 comprises a parent NN, designated M-NN 510, and three child NNs, designated S-NNs 520-1, 520-2, and 520-3. In one implementation, the parent NN is a multilayer perceptron (MLP) of size [3] + 4 ×

[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 VT. The subordinate NNs 520-1, 520-2, and 520-3 are MLPs of size [3] + 3 ×

[20] + [3], i.e., 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 VT.The higher-level NN 510 is responsible for predicting the PDE solutions in the region outside the edges of the battery pack, and the lower-level NNs 520-1, 520-2, and 520-3 are each responsible for predicting the PDE solutions at the BCs according to Eq. (22c), Eq. (22d), and Eq. (22e), respectively. In one implementation, the hyperparameters of the "hardness" β are s = 5 and β t = 10.

[0061] The HCNN 500 further includes a joining unit designated AS 530. Joining unit 530 comprises a parent joining unit, designated M-AS 530-4, and three child joining units, designated S-ASs 530-1, 530-2, and 530-3. S-ASs 530-1, 530-2, and 530-3 each take the outputs of the child NNs 520-1, 520-2, and 520-3 as input and receive the general solutions at the respective boundaries of the outer boundary, the cell boundary, and the cooling tube boundary. In one implementation, S-ASs 530-1, 530-2, and 530-3 each input the general solutions p̃. γou , p̃ γc,i p̃ γp,iat the three edges according to Eq. (23), Eq. (24) and Eq. (25). The higher-level assembly unit 530-4 takes the outputs of M-NN 510, S-ASs 530-1, 530-2 and 530-3 as input and obtains the solution of the PDEs according to Eq. (11). The solution of the PDEs of HCNN 500 is the temperature T and its two derivatives p(x, t) = ∇T. It should be noted that the output of HCNN 500 can only be the temperature T.

[0062] To train the HCNN 500, the loss can be obtained based on Eq. (13), and the parameters or weights of the HCNN 500's NNs can be updated based on the loss. In one implementation, the NN models are trained for, e.g., 5000 Adam iterations, followed by L-BFGS optimization until convergence. In another implementation, the mean squared error (MSE) is used for the loss function, and tanh is used for the activation function.

[0063] After training the HCNN 500, its output, i.e., the 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 Eq. (20). In an implementation, the HCNN 500 is trained in each iteration of the structure optimization of the object, such as the 2D battery pack, based on the current structural parameters, such as the positions and radii of the cells and cooling tubes. Therefore, improving the efficiency and accuracy of an HCNN significantly enhances the performance of the structure design process for the object, such as the 2D battery pack.

[0064] Fig. Figure 6 is a schematic diagram illustrating the structural optimization of a paddle blade according to the aspects of the disclosure.

[0065] As in Fig. As shown in 6, the blade edge γ af, which is located in that area, the structure to be optimized. The goal of structural optimization is to achieve the desired pressure distribution p. ref on the blade surface by changing the structural parameter θ.

[0066] The problem of structural optimization for the in Fig. The 6 shown paddle blade can be represented by J(θ)=∫γaf|p(x)−pref(x)|ds where p is the pressure, p ref The reference pressure is...

[0067] The controlling PDEs that describe the physical system with respect to the blade can be formulated according to physical laws; for example, the PDEs can be NS equations and given by: u(x)⋅∇u(x)=−∇p(x)+v∇2u(x) x∈Ω, ∇⋅u(x)=0, x∈Ω, u(x)=u0(x), x∈γil ∪ γtp ∪ γbt, p(x)=1, x∈γol, n(x)⋅u(x)=0, x∈γaf, 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, v is the viscosity of the fluid and is set to 1 / 50 in this example. u0(x) = (1,0). Vu is a Jacobian matrix, Vp(x) = (∂p / ∂x1, ∂p / ∂x2). In Fig. 6 stands for γ il for the inlet edge of the region, γ tp stands for the upper edge, γ bt stands for the lower edge, γ ol stands for the outlet edge, γ af stands for the edge of the shovel blade.

[0068] The structural dimensions include the shape of the blade, defined by the edge γ. af is shown. In order to optimize the structure of the blade based on Eq. (28), the controlling PDEs must be solved to obtain the physical quantity, i.e. the velocity u and the pressure p.

