Physical field prediction neural operator model for variable input measuring points and construction method

Abstract

The invention relates to a variable input measuring point-oriented physical field prediction neural operator model and a construction method thereof, the model constructed by the method adopts a two-layer architecture of a backbone network and a branch network, the backbone network encodes query coordinates of an output function by using a full-connection neural network, and the branch network encodes the query coordinates of the output function by using the full-connection neural network; and the branch network realizes adaptive coding of the measuring points at different positions by fusing the permutation invariance and the geometric feature extraction capability of the point cloud network. According to the model and the construction method disclosed by the invention, high-precision solution is kept, meanwhile, the change of the position of an input function measuring point can be flexibly adapted, and the strict limitation of a traditional neural operator model on the spatial consistency of input data is effectively overcome.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of physical field prediction, and particularly relates to a physical field prediction neural operator model with variable input measurement points and a construction method. BACKGROUND

[0002] Simulation of complex physical systems is the basis of engineering and scientific research, which can analyze and predict phenomena such as structural deformation, fluid dynamics and thermal processes. Partial differential equations, as mathematical models describing the spatiotemporal evolution of physical systems, are widely used in fluid mechanics, heat conduction, electromagnetism, structural mechanics and many other fields. For example, the Navier-Stokes equation describes fluid motion, the heat diffusion equation describes temperature distribution, and the Schrödinger equation describes quantum state evolution.

[0003] To solve these partial differential equations, traditional methods rely on numerical methods such as finite difference, finite element and finite volume methods, which approximate the true solution by iterative calculations on discretized grids and have achieved great success. However, traditional numerical methods often face high computational cost, discontinuous solution, and other problems. When faced with complex geometric domains or high-dimensional problems, the grid partitioning and iterative solution process requires huge computational resources, making it difficult to meet the needs of real-time prediction or large-scale parameter space exploration. In addition, numerical methods provide solutions at discrete points, and if continuous domain solutions are needed, additional interpolation operations may introduce errors.

[0004] Neural operator learning methods are a new technology for physical field prediction in continuous function space, which directly learns the mapping from the parameter space to the physical field solution space, achieving efficient generalization in continuous function space and significantly improving the model's adaptability to various resolutions, geometric shapes and parameter settings. However, traditional neural operator models have a significant limitation, which requires the input functions in the training set and test set to be discretized at fixed sensor locations. This limitation results in insufficient flexibility for traditional neural operator models when dealing with unstructured input data, especially when the input function has non-uniform sampling or sparse data, which may result in information loss or reduced accuracy. In particular, in the problem of physical field prediction in dynamically changing or unknown geometric domains, the requirement of fixed measurement point location discretization further limits the application in engineering practice.

[0005] Therefore, how to design a neural operator model that can adapt to input measurement points at any location without relying on fixed measurement point locations while maintaining high prediction accuracy is a key technical problem that needs to be solved in the field of physical field prediction. SUMMARY

[0006] One of the objectives of the present application is to at least solve one or more of the above problems existing in the prior art, in other words, one of the objectives of the present application is to provide a physical field prediction neural operator model and construction method for variable input measurement points, which meets one or more of the aforementioned needs.

[0007] In order to achieve the above-mentioned objectives of the present application, the present application adopts the following technical solutions:

[0008] In a first aspect, the present application provides a physical field prediction model construction method for variable input measurement points, comprising:

[0009] A trunk network is set, which is used to receive an output function query point, encode spatial coordinate features of the output function query point based on a multi-layer perception, extract the spatial coordinate features layer by layer through a multi-layer fully connected layer and a nonlinear activation function, and output the spatial coordinate features;

[0010] A branch network is set, which is used to receive an input function, the input function comprising a plurality of measurement point coordinates and measurement point values corresponding to the measurement point coordinates, wherein the measurement point coordinates are variable; the branch network encodes the input function comprising variable measurement point coordinates based on the permutation invariance and geometric feature extraction capability of the point cloud network for point cloud data, and outputs measurement point features;

[0011] Dot product operation is performed on the spatial coordinate features and the measurement point features to generate an output function;

[0012] The model learns the mapping relationship between the input function and the output function, thereby approximating the target operator.

