A neural operator modeling method based on a differential homeomorphism mapping network and application thereof

Abstract

The application discloses a neural operator modeling method based on a differential homeomorphism mapping network and application thereof, belongs to the technical field of artificial intelligence and computational mechanics, and is used for quickly solving partial differential equations for describing physical relations in a part machining deformation process under different complex part geometric domains, and comprises the following steps: a raw geometric domain of an input function is geometrically represented; mapping in different geometric domains to a reference domain is obtained through a differential homeomorphism neural network; a neural operator is constructed in the reference geometric domain, a mapping relation between an input function and an output function of the partial differential equation is established, and solving of the partial differential equation under different geometric domains is realized; and the above method and application thereof are applied to scenarios such as part residual stress field prediction deformation field.

Description

Technical Field

[0001] The present invention relates to the technical fields of artificial intelligence and computational mechanics, and in particular to a neural operator modeling method based on a differential homeomorphism mapping network and its application. Background Art

[0002] Partial differential equations (PDEs) are a crucial tool in scientific research and engineering design, enabling them to model the relationships between variables in scientific problems and analyze and predict the behavior of complex systems. However, PDEs for complex systems are often difficult to solve analytically. Numerical methods such as finite element and finite difference methods have been widely used to solve PDEs, but they still face computational cost and efficiency challenges due to the high modeling accuracy and solution precision required for complex problems.

[0003] Data-driven methods can achieve rapid solutions to target variables by modeling the correlation between complex variables in the system. Among them, the neural operator methods proposed in recent years, such as DeepONet and FNO, can model and quickly solve the relationship between partial differential equations. After training, the model can achieve efficient solution of partial differential equations under different input function conditions. However, it can only solve the problem of solving partial differential equations defined in a fixed domain. For the problem of solving partial differential equations defined in different geometric domains, the prior art can refer to the patent application announced as CN117648864A, which discloses a physical field operator construction method and application based on differential homeomorphism. By solving the harmonic mapping between the definition domain and the reference domain, the differential homeomorphism of different definition domains is mapped to the reference domain for operator learning. However, the above-mentioned neural operator modeling method is suitable for two-dimensional geometric mapping, and each definition domain needs to be solved separately. The computational efficiency is difficult to meet the scenarios of continuous changes and large number of iterations of complex geometric domains such as part design optimization. How to quickly solve the differential homeomorphism mapping of complex three-dimensional geometric domains to the reference domain has become a key bottleneck in the modeling of neural operators under different definition domains. Summary of the Invention

[0004] The purpose of the present invention is to provide a neural operator modeling method based on a differential homeomorphism mapping network and its application, so as to solve the problem that existing neural operator methods are difficult to efficiently model and solve partial differential equations on complex three-dimensional geometric domains.

[0005] To achieve the above objectives, the present invention provides a neural operator modeling method based on a differential homeomorphism mapping network for rapidly solving partial differential equations describing the physical relationships of part machining deformation processes in different complex part geometric domains, comprising the following steps:

[0006] S1. Input the original geometric domain of the part where the function is located , The geometric representation is , the input function is the physical quantity function that affects the part processing deformation from the physical relationship;

[0007] S2, through the differential homeomorphism neural network, the different geometric domains Mapping to a reference domain ,get ;

[0008] S3. In the reference domain A neural operator is constructed in the algorithm to establish a mapping relationship between the input function and the output function of the partial differential equation, so as to solve the partial differential equations in different geometric domains.

[0009] Preferably, the geometric representation in step S1 is in the form of a point cloud, and the original geometric representation of the complex geometric domain where the field function is located is represented in a discrete form as follows: :

[0010] ;

[0011] in, The first points; is the coordinate in three-dimensional space; is the total number of points in the point cloud.

[0012] Preferably, the input function in step S1 is preferably a residual stress field function, and the corresponding output function is a deformation field function.