[0069] From a disclosure perspective, the PDEs from Eq. (29a) to Eq. (29e) are reformulated by adding additional fields according to the HCNN model. The introduced additional fields are p1(x) = ∇u1, p2(x) = ∇u2, shown in Eq. (30c) and Eq. (30d), and the reformulated PDEs are: (u(x)−v∇)⋅(p1(x),p2(x))=−∇p(x), x∈Ω, ∇⋅u(x)=0, x∈Ω, p1(x)=∇u1,x∈Ω∪∂Ω, p2(x)=∇u2,x∈Ω∪∂Ω, u(x)=u0(x),x∈γil∪γtp∪γbt, p1(x)=1,x∈γol, n(x)⋅u(x)=0,x∈γaf,

[0070] The solution to the PDEs or reformulated PDEs is the velocity and the pressure (u(x), p(x)). For the pressure p γol (x) = 1 is the general solution at the outlet boundary p(x) exactly γ ol The approach for the pressure p is given by p^=pγol(x)+lγol(x)NNmain(x)[3] where [3] means that the third elements of the output of NN main (x) will be taken.

[0071] For the velocity u(x), the general solution at the inlet, upper and lower boundaries is u γ* (x) = u0(x) exactly γ * = γ il ∪ γ tp U γ bt The general solution at the blade edge γ af is given by uγaf=B(x)NNγ∗(x) where B(x) = [n2(x), -n1(x)] T according to Eq. (10b) and the output of NN γ* (x) is a scalar.

[0072] By combining u γ* and u γaf According to equation (11), the approach for velocity u can be obtained by u^=l∂Ω(x)NNmain(x)[1:2]+exp[−αγ∗lγ∗(x)]uγ∗(x)+exp[−αγaflγaf(x)]uγaf(x) where [1:2] means that the first two elements of the output of NN main (x) are taken and α γ* a γaf are defined similarly to those in Eq. (12).

[0073] The extended distance function l ∂ΩEq. (11) can be implemented in various ways according to the aspects of the disclosure. In one embodiment, a direct way to obtain l for the blade edge is available. γaf the calculation of the distance between x and the blade γ af However, this can be time-consuming, as the blade edge γ af It is complicated. In another implementation, the actual distance can be approximated using an MLP. For example, a NN with 3 hidden layers of width 30 can be trained with 1024 × 6 points located in the Fig. The area shown in Figure 6 is scanned. Some of the points are scanned within the boundary frame of the blade, and the rest are scanned along with their true distances, which can be calculated using the formula for the distance to a polygon.

[0074] For the outlet edge γ olIt is a vertical line, for example, the outlet edge x1 = a. The augmented distance function can be expressed as l γol (x) = a - x1 can be calculated, where x = (x1, x2).

[0075] For the collection of the inlet, top and bottom edges y * = γ il ∪ γ tp ∪ γ bt is the edge y * an open rectangle, such that the augmented distance function l γ* (x) can be constructed similarly to the case of the rectangle, as shown in Eq. (26), while ignoring the right-hand side. Furthermore, l ∂Ω (x) is calculated by using SoftMin of the distances to all edges, as shown in Eq. (27).

[0076] Fig. Figure 7 illustrates an architecture of an HCNN 700 for predicting PDE solutions of the blade according to the aspects of the revelation.

[0077] The HCNN 700 includes a parent NN, designated M-NN 710, and a child NN, designated S-NN 720. In one implementation, the parent NN is a multilayer perceptron (MLP) of size [2] + 6 ×

[50] + [7], which means 2 inputs, 6 hidden layers of width 50, and 7 outputs. The two inputs are 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. The child NN 720 is an MLP of size [2] + 4 ×

[40] + [1], which means 2 inputs, 4 hidden layers of width 40, and 1 output. As in Eq. (30e), Eq. As illustrated in equations (30f) and (30g), the BCs do not include the additional fields p1(x) and p2(x), which is why the subordinate NN 720 does not need to output the corresponding terms. The BCs of equations (30e) and (30g)Equation (30f) provides a constant speed and constant pressure at the corresponding edges, therefore the subordinate NN 720 does not need to provide an output for these BCs. Regarding the BC of Eq. (30g), the subordinate NN 720 must output if B(x) = [n2(x), -n1(x)]. T When used as the solution basis, only a scalar is provided, as illustrated in Eq. (32). The parent NN 710 is responsible for predicting the PDE solutions in the region outside the blade edge, and the child NN 720 is responsible for predicting the PDE solutions at the BC according to Eq. (30g). And the PDE solutions at the BCs according to Eq. (30e) and Eq. (30f) can be obtained analytically. In one implementation, the hyperparameters of the “hardness” β s = 5.