[0013] As a preferred embodiment, the branch network comprises a local feature transformation layer, a global feature aggregation layer and a feature alignment layer;

[0014] The local feature transformation layer comprises a plurality of convolution layers, which are used to map the measurement point coordinates and the measurement point values to the same high-dimensional feature space, and capture the relationship between the measurement point coordinates and the measurement point values;

[0015] The global feature aggregation layer realizes permutation invariance of input order by performing a maximum pooling operation on feature values in the convolution layer;

[0016] The feature alignment layer adjusts the global feature dimension using a multi-layer perception, so that the feature dimension of the measurement point features matches that of the spatial coordinate features.

[0017] As a preferred embodiment, the trunk network extracts the spatial features of the output function query point layer by layer through a plurality of fully connected layers and nonlinear activation functions.

[0018] As a preferred implementation, the training data set of the model is a four-tuple, whose structure is: measurement point coordinates, measurement point value, output function query point and target operator output value.

[0019] As a further preferred embodiment, the optimization of the model parameters adopts the mean square error loss function. By minimizing the loss function, the model can effectively learn the mapping relationship between the input function and the output function and approximate the target operator.

[0020] On the other hand, the present invention also provides a physical field prediction neural operator model for variable input measurement points, characterized in that it is constructed using any of the methods described above.

[0021] On the other hand, the present invention also provides an electronic device, comprising a memory, a processor, and a computer program stored in the memory and runnable on the processor, characterized in that when the processor executes the program, it implements a physical field prediction model construction method for variable input measurement points as described in any one of the above items.

[0022] On the other hand, the present invention also provides a computer-readable storage medium having a computer program stored thereon, characterized in that when the computer program is executed by a processor, it implements any of the above-mentioned methods for constructing a physical field prediction model for variable input measurement points.

[0023] Compared with the prior art, the physical field prediction model and construction method for variable input measurement points provided by the present invention have the following beneficial effects:

[0024] The model and construction method of the present invention are used to solve the problem of fixed measurement point positions of input functions in traditional neural operator models. A physical field prediction neural operator model based on a point cloud network is proposed. The neural operator model consists of a backbone network and a branch network. The backbone network uses a fully connected neural network to encode the query coordinates of the output function. The branch network integrates the permutation invariance and geometric feature extraction capabilities of the point cloud network to achieve adaptive encoding of the input function at any measurement point position. Finally, the outputs of the backbone network and the branch network are combined through a tensor product to generate a solution operator, thereby establishing a mapping relationship between the input function with variable measurement point positions and the corresponding PDE solution. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 2 is a schematic structural diagram of a physical field prediction model for variable input measuring points according to an embodiment of the present invention;

[0026] Figure 2 The following are the prediction results and errors of a one-dimensional ordinary differential equation example according to an embodiment of the present invention under two different input measurement point positions;

[0027] Figure 3are prediction results and error conditions of a two-dimensional diffusion reaction equation example of the embodiment of the present application under two groups of different input measurement point positions. DETAILED DESCRIPTION

[0028] The technical solutions in the embodiments of the present application will be apparently and completely described below with reference to the drawings in the embodiments of the present application.

[0029] The following description provides examples, and does not limit the scope, applicability, or examples set forth in the claims. Changes can be made in the function and arrangement of elements described without departing from the scope of the present content. Various examples can omit, substitute, or add various procedures or components as appropriate. For instance, the methods described can be performed in an order different than described, and various steps can be added, omitted, or combined. Also, features described with respect to some examples can be combined in other examples.

[0030] In order to facilitate understanding of the content of the present application, before introducing the specific embodiments of the present application, the existing neural operator physical field prediction method is described.

[0031] In the existing neural operator physical field prediction method, the mapping relationship between the input function and the output function is usually constructed by the following method, and the core idea is to learn an operator that can effectively learn the mapping between the input function and the output function.

[0032] Let S and U be Banach spaces of vector-valued functions, each element of these spaces is a vector-valued function, which can be specifically represented as:

[0033]

[0034] In the formula, S and U represent the input function space and the output function space respectively, and it is assumed that there is an operator G: S→U that maps the elements in the input function space S to the elements in the output function space U.

[0035] Specifically, the goal of operator learning in physical field prediction is to design an appropriately parameterized function G θ : S x Θ→U, which approximates the operator G on the given parameter space Θ, where θ∈Θ represents the network parameters of the parameterized function. By learning and optimizing the parameters θ on the given training data set, the parameterized operator G θ approximates the real operator G as much as possible, thereby effectively learning the mapping between the input function and the output function. By solving the parameterized PDE, the physical field prediction under different parameters is realized, and the optimal parameters can be obtained by minimizing the following formula,

[0036]

[0037] In the formula, the dataset Contains N sets of input and output function pairs, s obtained by sampling from the function space i Satisfy independent and identical distribution.

