A physical field prediction method based on a deep blurred physical information neural network

Abstract

The application discloses a physical field prediction method based on a deep fuzzy physical information neural network, and comprises the following steps: S1, collecting physical field data, pre-processing the data with noise to obtain a processed training data set D train and a test data set D test ; S2, constructing a deep fuzzy physical information neural network model; S3, inputting the physical field training data set with Gaussian noise and the corresponding partial differential equation, training the neural network model to obtain a trained neural network model; and S4, calculating the physical quantity under different space-time coordinates in the test data set through the trained neural network model to realize physical field prediction; the application solves the problem that the existing physical information neural network method does not consider the noise in the training data set, the training effect of the deep neural network model is poor under the interference of the noise data set, the physical information in the partial differential equation cannot be accurately learned, and the physical field prediction performance is seriously reduced.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of physical field prediction, and particularly relates to a physical field prediction method based on a deep fuzzy physical information neural network. BACKGROUND

[0002] With the rapid development of deep learning technology, remarkable progress has been made in various fields. In recent years, physical information neural networks (PINNs) have been widely used to solve partial differential equations (PDEs), with the aim of incorporating physical information into the loss function of neural networks using automatic differentiation. The effectiveness of PINNs has been proven in various disciplines, including optics, fluid dynamics, systems biology, and biomedical research. Specifically, PINNs have a certain degree of generalization and can be used to solve different forms of differential equations, such as integral differential equations, fractional differential equations, stochastic differential equations, and ordinary differential equations.

[0003] Although current PINNs have achieved outstanding results in physical field prediction, PINNs always assume that the data is completely accurate. However, considering that the data is simulated by commercial software, it is inherently biased from the true value of the physical world. This difference easily invalidates the above assumption, leading to ambiguity or uncertainty in the learning data representation. This results in poor generalization of the PINNs model and insufficient accuracy. That is, the existing PINNs method faces the problem of severe dependence on the accuracy of software simulation data. This means that the prediction performance of the existing method depends on the closeness of the simulation data to the true value of the physical world, making it difficult to guarantee the robustness of the method. SUMMARY

[0004] In view of the above problems of the prior art, the physical field prediction method based on a deep fuzzy physical information neural network provided by the present application solves the problem that the existing physical information neural network method does not consider the presence of noise in the training data set, and the training effect of the deep neural network model is poor under the interference of the noisy data set, which cannot accurately learn the physical information in the partial differential equation, resulting in a serious decline in the performance of physical field prediction.

[0005] In order to achieve the above-mentioned application purposes, the technical scheme adopted by the present application is as follows: a physical field prediction method based on a deep fuzzy physical information neural network, comprising the following steps:

[0006] S1, collecting physical field data, pre-processing the data with noise to obtain a processed training data set D train and a test data set D test ;

[0007] S2, constructing a deep fuzzy physical information neural network model;

[0008] S3, input the physical field training data set with Gaussian noise and its corresponding partial differential equation, train the neural network model, and obtain the trained neural network model;

[0009] S4, calculate the physical quantity under different space-time coordinates in the test data set by the trained neural network model, and realize physical field prediction;

[0010] The calculation formula of the physical field prediction in step S4 is as follows:

[0011] u pred =f(x,t)

[0012] Where x is the input spatial coordinate data vector, t is the input time coordinate vector, u pred is the predicted value of the physical quantity in the physical field extracted by the trained model. f is a deep fuzzy physical information neural network after training;

[0013] The beneficial effects of the present application are:

[0014] 1. By introducing partial differential equations into neural networks, the demand for training data is reduced, and the training of the model can be better realized with less training data;

[0015] 2. Introducing fuzzy representation into physical information neural network can greatly avoid the interference of data noise on training, and the fuzzy features obtained can also handle the uncertainty of the data itself, solve the problem of data mismatch caused by data noise and partial differential equations, and obtain better physical field prediction effect; In addition, the fusion of fuzzy features and deep features can realize more comprehensive feature representation from different angles, and reduce the problem of prediction performance decline caused by single feature representation;

[0016] 3. The method expands the application of fuzzy learning in the field of physical field prediction, and opens up a new application space for fuzzy learning.