[0078] The HCNN 700 further includes a merging unit designated AS 730. The merging unit 730 includes a higher-level merging unit designated M-AS 730-2 and a lower-level merging unit designated S-AS 730-1. The S-AS 730-1 takes the output of the lower-level NN 720 as input and obtains the general solution at the corresponding blade edge γ. af , the S-AS 730-1 gives the general solutions u γaf at the edge according to Eq. (30g) or Eq. (32). The higher-level assembly unit 730-2 takes the outputs of the M-NN 710 and the S-AS 730-1 as input and receives the solution of the PDEs according to Eq. (11) or according to Eq. (31) and Eq. (33). The solution of the PDEs of the HCNN 700 are the velocity u = (u1, u2), the pressure p, and the derivatives ∇u of velocity. It should be noted that the output of the HCNN 700 can only be the velocity u = (u1, u2) and the pressure p.

[0079] To train the HCNN 700, the loss can be obtained based on Eq. (13), and the parameters or weights of the HCNN 700's NNs can be updated based on the loss. In one implementation, the NN models are trained for, e.g., 5000 Adam iterations, followed by L-BFGS optimization until convergence. In another implementation, the mean squared error (MSE) is used for the loss function, and tanh is used for the activation function.

[0080] After training the HCNN 700, its output, i.e., the temperature T, is used to optimize the airfoil structure. For example, the airfoil shape can be optimized based on Eq. (28). In an implementation, the HCNN 700 is trained at each iteration of the structural optimization of the object, such as the airfoil, based on the current structural parameters, such as a set of control points consisting of splines representing the airfoil shape. Therefore, improving the efficiency and accuracy of an HCNN significantly enhances the performance of the structural design process for the object, such as the airfoil.

[0081] Fig. Figure 8 illustrates an architecture of an HCNN 800 for predicting PDE solutions of an object that is to be designed or manufactured according to the aspects of the revelation.

[0082] The HCNN 800 includes at least one parent NN, designated M-NN 810-1 to M-NN 810-m, and at least one child NN, designated S-NN 820-1 to S-NN 820-n. It should be noted that although a parent NN is used to predict all solutions u = (u1, ... , u n Since PDEs can be used, it is possible to predict respective parts of all solutions with multiple higher-level NNs. The one or more higher-level NNs 810-1 to 810-m are responsible for predicting the PDE solutions in the region outside the object's boundaries, and the one or more lower-level NNs 820-1 to 820-n are each responsible for predicting the PDE solutions at the respective boundaries according to corresponding BCs.

[0083] The HCNN 800 further includes a joining unit designated AS 830. The joining unit 830 includes a parent joining unit designated M-AS 830-k and at least one child joining unit designated S-AS 830-1 to S-AS 830-n. The one or more S-ASs 830-1 to 830-n each take the outputs of the child NNs 820-1 to 820-n as input and receive the general solutions at the respective boundaries of the object. The parent joining unit 830-k takes the outputs of the one or more parent NNs 810-1 to 810-m and the one or more child AS 830-1 to S-AS 830-n as input and receives the solution of the PDEs according to Eq. (11).

[0084] To train the HCNN 800, the loss can be obtained based on Eq. (13), and the parameters or weights of the NNs of the HCNN 800 can be updated based on the loss. In one implementation, the NN models are trained, for example, for a defined number of Adam iterations, followed by L-BFGS optimization until convergence.

[0085] After training the HCNN 800, the output, i.e., the prediction of the solution u = (u1, ... , u) n ) of PDEs, the trained HCNN 800 is used to optimize the structure of the object. Therefore, improving the efficiency and accuracy of an HCNN significantly increases the performance of the structural design process for the object to be manufactured.

[0086] Fig. Figure 9 illustrates an exemplary process for optimizing the structure of an object according to the principles of revelation. It should be noted that the process can be implemented using computers or processors.