[0038] However, the existing operator learning model for solving infinite-dimensional function space mapping, DeepONet, uses a two-layer neural network to approximate the target operator. Its architecture is more flexible, but it requires discretization at fixed measurement points, which limits its application in practical scenarios.

[0039] The embodiment of the present invention provides a method for constructing a physical field prediction neural operator model for variable input measurement points. The structural diagram of the physical field prediction neural operator model for variable input measurement points is shown in FIG. Figure 1 As shown in Figure 1, the model mainly consists of a backbone network and a branch network. The backbone network is responsible for encoding the query coordinates of the output function, while the branch network is used to encode the measured point values ​​and locations of the input function. The two networks are combined through dot product to learn the mapping relationship between the input function and the output function, thereby approximating the target operator. The model construction method includes the following steps:

[0040] A backbone network is set up, which is used to receive output function query points, encode the spatial coordinate features of the output function query points based on a multi-layer perceptron, extract the spatial coordinate features layer by layer through multiple fully connected layers and nonlinear activation functions, and output the spatial coordinate features.

[0041] Specifically, the backbone network encodes the spatial coordinate features of the output function query point based on the multi-layer perceptron. Given a query point of the output function Where d represents the spatial dimension of the output function. The backbone network extracts the spatial features of the output query coordinates layer by layer through L fully connected layers and nonlinear activation function σ to obtain Z trunk =f θ (y) = h L , thereby effectively expressing the spatial information of the query point, where the output of the lth layer is:

[0042] h l (y)=σ1(w l ·h l-1 +b l )

[0043] Where: w l with b l are the learnable weight matrix and bias parameters of the lth layer, respectively, and h0 = y is the original coordinate of the input query point. After multiple layers of nonlinear transformation, the backbone network can extract and combine local and global features of spatial coordinates layer by layer, capturing more complex spatial structure information.

[0044] The branch network is configured to receive an input function, the input function including a plurality of measurement point coordinates and measurement point values corresponding to the measurement point coordinates, wherein the measurement point coordinates are variable; the branch network encodes the input function including the variable measurement point coordinates based on the permutation invariance and the geometric feature extraction capability of the point cloud network on the point cloud data, and outputs measurement point features.

[0045] The input function represents a discretized sampling of a physical field input function, including a plurality of measurement point coordinates and measurement point values corresponding to the measurement point coordinates, and the input function can be unstructured, i.e., the number and spatial positions of the measurement points can be arbitrarily changed. Taking a heat conduction problem as an example, the measurement point positions can be the spatial coordinates of a plurality of randomly distributed points on the surface of an object, and the measurement point values are the initial temperature values corresponding to these coordinate points.

[0046] Specifically, the branch network mainly includes a local feature transformation layer, a global feature aggregation layer and a feature alignment layer connected in sequence.

[0047] The local feature transformation layer includes K convolutional layers, which are used to map the measurement point position x i and the measurement point value s i of each independent input measurement point to a high-dimensional feature space, so as to capture the relationship between the spatial coordinate and the function value, and through these convolutional layers, the network can learn the complex nonlinear relationship between the measurement point position and the measurement point value, and obtain a high-level feature representation Z local =g θ (X)=m K In the kth convolutional layer, the output can be represented as:

[0048] m k =σ2(Conv k (m k-1 ))

[0049] On this basis, the global feature aggregation layer realizes the permutation invariance of the input order through a max-pooling operation Maxpool,

[0050] Z global =Maxpool(Z local )

[0051] By performing a max-pooling operation on all values in Z local , the output Z global remains the same regardless of the arrangement of the input measurement points, as long as the maximum feature value is the same.

[0052] Further, the feature alignment layer adjusts the global feature dimension using a multi-layer perceptron to match the feature dimension of the backbone network. Through this alignment mechanism, the network can accurately dock the spatial coordinate information of the input function with the spatial coordinates of the output function query point, thereby effectively learning the mapping relationship between the input function and the output function as

[0053] Z branch = MLP(Z global )

[0054] After the backbone network and the branch network, the model receives the spatial coordinate features and the measurement point features of the backbone network and the branch network, and performs a dot product operation on the spatial coordinate features and the measurement point features to generate an output function.

[0055] Then, the model construction method makes the model learn the mapping relationship between the input function and the output function, thereby approximating the target operator.