[0017] Further, the step 1 comprises the following steps:

[0018] S11, divide the data set with noise into a training data set D train and a test data set D test according to a certain proportion;

[0019] S12, perform maximum and minimum normalization on all time and space coordinate data in the training data set D train and the test data set D test to obtain the processed physical field data set;

[0020] The normalization formula in step S12 is as follows:

[0021]

[0022] where x' is the normalized spatial coordinate value, t' is the normalized time coordinate value, u' is the normalized physical quantity value in the physical field, x is the original spatial coordinate value, t is the original time coordinate value, u is the original physical quantity value in the physical field, x max is the maximum value of the spatial coordinate, x min is the minimum value of the spatial coordinate, t max is the maximum value of the time coordinate, t min is the minimum value of the time coordinate, u max is the maximum value of the physical quantity in the physical field, u min is the minimum value of the physical quantity in the physical field.

[0023] S13, taking the processed image data set as a new training data set and a test data set;

[0024] Further, the step 2 comprises the following steps:

[0025] S21, using a randomly initialized fully connected layer network as a deep feature extractor f0 for the deep fuzzy physical information network model, for extracting deep features;

[0026] S22, using a fuzzy network based on a Gaussian membership function as a fuzzy feature extractor f1 for the deep fuzzy physical information network model, for extracting fuzzy features;

[0027] S23, using a randomly initialized fully connected layer network as a physical field predictor f2 for the deep fuzzy physical information network model, for predicting the physical quantity in the physical field;

[0028] S24, connecting the deep feature extractor and the fuzzy feature extractor together with the physical field predictor, to construct a deep fuzzy physical information network model M0={f0, f1, f2}.

[0029] Further, the step 3 comprises the following steps:

[0030] S31, inputting training data into the deep feature extractor f0 and the fuzzy feature extractor f1 of the model to obtain deep features DF and fuzzy features FF respectively, and the update formula is as follows:

[0031] DF=f0(x, t)

[0032] FF=f1(x, t)

[0033] where f0 and f1 are the deep feature extractor and the fuzzy feature extractor in the model respectively, DF is the deep feature extracted by the deep feature extractor f0, and FF is the fuzzy feature extracted by the fuzzy feature extractor f1.

[0034] S32, the depth feature DF extracted by the depth feature extractor f0 and the fuzzy feature FF extracted by the fuzzy feature extractor f1 are fused to update the fused feature TF, and the update formula is as follows:

[0035] TF = concat (DF, FF)

[0036] Where concat represents splicing the depth feature DF and the fuzzy feature FF, and TF represents the obtained fused feature.

[0037] S33, input the fused feature into the physical field predictor f2 of the model to obtain the final physical quantity prediction value.

[0038] The update formula in step S33 is as follows:

[0039] u pred = f2 (TF)

[0040] Where u pred represents the final physical quantity prediction value, f2 represents the physical field predictor, and TF represents the fused feature.

[0041] S34, calculate the loss of data and physics to update the model parameters and perform training.

[0042] Further, step S31 is specifically:

[0043] S311, input the normalized spatio-temporal coordinate data into the fuzzy membership function layer, and capture the membership of the fuzzy set through the Gaussian membership, specifically, the i-th fuzzy neuron u(i): R→[0, 1] maps the k-th input to the fuzzy membership;

[0044] The calculation formula of the i-th fuzzy neuron in step S311 mapping the k-th input to the fuzzy membership is as follows:

[0045]

[0046] Where u i is the Gaussian membership function, whose mean is μ and variance is δ 2 , represents the i-th neuron of the l-th layer.

[0047] S312, input the obtained fuzzy membership into the fuzzy rule layer, fuse the membership relationship of the fuzzy membership function layer, and perform ‘AND’ fuzzy logic operation to obtain the final fuzzy feature;

[0048] The calculation formula of fusing the membership relationship and obtaining the final fuzzy feature in step S312 is as follows:

[0049]

[0050] wherein Ω i (∑Ω i represents a set of neurons of the fuzzy membership function layer connected to the fuzzy rule neuron;

[0051] The above further scheme has the beneficial effects that:

[0052] 1. Representing the input data from another perspective can enable the model to learn more comprehensive feature information, thereby enhancing the generalization of the model;

[0053] 2. By introducing fuzzy representation, the uncertainty of the data itself is processed, thereby greatly reducing the influence of noise on the accuracy of physical field prediction.

[0054] S313, input the normalized spatio-temporal coordinate data into a deep neural network layer to convert it into a high-dimensional representation;

[0055] The calculation formula for obtaining the high-dimensional representation in step S313 is as follows:

[0056]

[0057] wherein Tanh represents a tanh activation function, represents the weight of the i-th neuron of the l-th layer, represents the bias of the i-th neuron of the l-th layer.

[0058] Further, step S33 is specifically:

[0059] S331, after obtaining the fused overall features, mapping them to the output through a fully connected layer.