[0087] In Block 910, structural quantities are received by an HCNN framework, and these structural quantities are used to describe the boundaries of the object. The HCNN comprises at least one primary NN, at least one secondary NN, and a joining unit, where PDEs are formulated to characterize a physical system with respect to the object using the object's physical quantities.

[0088] In block 920, a first prediction of solutions to reformulated PDEs is generated by at least one primary neural network. The reformulated PDEs are obtained by replacing gradients of the physical quantities in the PDEs with additional quantities, where the first prediction of the solutions of the reformulated PDEs includes a prediction of the object's physical quantities and the additional quantities that represent gradients of the object's physical quantities.

[0089] In Block 930, a second prediction of solutions to BCs of the reformulated PDEs is generated by at least one secondary NN. The BCs are formulated to characterize the physical system with respect to the boundaries of the object, and the second prediction of the solutions to the BCs of the reformulated PDEs includes a prediction of the physical quantities of the object.

[0090] In Block 940, a final prediction of solutions of the reformulated PDEs is obtained by combining the first and second predictions of a solution using the merging unit. The final prediction of solutions of the reformulated PDEs represents a prediction of the object's physical properties.

[0091] In block 950, the structural quantities representing the edges of the object are updated based on the physical quantities of the final prediction.

[0092] According to one embodiment, the BCs of the edges of the object comprise one or more of a Dirichlet BC, Neumann BC and Robin BC.

[0093] According to one embodiment, the primary NN corresponds to the region outside the edges of the object, and the at least one secondary NN corresponds to at least one of the edges of the object.

[0094] According to one embodiment, the second prediction of the solutions of the BCs of the reformulated PDEs further includes a prediction of the additional quantities that represent gradients of the physical quantities of the object.

[0095] According to one embodiment, generating a second prediction of solutions of BCs of the reformulated PDEs in block 930 further comprises: for each of at least one boundary corresponding to the at least one secondary NN, generating a corresponding part of the second prediction based on constraints of the BC of the boundary and an output of a corresponding secondary NN.

[0096] According to one embodiment, generating a second prediction of solutions of BCs of the reformulated PDEs in block 930 further comprises: for each of the at least one boundary corresponding to the at least one secondary NN, generating the corresponding part of the second prediction further based on a basis matrix or basis vector of the null space for the boundary, wherein the basis matrix or basis vector is obtained based on a unit normal of the boundary.

[0097] According to one embodiment, obtaining a final prediction of solutions of the reformulated PDEs in block 940 further comprises: combining the first prediction of a solution and the second prediction of a solution based on a distance of a sampled point of the object to the edges of the object.

[0098] According to one embodiment, the joining of the first prediction of a solution and the second prediction of a solution in block 940 further comprises: joining respective parts of the second prediction according to the edges of the object by ensuring that a part of the second prediction corresponding to any edge at the other edges decreases to such an extent that it has an insignificant effect on the other edges.

[0099] According to one embodiment, process 900 further comprises: obtaining a first discrepancy of the reformulated PDEs based on the final prediction of solutions, including the physical quantities of the object and the gradients of the physical quantities; obtaining a second discrepancy between the physical quantities and the gradients of the physical quantities in the final prediction of solutions; obtaining a loss based on the first discrepancy and the second discrepancy and updating the at least one primary NN and the at least one secondary NN based on the loss.

[0100] According to one embodiment, the steps of generating a first prediction, generating a second prediction, obtaining a final prediction, obtaining a loss, and updating the at least one primary NN and the at least one secondary NN are repeated until a training requirement is met.

[0101] According to one embodiment, these are at least one primary NN and at least one secondary NN MLP-NNs.

[0102] According to one embodiment, the object is a bipolar plate for a fuel cell, a part of a car, a part of an aircraft, a part of a building, pipes of a reactor, flow guide vanes, etc., which are to be manufactured or designed.

[0103] According to one embodiment, the structural dimensions include at least one of position, radius, width, height, length, anchor points, etc. The physical dimensions include at least one of velocity, pressure, temperature, etc. Fig. Figure 10 illustrates an exemplary process for predicting the physical quantities of an object according to the principles of revelation. It should be noted that the process can be implemented using computers or processors.

[0104] In Block 1010, structural quantities are received by an HCNN framework, and these structural quantities are used to describe the boundaries of the object. The HCNN comprises at least one primary NN, at least one secondary NN, and a joining unit, where PDEs are formulated to characterize a physical system with respect to the object using the object's physical quantities.

