A method for constructing physical field operators based on differential homeomorphism and application thereof

Abstract

A physical field operator construction method based on differential homeomorphism and application, different definition domains D i under partial differential equations are mapped into the same input domain through differential homeomorphism mapping, based on the generalized invariance principle, the invariance of physical laws before and after mapping is ensured, that is, partial differential equations under different definition domains D i can share physical laws in the input domain, and then a unified solution function operator is learned in the input domain through the neural operator class method, and finally the solution function operator corresponding to the partial differential equation can be given for the partial differential equation under different definition domains.

Description

Technical Field

[0001] This invention belongs to the fields of artificial intelligence and computational mechanics, and in particular relates to a data-driven method for solving partial differential equations of physical fields under different domains, specifically a method for constructing physical field operators based on differential homeomorphisms and its application. Background Technology

[0002] In science and engineering, many problems can be modeled using partial differential equations (PDEs), and solving PDEs is crucial for solving these problems. However, many PDEs lack analytical solutions under complex boundary and initial conditions. Therefore, over the past few decades, numerical methods such as finite difference (FD), finite volume (FV), and finite element (FE) methods have been used to obtain approximate solutions to PDEs. However, these numerical methods are computationally inefficient when dealing with complex or repetitive scenarios.

[0003] In recent years, with the rapid development of computer and artificial intelligence technologies, machine learning methods have gradually attracted attention in the field of computational mechanics due to their powerful modeling capabilities for complex nonlinear systems. Various neural operator-based methods have been developed and applied to solving partial differential equations in fields including fluid dynamics and heat transfer. Well-trained neural operator models are orders of magnitude faster than traditional numerical methods. Furthermore, researchers have extended neural operator-based methods to solve partial differential equations with fixed irregular domains by using coordinate transformations and parameterization. For example, elliptical coordinate transformation maps the domain of a quadrilateral to a unit rectangular domain, thus extending neural operator-based methods to solving partial differential equations with fixed irregular domains. However, in many real-world scenarios, such as injection molding, additive manufacturing, and subtractive manufacturing in the manufacturing industry, aerodynamic design based on computational fluid dynamics (CFD), and spatial transformation of fluids, the domains of their partial differential equation systems are not only irregular but also constantly changing. The solution operators for partial differential equations differ under different domains, but existing neural operator-based methods can only fit a single operator. This limits the application of the solution function operators obtained by existing methods to fixed domains, making it impossible to generalize to other domains after model training, and thus unsuitable for scenarios with varying domains. Therefore, establishing a general machine learning model for scenarios with varying domains remains a challenge. Summary of the Invention

[0004] The purpose of this invention is to address the problem that existing machine learning methods can only fit a single operator, and that the stability and generalization performance of the model are difficult to guarantee when the domain of the partial differential equation of the physical field changes. This invention proposes a physical field operator construction method based on differential homeomorphism, and has been successfully applied to the prediction of machining deformation of aerospace structural components. It first uses differential homeomorphism mapping to connect different domains D... i Mapping partial differential equations to the same input domain In this context, based on the principle of generalized invariance, the invariance of physical laws before and after the mapping is guaranteed, i.e., different domains D... i The partial differential equations under the following conditions can be used in the input domain. Sharing physical laws, and thus in the input domain The method learns a unified solution function operator through neural operator classes. Ultimately, when solving partial differential equations with different domains, this solution operator... The solution function can be given accurately in both cases.

[0005] The technical solution of this invention is:

[0006] A method for constructing physical field operators based on differential homeomorphisms and its application. The overall idea is as follows: First, for the partial differential equations of physical fields in different domains D... i The problem of solving the domain D is first solved by using a set of differential homeomorphisms F to define the different domains D. i Mapped to the same input domain In, that is

[0007]

[0008] Secondly, in the input domain In this paper, neural operators are used to construct solution operators for partial differential equations.

[0009]

[0010]

[0011] in, It is the parameter function of the partial differential equation under this physical field. These are the boundary conditions for a partial differential equation. F is the solution function of the partial differential equation. -1 It is the inverse mapping of the differential homeomorphism F.

[0012] Finally, the learned operators are used. For different domains D i Solve the partial differential equation in ∈D to obtain the solution function.

[0013] Furthermore, the partial differential equations in the scheme contain solution functions. Parameter function Boundary conditions And defined in the domain D i In this context, its operator form can be expressed as:

[0014] Gu i (x)=f(x), x∈D i (Formula 4)

[0015]

[0016] Where G is the partial differential operator, B is the boundary operator, and D is the boundary operator. i The domain of the partial differential equation is the range of values ​​for the independent variable x.