[0056] The feature tensor of the backbone network and the branch network generates a predicted value through a dot product operation as:

[0057]

[0058] In the above structure, the backbone network is responsible for encoding the query coordinates of the output function, and the branch network is used to encode the measurement point values and the measurement point positions of the input function. The dot product operation layer combines the two networks through dot product, so that the model can learn the mapping relationship between the input function and the target operator output value, thereby approximating the target operator.

[0059] The model of the present embodiment can flexibly adapt to changes in the measurement point positions of the input function while maintaining high-precision solving, with an average normalized mean square error of 10 -2 level, providing a reliable basis for prediction, which helps to accurately solve partial differential equations.

[0060] The model constructed by the above method is composed of a backbone network and a branch network. The backbone network encodes the query coordinates of the output function using a fully connected neural network. The branch network realizes adaptive encoding of the input function under any measurement point position by fusing the permutation invariance of the point cloud network and the geometric feature extraction capability. Finally, the output of the backbone network and the branch network is combined through tensor product to generate a solver, thereby establishing the mapping relationship from the input function with variable measurement point positions to the corresponding PDE solution. After learning the mapping relationship between the input function and the output function through the training data set, the model can approximate the target operator and realize variable measurement point position physical field prediction of the input function.

[0061] The present embodiment also provides a training data set for training the model:

[0062] The training data set is a four-tuple, and its structure is: measurement point coordinates, measurement point values, output coordinate query point, and target operator output value. The structure of this training data set allows the model to learn the mapping from the function space to the output space.

[0063] As an example, the training set of the method can be composed of a set of four-tuples [x, u, y, G(u)(y)], and its structure can be represented as:

[0064]

[0065] The optimization of the model parameters uses the Mean Squared Error (MSE) loss function. By minimizing the loss function, the model can effectively learn the mapping relationship between the input function and the output function, approximating the target operator.

[0066] The embodiment also provides an example of effectiveness verification, Figure 2 and Figure 3 respectively show the prediction results and error conditions of one-dimensional ordinary differential equation examples and two-dimensional diffusion reaction equation examples under two different input measurement point positions. As shown in the figures, under different input measurement point positions, the predicted solution of the model of the embodiment is consistent with the reference solution, and the absolute error is within 10 -2 horizontal.

[0067] Another embodiment of the application provides a physical field prediction neural operator model for variable input measurement points, which is generated by applying the variable input measurement point physical field prediction model construction method shown in the above embodiment, and its structure is as shown in Figure 1 includes:

[0068] The backbone network is used to receive the output function query point, encode the spatial coordinate features of the output function query point based on the multi-layer perception, extract the spatial coordinate features layer by layer through the multi-layer fully connected layer and the nonlinear activation function, and output the spatial coordinate features.

[0069] The branch network is used to receive the input function, and the input function includes a plurality of measurement point coordinates and measurement point values corresponding to the measurement point coordinates, wherein the measurement point coordinates are variable. The branch network encodes the input function including the variable measurement point coordinates based on the point cloud network's permutation invariance and geometric feature extraction capability, and outputs the measurement point features.

[0070] The dot product operation layer is after the backbone network and the branch network, receives the spatial coordinate features and the measurement point features of the backbone network and the branch network, performs dot product operation on the spatial coordinate features and the measurement point features, and generates the output function.

[0071] The main network encodes the query coordinates of the output function by using a fully connected neural network, and the branch network realizes adaptive coding of the input function at any measuring point position by fusing the permutation invariance and geometric feature extraction capability of the point cloud network. Finally, the output of the main network and the branch network is combined by tensor product to generate the solver, thereby establishing the mapping relationship from the input function with variable measuring point position to the corresponding PDE solution. After learning the mapping relationship between the input function and the output function through the training set, the model can approximate the target operator to realize the variable measuring point position physical field prediction of the input function.

[0072] Another embodiment of the application also provides an electronic device, which can include but is not limited to a processor 101, a memory 102, and optionally a communication interface 103 and an input / output interface 104, etc. The processor can be a central processing unit (CPU), a digital signal processor (DSP), a programmable logic device (FPGA), an application specific integrated circuit (ASIC) or other processing cores capable of executing instructions. In this embodiment, the processor 101 is the control center of the electronic device, responsible for running the computer programs stored in the memory 102 to execute one or more steps of the variable input measuring point physical field prediction method in the above embodiments. For example, the processor 101 can calculate the corresponding control instruction according to the data collected from the power grid (obtained through the communication interface 103 or the input / output interface 104) and the preset control target (such as the target exchange power of the tie line).