[0105] In block 1020, a first prediction of solutions to reformulated PDEs is generated by at least one primary neural network. The reformulated PDEs are obtained by replacing gradients of the physical quantities in the PDEs with additional quantities, where the first prediction of the solutions of the reformulated PDEs includes a prediction of the object's physical quantities and the additional quantities that represent gradients of the object's physical quantities.

[0106] In block 1030, a second prediction of solutions to the BCs of the reformulated PDEs is generated by at least one secondary NN. The BCs are formulated to characterize the physical system with respect to the boundaries of the object, and the second prediction of the solutions to the BCs of the reformulated PDEs includes a prediction of the physical quantities of the object.

[0107] In block 1040, a final prediction of solutions of the reformulated PDEs is obtained by combining the first and second predictions of a solution using the merging unit. The final prediction of solutions of the reformulated PDEs represents a prediction of the object's physical properties.

[0108] According to one embodiment, the BCs of the edges of the object comprise one or more of a Dirichlet BC, Neumann BC and Robin BC.

[0109] According to one embodiment, the primary NN corresponds to the region outside the edges of the object, and the at least one secondary NN corresponds to at least one of the edges of the object.

[0110] According to one embodiment, the second prediction of the solutions of the BCs of the reformulated PDEs further includes a prediction of the additional quantities that represent gradients of the physical quantities of the object.

[0111] According to one embodiment, generating a second prediction of solutions of BCs of the reformulated PDEs in block 1030 further comprises: for each of at least one boundary corresponding to the at least one secondary NN, generating a corresponding part of the second prediction based on constraints of the BC of the boundary and an output of a corresponding secondary NN.

[0112] According to one embodiment, generating a second prediction of solutions of BCs of the reformulated PDEs in block 1030 further comprises: for each of the at least one boundary corresponding to the at least one secondary NN, generating the corresponding part of the second prediction further based on a basis matrix or basis vector of the null space for the boundary, wherein the basis matrix or basis vector is obtained based on a unit normal of the boundary.

[0113] According to one embodiment, obtaining a final prediction of solutions of the reformulated PDEs in block 1040 further comprises: combining the first prediction of a solution and the second prediction of a solution based on a distance of a sampled point of the object to the edges of the object.

[0114] According to one embodiment, the joining of the first prediction of a solution and the second prediction of a solution in block 1040 further comprises: joining respective parts of the second prediction according to the edges of the object by ensuring that a part of the second prediction corresponding to any edge at the other edges decreases to such an extent that it has an insignificant influence on the other edges.

[0115] According to one embodiment, process 1000 further comprises: obtaining a first discrepancy of the reformulated PDEs based on the final prediction of solutions, including the physical quantities of the object and gradients of the physical quantities; obtaining a second discrepancy between the physical quantities and the gradients of the physical quantities in the final prediction of solutions; obtaining a loss based on the first discrepancy and the second discrepancy and updating the at least one primary NN and the at least one secondary NN based on the loss.

[0116] According to one embodiment, the steps of generating a first prediction, generating a second prediction, obtaining a final prediction, obtaining a loss, and updating the at least one primary NN and the at least one secondary NN are repeated until a training requirement is met.

[0117] According to one embodiment, these are at least one primary NN and at least one secondary NN MLP-NNs.

[0118] According to one embodiment, the object is a bipolar plate for a fuel cell, a part of a car, a part of an aircraft, a part of a building, pipes of a reactor, flow guide vanes, etc., which are to be manufactured or designed.

[0119] According to one embodiment, the structural dimensions include at least one of position, radius, width, height, length, anchor points, etc. The physical dimensions include at least one of velocity, pressure, temperature, etc.

[0120] Fig. Figure 11 illustrates an exemplary computing system according to the aspects of the disclosure. The computing system 1100 can comprise at least one processor 1110. The computing system 1100 can further comprise at least one storage device 1120. The storage device 1120 can store computer-executable instructions which, when executed, cause the processor 1110 to perform operations according to the embodiments of the present disclosure, as described in connection with Fig. The steps described in points 1 to 10 are to be carried out.