[0013] Preferably, the input of the diffeomorphic neural network in step S2 is the geometric representation of the original domain , or the original domain geometry representation and target geometry representation , the output is Mapping to reference domain Geometric representation of , diffeomorphic neural networks Expressed as:

[0014] ;

[0015] ;

[0016] in, Diffeomorphic mappings learned for diffeomorphic neural networks.

[0017] Preferably, the differential homeomorphism neural network guides the training update of the network through the total loss constraint of the neural network, wherein the total loss constraint of the neural network includes the differential homeomorphism constraint and the geometric approximation constraint, and the corresponding neural network loss term is:

[0018] ;

[0019] in, is the total loss of the diffeomorphic neural network; is the diffeomorphism loss term; is the geometric approximation loss term.

[0020] Diffeomorphism loss term The constraints consist of the reversibility constraints and smoothness constraints of the mapping, and the corresponding neural network loss term is:

[0021] ;

[0022] in, is the reversibility loss term; is the smoothness loss term.

[0023] Preferably, the reversibility constraint makes the mapping converge to a bijection, and the reversibility loss term is expressed as:

[0024] ;

[0025] in, is the mapping reversibility evaluation function; is the coefficient of the reversibility loss term;

[0026] The smoothness constraint makes the mapping converge to a differentiable mapping, and the smoothness loss term is expressed as:

[0027] ;

[0028] in, is the evaluation function of mapping smoothness; is the coefficient of the smoothness loss term.

[0029] Preferably, the geometric approximation constraint is and The distance between them, the geometric approximation loss term is expressed as:

[0030] ;

[0031] in, is the geometric distance calculation function; is the coefficient of the geometric approximation loss term; the geometric approximation constraint makes Converge to the reference domain .

[0032] Preferably, the partial differential equation in step S3 is defined in the original geometric domain In the example, the input function is the parameter function , boundary conditions Any one or combination of the above, the output function is the solution function , its operator form is expressed as:

[0033] ;

[0034] in, Define solution operators for partial differential equations in infinite dimensional space; are the function spaces of parameter function, boundary conditions and solution function respectively.

[0035] Preferably, the neural operator in step S3 uses a parameter Neural network model construction function mapping , a solution operator that maps the input function space of the approximate partial differential equation to the solution function space , expressed as:

[0036] ;

[0037] in, For parameters The parameter space to which it belongs.

[0038] The present invention also provides an application of a neural operator modeling method based on a differential homeomorphism mapping network in the rapid prediction scenario of the processing deformation field of various geometrically complex structural parts such as aircraft frames and beams. The corresponding partial differential equation input function is a physical quantity function that affects the processing deformation of parts from a physical relationship, preferably a residual stress field function, and the corresponding solution function is a deformation field function.

[0039] Therefore, the present invention adopts the above-mentioned neural operator modeling method based on differential homeomorphism mapping network and its application, which has the following beneficial effects:

[0040] (1) Based on the differential homeomorphism mapping neural network, the differential homeomorphism mapping from different geometric domains to the reference domain can be quickly solved;

[0041] (2) Representing geometry as a point cloud, suitable for modeling three-dimensional geometry using differential homeomorphism mapping operators;

[0042] (3) The constructed neural operator can map partial differential equations in different domains to the same reference domain for generalized solution.

[0043] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 This is a schematic diagram of a neural operator modeling method based on a differential homeomorphism mapping network according to the present invention;

[0045] Figure 2 This is an overall flow chart of a neural operator modeling method based on a differential homeomorphism mapping network according to the present invention;

[0046] Figure 3 This is a diagram showing the rapid prediction effect of deformation field in aircraft structural parts processing using an embodiment of the present invention. DETAILED DESCRIPTION

[0047] The following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but rather merely represents selected embodiments of the present invention. All other embodiments derived by persons of ordinary skill in the art based on the embodiments of the present invention without inventive effort shall fall within the scope of protection of the present invention.