[0073] The memory can be any type of volatile or non-volatile storage medium, such as random access memory (RAM), read-only memory (ROM), flash memory (Flash Memory), hard disk drive (HDD), solid state drive (SSD), etc. The memory is used to store the operating system, various data and computer programs.

[0074] The computer program stored on the memory, when the instructions of the computer program are executed by the processor, causes the electronic device to execute the variable input measuring point physical field prediction method of the preceding embodiments.

[0075] The embodiment of the application also provides a computer readable storage medium. The computer readable storage medium has a computer program stored thereon.

[0076] The computer program includes a series of instructions, which when loaded and executed by the processor of the electronic device, can enable the electronic device to implement the variable input measuring point physical field prediction method of the preceding embodiments.

[0077] Those skilled in the art can realize the units and algorithm steps of each example described in connection with the embodiments disclosed herein can be realized in electronic hardware, or a combination of computer software and electronic hardware. Whether the functions are performed in hardware or software depends on the specific application and design constraints. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation does not need to be considered to be beyond the scope of the present application.

[0078] The above description is merely illustrative of the exemplary embodiments of the present disclosure and does not limit the scope of the present disclosure. Any equivalent variations and modifications of the present disclosure, which are based on the teachings of the present disclosure, are to be considered within the scope of the present disclosure. Embodiments of the present disclosure would be readily apparent to those skilled in the art from the description of the present disclosure. The present application is intended to cover any variations or modifications thereof which follow the principles of the present disclosure and include common general knowledge or custom of the art not specifically mentioned. The scope of the present disclosure is defined by the appended claims rather than the description and embodiments.

Claims

1. A method for constructing a physical field prediction neural operator model for variable input measurement points, characterized in that: include: Setting a backbone network, the backbone network is used to receive an output function query point, encode the spatial coordinate features of the output function query point based on a multi-layer perceptron, extract the spatial coordinate features layer by layer through multiple fully connected layers and nonlinear activation functions, and output the spatial coordinate features; A branch network is provided, the branch network being configured to receive an input function, the input function including a plurality of measurement point coordinates and measurement point values ​​corresponding to the measurement point coordinates, wherein the measurement point coordinates are variable; the branch network encodes the input function of the variable measurement point coordinates based on the point cloud network's permutation invariance and geometric feature extraction capabilities for point cloud data, and outputs measurement point features; Performing a dot product operation on the spatial coordinate feature and the measurement point feature to generate an output function; The model learns the mapping relationship between the input function and the output function, thereby approaching the target operator.

2. The physical field prediction neural operator model for variable input measurement points according to claim 1, characterized in that: The branch network includes a local feature transformation layer, a global feature aggregation layer and a feature alignment layer; The local feature transformation layer includes several convolutional layers for mapping the measurement point coordinates and the measurement point values ​​into the same high-dimensional feature space to capture the relationship between the measurement point coordinates and the measurement point values; The global feature aggregation layer performs a maximum pooling operation on the feature values ​​in the convolutional layer to achieve permutation invariance to the input order; The feature alignment layer uses a multi-layer perceptron to adjust the global feature dimension so that the feature dimension of the measurement point feature matches the feature dimension of the spatial coordinate feature.

3. The physical field prediction neural operator model for variable input measurement points according to claim 1, characterized in that: The backbone network extracts the spatial features of the output function query points layer by layer through a number of fully connected layers and nonlinear activation functions.

4. The physical field prediction neural operator model for variable input measurement points according to claim 1, characterized in that: The training data set of the model is a four-tuple, whose structure is: measurement point coordinates, measurement point value, output function query point and target operator output value.

5. The physical field prediction neural operator model for variable input measurement points according to claim 4, characterized in that: The mean square error loss function is used to optimize the model parameters. By minimizing the loss function, the model can effectively learn the mapping relationship between the input function and the output function and approach the target operator.

6. A physical field prediction neural operator model for variable input measurement points, characterized by: Constructed using the method according to any one of claims 1 to 5.

7. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the method for constructing a physical field prediction neural operator model for variable input measurement points as described in any one of claims 1 to 5 is implemented.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the method for constructing a physical field prediction neural operator model for variable input measurement points as described in any one of claims 1 to 5 is implemented.

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