[0060] The calculation formula for mapping the fused features to the output in step S331 is as follows:

[0061]

[0062] wherein o d and o f represent the outputs of the deep neural representation and the fuzzy logic representation respectively. w d and w f represent the fused weights.

[0063] The above further scheme has the beneficial effects that:

[0064] 1. By fusing the features, the network can process the input data from a more comprehensive perspective, thereby improving the accuracy of physical field prediction.

[0065] Further, the step S34 is specifically:

[0066] S341, calculating data loss L according to the physical quantity value in the real physical field and the physical quantity value obtained by network prediction Data , calculating physical loss L according to the physical quantity value obtained by network prediction and the corresponding partial differential equation PDE , obtaining total loss L by summing the two losses Total ;

[0067] The loss function calculation formula in the step S341 is as follows:

[0068]

[0069] Wherein, N d and N p respectively represent the number of outputs and the number of control equations. o represents the value of the physical quantity predicted by the neural network, o i represents the value of the real physical quantity, and ε is the physical loss which changes with the change of the control equation.

[0070] S342, using Adam optimizer and L-BFGS optimizer to optimize model parameters according to the trained data loss L Data and physical loss L PDE , complete the training of the deep model, and obtain the trained model M0={f0,f1,f2}.

[0071] The calculation formula of the optimized fuzzy deep neural network parameters in the step S342 is as follows:

[0072]

[0073] Wherein, θ is the parameter of the encoder, AdamOptimizer is the Adam back propagation optimizer, and L-BFGSOptimizer is the L-BFGS back propagation optimizer. BRIEF DESCRIPTION OF DRAWINGS

[0074] Figure 1 The method flowchart of the present application. DETAILED DESCRIPTION

[0075] The technical solutions in the embodiments of the present application will be described below with reference to the drawings in the embodiments of the present application.

[0076] Specifically, the method comprises the following steps:

[0077] S1, collecting physical field data, pre-processing the data with noise to obtain processed training data set D train and test data set Dtest ; further, the method comprises the following steps:

[0078] S11, dividing the data set with noise into a training data set D train and a test data set D test ;

[0079] S12, performing maximum and minimum normalization on all time and space coordinate data in the training data set D train and the test data set D test to obtain a processed physical field data set, wherein the normalization formula is as follows:

[0080]

[0081] wherein x' is the normalized spatial coordinate value, t' is the normalized time coordinate value, u' is the normalized physical quantity value in the physical field, x is the original spatial coordinate value, t is the original time coordinate value, u is the original physical quantity value in the physical field, x max is the maximum spatial coordinate value, x min is the minimum spatial coordinate value, t max is the maximum time coordinate value, t min is the minimum time coordinate value, u max is the maximum physical quantity value in the physical field, and u min is the minimum physical quantity value in the physical field.

[0082] S13, taking the processed image data set as a new training data set and a test data set;

[0083] S2, constructing a deep fuzzy physical information neural network model; further, the method comprises the following steps:

[0084] S21, using a randomly initialized fully connected layer network as a deep feature extractor f0 for the deep fuzzy physical information network model, for extracting deep features;

[0085] S22, using a fuzzy network based on a Gaussian membership function as a fuzzy feature extractor f1 for the deep fuzzy physical information network model, for extracting fuzzy features;

[0086] S23, using a randomly initialized fully connected layer network as a physical field predictor f2 for the deep fuzzy physical information network model, for predicting the physical quantity in the physical field;

[0087] S24, connecting the deep feature extractor and the fuzzy feature extractor together with the physical field predictor to construct a deep fuzzy physical information network model M0 = {f0, f1, f2}.

[0088] S3, input the physical field training data set with Gaussian noise and its corresponding partial differential equation, train the neural network model, and obtain the trained neural network model;

[0089] Further, the step 3 comprises the following steps:

[0090] S31, input the training data into the deep feature extractor f0 and the fuzzy feature extractor f1 of the model to obtain the deep feature DF and the fuzzy feature FF, and the update formula is as follows:

[0091] DF = f0(x, t)

[0092] FF = f1(x, t)

[0093] Where f0 and f1 are the deep feature extractor and the fuzzy feature extractor in the model respectively, DF is the deep feature extracted by the deep feature extractor f0, and FF is the fuzzy feature extracted by the fuzzy feature extractor f1.