[0121] The embodiments of the present disclosure may be embodied in a computer-readable medium, such as a non-volatile computer-readable medium. The non-volatile computer-readable medium may include instructions which, when executed, cause one or more processors to perform operations according to the embodiments of the present disclosure, as described in connection with Fig. The steps described in points 1 to 10 are to be carried out.

[0122] The embodiments of the present disclosure may be embodied in a computer program product comprising computer-executable instructions which, when executed, cause one or more processors to perform operations according to the embodiments of the present disclosure as described in connection with Fig. The steps described in points 1 to 10 are to be carried out.

[0123] It is understood that all operations in the above-described procedures are merely exemplary and that the present disclosure is not limited to operations in the procedures or sequences of these operations and should cover all other equivalents under the same or similar concepts.

[0124] It is also understood that all modules in the aforementioned facilities can be implemented in various ways. These modules can be implemented as hardware, software, or a combination of both. Furthermore, any of these modules can be functionally subdivided into submodules or combined.

[0125] The foregoing description is provided to enable a person skilled in the art to carry out the various aspects described herein. Various modifications of these aspects will be readily apparent to the person skilled in the art, and the general principles defined herein can 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 this disclosure, which are known or subsequently become known to the person skilled in the art, are expressly included herein by reference and are intended to be encompassed by the claims. QUOTES INCLUDED IN THE DESCRIPTION

[0000] This list of documents cited by the applicant was automatically generated and is included solely for the reader's convenience. The list is not part of the German patent or utility model application. The DPMA accepts no liability for any errors or omissions. Cited non-patent literature

[0000] Journal of Computational Physics, 378:686-707, 2019

[0021] Journal of Computational Physics, 428:110085, 2021

[0041]