[0017] The solution function operator of a partial differential equation refers to the parameter function of the partial differential equation. and corresponding boundary conditions to solution function mapping Its operator form can be expressed as:

[0018]

[0019] Furthermore, the differential homeomorphism F is preferably a harmonic map f, and the harmonic map f establishes different domains D. i To the same input domain C between 1 A smooth map, or harmonic map f, is obtained by minimizing the harmonic energy. Minimizing the harmonic energy is equivalent to solving the Laplace differential equation of the harmonic map, i.e.:

[0020] Δf = 0 (Formula 7)

[0021] Where Δ is the Laplace operator. This equation is generally solved in a computer using an approximate discrete form, which is:

[0022]

[0023] in The weight of the remaining cutting edge.

[0024] Furthermore, neural operator-based methods refer to methods that utilize neural operators with parameter θ. As a surrogate model, it learns from the data and approximates nonlinear computation. Among them, neural operators Composed of coding networks and multi-layer nonlinear operator layers Composition can be expressed as:

[0025]

[0026] In the formula It is a fully connected neural network used to encode a high-dimensional space into a low-dimensional solution function space. It is a fully connected neural network used to encode low-dimensional input functions into a high-dimensional space. Each nonlinear operator layer has the same structure, which can be expressed as:

[0027] (Formula 10)

[0028] Where σ is a nonlinear activation function, and I(x) is the input function encoded into a high-dimensional space. For linear operators, b N For bias operators, It is a kernel integral operator.

[0029] In the above schemes, different domains D i ∈D have the same genus.

[0030] The steps of this invention in predicting deformation of aerospace structural components are as follows:

[0031] Step 1) Perform machining deformation mechanism analysis and determine the domain D i The machining process of aerospace structural components is a material removal process. As the material is removed, the workpiece will deform under the action of an unbalanced residual stress field. The relationship between the residual stress field and the workpiece deformation can be expressed as:

[0032]

[0033]

[0034] In the formula σ i (x, y, z) represents the residual stress field of the workpiece; E i (x, y, z) represents the stiffness of the workpiece; u i (x, y, z) represents the machining deformation of the workpiece; the domain is D. i Let B be the geometry of the remaining material on the workpiece, which changes continuously as machining progresses; B is the boundary operator, specifying the clamping state of the workpiece. The operator form of the above equation can be expressed as:

[0035]

[0036] in It is the solver of formula 11.

[0037] Step 2) Domain Mapping: Sample workpieces D under different geometries to obtain training set D1. For workpieces with different geometric shapes in training set D1, firstly, use harmonic mapping f to map the equations under different domains D1 to the same input domain. In the middle, the input field Given a unit square domain; calculate the harmonic mapping f;

[0038] Step 3) Building and training the neural operator model: in a unit square domain Uniform sampling is performed to obtain a unit square domain. Grid points under uniform sampling Through the inverse mapping f of the harmonic mapping f -1 Obtain the coordinates v(x, y, z) of the workpiece in its domain, i.e.:

[0039]

[0040] Then, the stress function σ corresponding to the coordinate point v is obtained. i (v) Bending stiffness E i (v), solution function u i (v), and use these data as training data to perform Fourier neural operator modeling. The training of the Fourier neural operator is completed when the model converges. The operator that can approximate the workpiece deformation equation in Formula 13

[0041] Step 4) Prediction of machining deformation for workpieces with different geometries: For workpieces D under different geometries i ∈D2, where D2∈D_object. First, the harmonic mapping f obtained in step 2 is used, and then in workpiece D... i The corresponding unit square domain Sampling is performed to obtain the unit square domain The stress function σ in discrete form i (v) Bending stiffness E i (v) Using the above function as a Fourier neural operator The input, Fourier neural operator The corresponding deformation function u will be given. i That is, the workpiece is in D i Deformation under geometry.

[0042] The beneficial effects of this invention are:

[0043] 1. Under the principle of generalized invariance, the physical field operator construction method based on differential homeomorphism obtained by this invention has strong generalization performance and can generalize to partial differential equation solving problems in domains outside the training set;

[0044] 2. This method is based on differential homeomorphism, which can map partial differential equations under irregular and non-fixed domains to a fixed and regular new domain, thus alleviating the requirements of machine learning methods on data input and output formats; Attached Figure Description

[0045] Figure 1 A schematic diagram illustrating the overall approach to operator construction and application based on differential homeomorphisms. Detailed Implementation

[0046] The present invention will be further described below with reference to the accompanying drawings and examples, but the present invention is not limited to these embodiments.