[0048] See also Figures 1-2 A neural operator modeling method based on a differential homeomorphism mapping network is used to quickly solve the partial differential equations describing the physical relationship of the part processing deformation process in complex parts and variable geometry domains. It includes the following steps:

[0049] S1. Input the original geometric domain of the part where the function is located , The geometric representation is , the input function is the physical quantity function that affects the part processing deformation from the physical relationship, preferably the residual stress field function, and the corresponding output function is the deformation field function;

[0050] S2, through the differential homeomorphism neural network, the different geometric domains Mapping to a reference domain ,get ;

[0051] S3. In the reference domain A neural operator is constructed in the algorithm to establish a mapping relationship between the input function and the output function of the partial differential equation, so as to solve the partial differential equations in different geometric domains.

[0052] This embodiment uses the above method to apply to the problem of machining deformation prediction of aircraft structural parts as an example, specifically including:

[0053] Aircraft structural components have a variety of part characteristics and complex initial residual stress field distributions. As the material is removed during machining, the balance of the initial residual stress is disrupted, forming internal torques and causing deformation. After machining is complete and the fixture constraints are removed, the unbalanced stress field is redistributed, causing the part to deform. The mechanical relationship between the residual stress field of the part and the machining deformation field can be expressed as a partial differential equation:

[0054] ;

[0055] ;

[0056] in, Represents the geometric domain during part processing, is the initial residual stress field distribution function of the part, is a coefficient determined by the mechanical properties of the material and the geometry of the part, is the distribution function of part processing deformation, Represents a clamping constraint point on a part.

[0057] The data involved in model training include part geometry, residual stress field, and machining deformation field. The obtained geometry, stress field, and deformation field data are all represented geometrically in the form of point clouds:

[0058] ;

[0059] in, The first Points, are coordinates in three-dimensional space, is the stress value at the corresponding position, is the deformation value of the corresponding position, is the total number of points in the point cloud.

[0060] Diffeomorphic Mapping Network Construction: From Point Cloud Data Extract the geometric part As the input of the differential homeomorphism mapping network DiffNet, the model fits a mapping during training , through the loss term constraint, the model realizes a differential homeomorphism mapping and represents the original geometry Align to reference domain Target geometry representation :

[0061] ;

[0062] The diffeomorphism constraints are the reversibility and smoothness constraints of the mapping. The reversibility constraint makes the mapping converge to a bijection, and the reversibility loss term is obtained by the Jacobian determinant Calculate and express it as:

[0063] ;

[0064] in, is the mapping reversibility evaluation function, is the coefficient of the reversibility loss term;

[0065] The smoothness constraint makes the mapping converge to a differentiable mapping, and the resulting smoothness loss term is calculated by the Sobolev regularization term of the mapping. Calculate and express it as:

[0066] ;

[0067] wherein, is a mapping smoothness evaluation function, is a coefficient of the smoothness loss term;

[0068] The reversibility constraint and the smoothness constraint jointly make the differential homeomorphism mapping neural network realize a differential homeomorphism.

[0069] The geometric approximation constraint is the distance between and The geometric approximation loss term is calculated by the Sinkhorn distance function :

[0070] ;

[0071] wherein, is a coefficient of the geometric approximation loss term; the geometric approximation constraint makes the original geometric representation mapped to the geometric representation in the reference domain by the differential homeomorphism mapping neural network.

[0072] Neural operator model construction: the stress field part of the input function and the reference domain geometric representation obtained by the differential homeomorphism mapping network are taken as the input of the neural operator, the deformation field part is taken as the output of the neural operator, and the model is trained. The neural operator fits the mapping relationship between functions in the partial differential equation through the kernel integral operation, and outputs the prediction of the deformation field function :

[0073] ；

[0074] and further calculate the loss function with the deformation field label data to complete the training of the neural operator model. The deformation field prediction effect of part machining is shown in Figure 3 .

[0075] Therefore, the neural operator modeling method based on the differential homeomorphism mapping network and the application thereof are adopted, aiming at the problem that the existing neural operator method is difficult to efficiently model and solve the partial differential equation problem on a complex three-dimensional geometric domain, and the neural operator fitting partial differential equation solver efficiently calculates the function relationship.