Claims

[1] Computer-implemented method for optimizing the structure of an object, comprising: Receiving structural quantities by a constraint neural network (NN) framework, wherein the structural quantities are used to describe edges of the object, the constraint NN framework comprising at least one primary NN, at least one secondary NN, and a stitching unit, wherein partial differential equations (PDEs) are formulated to characterize a physical system with respect to the object with physical quantities of the object; Generating a first prediction of solutions of reformulated PDEs by the at least one primary NN, wherein the reformulated PDEs are obtained by replacing gradients of the physical quantities in the PDEs with additional quantities, wherein the first prediction of the solutions of the reformulated PDEs comprises a prediction of the physical quantities of the object and the additional quantities representing gradients of the physical quantities of the object; Generating a second prediction of solutions to boundary conditions (BCs) of the reformulated PDEs by at least using the at least one secondary NN, wherein the BCs are formulated to characterize the physical system with respect to the boundaries of the object, wherein the second prediction of the solutions of the BCs of the reformulated PDEs comprises a prediction of the physical quantities of the object; Obtaining a final prediction of solutions of the reformulated PDEs by merging the first prediction of a solution and the second prediction of a solution by the merging unit, wherein the final prediction of solutions of the reformulated PDEs represents a prediction of the physical quantities of the object; and Update the structural quantities representing the edges of the object based on the physical quantities of the final prediction. [2] The method of claim 1, wherein the BCs of the edges of the object comprise one or more of a Dirichlet BC, Neumann BC and Robin BC. [3] The method of claim 1, wherein the primary NN corresponds to a region outside the edge of the object, wherein the at least one secondary NN corresponds to at least one of the edges of the object. [4] The method of claim 1, wherein the second prediction of the solutions of the BCs of the reformulated PDEs further comprises a prediction of the additional quantities representing gradients of the physical quantities of the object. [5] The method of claim 2, wherein generating a second prediction of solutions of BCs of the reformulated PDEs further comprises: for each of at least one edge corresponding to the at least one secondary NN, generating a corresponding part of the second prediction based on constraints on the BC of the edge and an output of a corresponding secondary NN. [6] The method of claim 5, wherein generating a second prediction of solutions of BCs of the reformulated PDEs further comprises: for each of the at least one edge corresponding to the at least one secondary NN, generating the corresponding part of the second prediction further based on a basis matrix or a basis vector of the null space for the edge, wherein the basis matrix or the basis vector is obtained based on a unit normal of the edge. [7] The method of claim 2, wherein obtaining a final prediction of solutions of the reformulated PDEs by merging the first prediction of a solution and the second prediction of a solution by the merging unit further comprises: Merging the first prediction of a solution and the second prediction of a solution based on a distance of a sampled point of the object to the edges of the object. [8] The method of claim 7, wherein merging the first prediction of a solution and the second prediction of a solution further comprises merging respective portions of the second prediction corresponding to the edges of the object by ensuring that a portion of the second prediction corresponding to any edge falls off at the other edges to an extent that it has an insignificant influence on the other edges. [9] The method of claim 2, further comprising: Obtaining a first discrepancy of the reformulated PDEs based on the final prediction of solutions, including the physical quantities of the object and gradients of the physical quantities; Obtaining a second discrepancy between the physical quantities and the gradients of the physical quantities in the final prediction of solutions; Obtaining a loss based on the first discrepancy and the second discrepancy and Updating the at least one primary NN and the at least one secondary NN based on the loss. [10] The method of claim 9, further comprising: Repeating the steps of generating a first prediction, generating a second prediction, obtaining a final prediction, obtaining a loss, and updating the at least one primary NN and the at least one secondary NN until a training requirement is met. [11] The method of claim 1, wherein the at least one primary NN and the at least one secondary NN are multilayer perceptron NNs (MLP-NNs). [12] The method of claim 1, wherein the object is one of a bipolar plate for a fuel cell, a part of a car, a part of an aircraft, a part of a building, pipes of a reactor, flow baffles. [13] The method of claim 1, wherein the structural quantities comprise at least one of position, radius, width, height, length, anchor points and the physical quantities comprise at least one of velocity, pressure, temperature. [14] A computer-implemented method for predicting physical quantities of an object, comprising: Receiving structural quantities by a constraint neural network (NN) framework, wherein the structural quantities are used to describe edges of the object, the constraint NN framework comprising at least one primary NN, at least one secondary NN, and a stitching unit, wherein partial differential equations (PDEs) are formulated to characterize a physical system with respect to the object with physical quantities of the object; Generating a first prediction of solutions of reformulated PDEs by the at least one primary NN, wherein the reformulated PDEs are obtained by replacing gradients of the physical quantities in the PDEs with additional quantities, wherein the first prediction of the solutions of the reformulated PDEs comprises a prediction of the physical quantities of the object and the additional quantities representing gradients of the physical quantities of the object; Generating a second prediction of solutions to boundary conditions (BCs) of the reformulated PDEs by at least using the at least one secondary NN, wherein the BCs are formulated to characterize the physical system with respect to the boundaries of the object, wherein the second prediction of the solutions of the BCs of the reformulated PDEs comprises a prediction of the physical quantities of the object; Obtaining a final prediction of solutions of the reformulated PDEs by merging the first prediction of a solution and the second prediction of a solution by the merging unit, wherein the final prediction of solutions of the reformulated PDEs represents a prediction of the physical quantities of the object. [15] The method of claim 14, wherein the BCs of the edges of the object comprise one or more of a Dirichlet BC, Neumann BC and Robin BC. [16] The method of claim 15, wherein generating a second prediction of solutions of BCs of the reformulated PDEs further comprises: for each of the at least one edge corresponding to the at least one secondary NN, generating a corresponding part of the second prediction based on constraints on the BC of the edge and an output of a corresponding secondary NN. [17] The method of claim 15, wherein obtaining a final prediction of solutions of the reformulated PDEs by merging the first prediction of a solution and the second prediction of a solution by the merging unit further comprises: Merging the first prediction of a solution and the second prediction of a solution based on a distance of a sampled point of the object to the edges of the object. [18] The method of claim 13, further comprising: Obtaining a first discrepancy of the reformulated PDEs based on the final prediction of solutions, including the physical quantities of the object and the gradients of the physical quantities; Obtaining a second discrepancy between the physical quantities and the gradients of the physical quantities in the final prediction of solutions; Obtaining a loss based on the first discrepancy and the second discrepancy and Updating the at least one primary NN and the at least one secondary NN based on the loss. [19] Computer system comprising: one or more processors and one or more storage devices having stored thereon computer-executable instructions which, when executed, cause the one or more causing the plurality of processors to perform the operations of the method according to any one of claims 1 to 18. [20] One or more computer-readable storage media having stored thereon computer-executable instructions which, when executed, cause one or more processors to perform the operations of the method of any one of claims 1 to 18.

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