[0047] The following section uses the prediction of machining deformation of aerospace structural components as an example to provide a detailed explanation of the operator construction method and its application based on differential homeomorphism. The overall concept diagram is shown below. Figure 1 As shown, the specific steps include:

[0048] 1. Analysis of Deformation Mechanism During Machining: The machining process of aerospace structural components involves material removal. As material is removed, the original residual stress equilibrium state inside the workpiece is disrupted. Under the action of an unbalanced residual stress field, the workpiece deforms, and the geometry of the remaining material in the workpiece forms the domain D for solving the problem. i Its deformation changes continuously as processing progresses. Based on the workpiece state and clamping layout, the deformation of the structural component can be simplified and analyzed using a cantilever beam model from typical mechanics, as shown in the figure. The workpiece thickness is relatively small relative to its length (x), so the deformation is mainly caused by the bending moment M. i The bending along the thickness direction z caused by (x), and the unbalanced residual stress σ x The bending moment generated by (x, z) can be expressed as:

[0049]

[0050] Furthermore, since the ratio of the workpiece thickness to the minimum dimension of the material surface is less than 0.2, the deformation mechanics model of the workpiece is established using thin plate theory. The relationship between workpiece deformation and bending moment can be expressed by the equilibrium equation based on thin plate theory as follows:

[0051]

[0052]

[0053] Among them B i (x) represents the bending stiffness, which is related to the workpiece geometry D. i Variables related to workpiece material properties (elastic modulus E and Poisson's ratio μ), u i (x) represents the part in geometric shape D. i The solution function of the deformable field under the given conditions. It is a biharmonic operator, M i (x) represents the torque distribution along the length x of the workpiece. The operator form of the above equation can be expressed as:

[0054]

[0055] in It is the solver of Equation 12, specifically the parametric function σ of the residual stress field. x (x, z), bending stiffness function gi (x) to the solution function u of the deformed field i The mapping of (x).

[0056] 2. Domain Mapping: For workpieces with different geometries, the workpieces with different geometries are first parameterized using triangular meshes, and then the different domains D are mapped using a harmonic mapping f. i The equations below are mapped to the same input domain In the middle, the input field It is a unit square domain.

[0057] The Laplace differential equation for the harmonic mapping f is established. The Laplace operator on the mesh can be defined in discrete form and can be calculated from each vertex of the triangular mesh. The formula for calculating the Laplace operator for a triangular mesh is as follows:

[0058]

[0059] In the formula, Δf is the Laplace operator, f i and f j These are the triangular mesh vertices i and j mapped by the harmonic mapping f into the unit square domain. The coordinates in w ij To connect the edge e between vertex i and vertex j of the triangular mesh ij The weight.

[0060] In the triangular mesh, edge e ij weight w ij The cotangent formula is used to calculate the cotangent, which is as follows:

[0061]

[0062] Equation 15 can be further improved by using a weighted average of the function values ​​at adjacent vertices. The Laplace operator can be expressed using cotangent weights as follows:

[0063] Lx = 0 (Formula 17)

[0064] Where x = {…f i …} represents the unit square domain formed by harmonic mapping of all triangular mesh nodes. The matrix consists of coordinates, where L is the Laplace matrix, and the elements of the matrix can be obtained using cotangent weights.

[0065]

[0066] The harmonic mapping f and its domain D can be calculated using Equation 15. i The mapping of the following to the unit square domain The coordinate x after that.

[0067] 3. Neural Operator Model Construction and Training: Samples are taken from workpieces D under different geometries to obtain a training set D1. The solution function u of the workpiece deformation field in training set D1 is obtained using the finite element method, and then applied in a unit square domain. Sampling is performed to obtain the unit square domain The residual stress field parameter function σ in discrete form x Bending stiffness g i The solution function u of the deformation field i Then, a Fourier neural operator model is developed using the training set data. The specific training methods are as follows:

[0068] The harmonic mapping f obtained in step 2 is used in the unit square domain. Uniform sampling is performed to obtain a unit square domain. Grid points under uniform sampling Through the inverse mapping f of the harmonic mapping f -1 Obtain the corresponding coordinate point v(x, z) in the workpiece's domain, i.e.:

[0069]

[0070] Then, the stress σ corresponding to coordinate point v is obtained. x (v), Bending stiffness g i (v) The solution function u is obtained by solving Equation 12 using the finite element model. i (v).

[0071] u i (v)=FEM(σ x (v), g i (v)) (Formula 20)

[0072] At this point, the residual stress field function σ in discrete form is further... x Bending stiffness g i and deformation field function u i All of these can be represented using matrices, and Fourier neural operator models can be built in a Python environment. The input to the model is the residual stress field function σ corresponding to different geometric workpieces in the training set D1. x Bending stiffness g i The output is the deformation field function u of the workpiece. i The model training process samples the Adam optimizer. When the model reaches convergence, the Fourier neural operator... The operator that can approximate the workpiece deformation equation in Formula 13

[0073] 4. Prediction of machining deformation for parts with different geometries: For workpieces D under different geometries... i ∈D2, where D2∈D and First, the harmonic mapping f obtained in step 2 is applied to the unit square domain. Sampling is performed to obtain the unit square domain The stress function σ in discrete form x Bending stiffness g i and deformation function u i Finally, the stress function σ x Bending stiffness g i and deformation function u i As a Fourier neural operator The input, Fourier neural operator The corresponding deformation function u will be given. i That is, the workpiece is in D i Deformation under geometry.