[0076] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A neural operator modeling method based on a differential homeomorphism mapping network is used to quickly solve partial differential equations describing the physical relationship of the part processing deformation process in different complex part geometric domains, characterized by: The following steps are involved: S1. Input the original geometric domain of the part where the function is located , The geometric representation is , the input function is the physical quantity function that affects the part processing deformation from the physical relationship; S2, through the differential homeomorphism neural network, the different geometric domains Mapping to a reference domain ,get ; S3. In the reference domain Construct neural operators in the model, establish the mapping relationship between the input function and the output function of the partial differential equation, and solve the partial differential equations in different geometric domains; The partial differential equation in step S3 is defined in the original geometric domain In the example, the input function is the parameter function , boundary conditions Any one or combination of the above, the output function is the solution function , its operator form is expressed as: ； in, Define solution operators for partial differential equations in infinite dimensional space; are the function spaces of parameter function, boundary conditions and solution function respectively; The neural operator in step S3 uses a Neural network model construction function mapping , a solution operator that maps the input function space of the approximate partial differential equation to the solution function space , expressed as: ； in, For parameters The parameter space to which it belongs.

2. The neural operator modeling method based on a differential homeomorphism mapping network according to claim 1, characterized in that: The geometric representation in step S1 is in the form of a point cloud, and the original geometric representation of the complex geometric domain where the field function is located in a discrete form is : ； in, The first points; is the coordinate in three-dimensional space; is the total number of points in the point cloud.

3. The neural operator modeling method based on a diffeomorphic mapping network according to claim 1, characterized in that: The input function in step S1 is the residual stress field function, and the output function is the deformation field function.

4. The neural operator modeling method based on a differential homeomorphism mapping network according to claim 1, characterized in that: The input of the diffeomorphic neural network in step S2 is the original geometric representation , or the original geometric representation and target geometry representation , the output is Mapping to reference domain Geometric representation of , diffeomorphic neural networks Expressed as: ； ； in, Diffeomorphic mappings learned for diffeomorphic neural networks.

5. The neural operator modeling method based on a differential homeomorphism mapping network according to claim 4, characterized in that: The differential homeomorphism neural network guides the training update of the network through the total loss constraint of the neural network. The total loss constraint of the neural network includes the differential homeomorphism constraint and the geometric approximation constraint. The corresponding neural network loss term is: ； in, is the total loss of the diffeomorphic neural network; is the diffeomorphism loss term; is the geometric approximation loss term; Diffeomorphism loss term The constraints consist of the reversibility constraints and smoothness constraints of the mapping, and the corresponding neural network loss term is: ； in, is the reversibility loss term; is the smoothness loss term.

6. The neural operator modeling method based on a differential homeomorphism mapping network according to claim 5, characterized in that: The reversibility constraint makes the mapping converge to a bijection, and the reversibility loss term is expressed as: ； in, is the mapping reversibility evaluation function; is the coefficient of the reversibility loss term; The smoothness constraint makes the mapping converge to a differentiable mapping, and the smoothness loss term is expressed as: ； in, is the evaluation function of mapping smoothness; is the coefficient of the smoothness loss term.

7. The neural operator modeling method based on a differential homeomorphism mapping network according to claim 5, characterized in that: The geometric approximation constraint is and The distance between them, the geometric approximation loss term is expressed as: ； in, is the geometric distance calculation function; is the coefficient of the geometric approximation loss term; the geometric approximation constraint makes Converge to the reference domain .

8. An application of the neural operator modeling method based on a diffeomorphic mapping network as claimed in any one of claims 1 to 7 in a scenario of rapid prediction of machining deformation fields of various geometrically complex structural parts, characterized in that: The corresponding partial differential equation input function is a physical quantity function that affects the part processing deformation from a physical relationship, and the corresponding solution function is a deformation field function.

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