[0074] The parts not covered in this invention are the same as or can be implemented using existing technologies.

Claims

1. A method for constructing physical field operators based on differential homeomorphism, characterized in that: For partial differential equations of physical fields in different domains The following problem is solved first through a set of differential homeomorphisms. Different domains Mapped to the same input domain In, that is (Official 1) Furthermore, in the input domain In this paper, neural operators are used to construct solution operators for partial differential equations. , (Official 2) (Official 3) in, It is the parameter function of the partial differential equation under this physical field. These are the boundary conditions for a partial differential equation. It is the solution function of the partial differential equation. It is a differential homeomorphism mapping The inverse mapping; finally, the learned operator is used. For different domains The partial differential equations in the equations are solved to obtain the solution function; the steps for predicting deformation of aerospace structural components are as follows: Step 1) Perform machining deformation mechanism analysis and determine the domain of definition. The machining process of aerospace structural components involves material removal. As material is removed, the workpiece deforms under the influence of an unbalanced residual stress field. The relationship between the residual stress field and the workpiece deformation is as follows: (Official 11) (Official 12) In the formula This represents the residual stress field of the workpiece. For the stiffness of the workpiece; For the machining deformation of the workpiece; domain The geometry of the remaining material in the workpiece changes continuously as processing progresses; The boundary operator specifies the clamping state of the workpiece; the operator form of the above equation is: (Official 13) in It is the solver of formula 11; Step 2) Domain mapping: For workpieces under different geometries Sampling is performed to obtain the training set. For the training set Workpieces of different geometric shapes are first processed through harmonic mapping. Different domains The equations below are mapped to the same input domain In the middle, the input field Given a unit square domain; calculate the harmonic mapping. ; Step 3) Building and training the neural operator model: in a unit square domain Uniform sampling is performed to obtain a unit square domain. Grid points under uniform sampling Through harmonic mapping inverse mapping Obtain the corresponding coordinate points in the workpiece domain. ,Right now: (Official 14) Thus, the coordinates are obtained. Corresponding stress function Bending stiffness Solution function And use this data as the training set to develop a Fourier neural operator model. The training of the Fourier neural operator is completed when the model converges. Approximating the solver of the workpiece deformation equation in Formula 13 ; Step 4) Prediction of machining deformation for workpieces with different geometries: For workpieces with different geometries ,in and First, the harmonic mapping obtained in step 2... and in the workpiece The corresponding unit square domain Sampling is performed to obtain the unit square domain Stress function in discrete form Bending stiffness The above function is used as a Fourier neural operator. The input, Fourier neural operator The corresponding deformation function will be given. That is, the workpiece is Deformation under geometry.

2. The method according to claim 1, characterized in that: The partial differential equations mentioned include solution functions , parameter function Boundary conditions And defined in the domain In this context, its operator form is: (Official 4) (Official 5) in For partial differential operators, For boundary operators, The independent variable in a partial differential equation The range of values ​​for is the domain of the partial differential equation; The so-called solver refers to the parameter function of the partial differential equation. and corresponding boundary conditions to solution function mapping Its operator form is: (Official 6).

3. The method according to claim 1, characterized in that: The aforementioned differential homeomorphism mapping Choosing harmonic mapping Harmonic mapping Different domains were established To the same input domain Between Smooth mapping, harmonic mapping Minimizing the harmonic energy is achieved by minimizing the harmonic energy. The process of minimizing the harmonic energy is equivalent to solving the Laplace differential equation of the harmonic mapping, i.e.: (Official 7) in It is the Laplace operator; the approximate discrete form of this equation is used to solve it in a computer, and its approximate discrete form is: (Official 8) in The weight of the remaining cutting edge.

4. The method according to claim 1, characterized in that: The aforementioned neural operator class method refers to a method that uses parameters... neural operators As a surrogate model, it learns from the data and approximates nonlinear operators. ;In which neural operators Composed of coding networks and multi-layer nonlinear operator layers Composition, expressed as: (Official 9) in It is a fully connected neural network used to encode a high-dimensional space into a low-dimensional solution function space; It is a fully connected neural network used to encode low-dimensional input functions into a high-dimensional space; each nonlinear operator layer has the same structure, expressed as: (Official 10) in It is a non-linear activation function. The input function is encoded into a high-dimensional space. For linear operators, For bias operators, It is a kernel integral operator.

5. The method according to claim 1, characterized in that: The different domains mentioned above have the same genus.

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