[0094] Further, the step S31 is specifically:

[0095] S311, input the normalized spatio-temporal coordinate data into the fuzzy membership function layer, and capture the membership of the fuzzy set through the Gaussian membership, specifically, the i-th fuzzy neuron u(i): R→[0, 1] maps the k-th input to the fuzzy membership;

[0096] The calculation formula of the i-th fuzzy neuron in the step S311 mapping the k-th input to the fuzzy membership is as follows:

[0097]

[0098] Where u i is the Gaussian membership function, whose mean is μ and variance is δ 2 , represents the i-th neuron of the l-th layer.

[0099] S312, input the obtained fuzzy membership into the fuzzy rule layer, fuse the membership relation of the fuzzy membership function layer, and perform ‘AND’ fuzzy logic operation to obtain the final fuzzy feature;

[0100] The calculation formula of fusing the membership relation and obtaining the final fuzzy feature in the step S312 is as follows:

[0101]

[0102] Where, Ω i (∑Ω i =N·d) represents the neuron set of the fuzzy membership function layer connected to the fuzzy rule neuron.

[0103] S313, input the normalized spatio-temporal coordinate data into the deep neural network layer, convert it to a high-dimensional representation, the calculation formula is as follows:

[0104]

[0105] wherein Tanh represents a tanh activation function, represents the weight of the i-th neuron of the l-th layer, represents the bias of the i-th neuron of the l-th layer.

[0106] S32, the deep feature extractor f0 extracts the deep feature DF and the fuzzy feature extractor f1 extracts the fuzzy feature FF to update the fused feature TF, the update formula is as follows:

[0107] TF = concat (DF, TF)

[0108] Where concat represents concatenating the deep feature DF and the fuzzy feature FF, and TF represents the obtained fused feature.

[0109] S33, input the fused feature into the physical field predictor f2 of the model to obtain the final physical quantity prediction value, the update formula is as follows:

[0110] u pred = f2 (TF)

[0111] Where u pred represents the final physical quantity prediction value, f2 represents the physical field predictor, and TF represents the fused feature. Further, step S33 is specifically:

[0112] S331, after obtaining the fused overall feature, map them to the output through a fully connected layer, the calculation formula is as follows:

[0113]

[0114] Where o d and o f represent the output of the deep neural representation and the fuzzy logic representation respectively. w d and w f represent the fused weights.

[0115] S34, calculate the data and physical loss to update the model parameters and train.

[0116] Further, step S34 is specifically:

[0117] S341, calculate the data loss L according to the physical quantity value in the real physical field and the physical quantity value obtained by the network predictionData , the physical loss L is calculated according to the physical quantity value obtained by network prediction and the corresponding partial differential equation PDE , the total loss L is obtained by summing the two losses Total ;

[0118] The loss function calculation formula in the step S341 is as follows:

[0119]

[0120] Wherein, N d and N p respectively represent the number of outputs and the number of control equations. represents the value of the physical quantity predicted by the neural network, o i represents the value of the real physical quantity, and ε is the physical loss which changes with the control equation.

[0121] S342, using Adam optimizer and L-BFGS optimizer, according to the data loss L Date and the physical loss L PDE Optimize the model parameters to complete the training of the deep model and obtain the trained model M0={f0,f1,f2}.

[0122] The calculation formula of the optimized fuzzy deep neural network parameters in the step S342 is as follows:

[0123]

[0124] Wherein, θ is the parameter of the encoder, AdamOptimizer is Adam back propagation optimizer, and L-BFGSOptimizer is L-BFGS back propagation optimizer.

[0125] S4, calculate the physical quantity under different space-time coordinates in the test data set through the trained neural network model to realize the prediction of physical field, specifically, the calculation formula is as follows:

[0126] u pred =f(x,t)

[0127] Wherein, x is the input space coordinate data vector, t is the input time coordinate vector, u pred is the predicted value of the physical quantity in the physical field extracted by the trained model. F is the trained deep fuzzy physical information neural network.

[0128] In the embodiments of the present application, it should be understood that the disclosed method can be realized in electronic hardware or in combination of computer software and electronic hardware. Whether the functions are realized in hardware or software mode depends on the specific application and design constraints of the technical solution. The skilled person can use different methods to realize the described functions for each specific application, but such implementation should not be considered beyond the scope of the present application.

[0129] In the embodiments of the present application, it should be understood that the disclosed method can be realized in other ways. For example, the above-described method embodiments are only schematic. For example, the division of the method step units is only a logical function division. There can be another division in actual implementation. For example, more or fewer steps and details can be included. Or some steps can be omitted or not executed. The above description is only a preferred embodiment of the present application, and is not used to limit the present application. For the skilled person in the art, the embodiments of the present application can have various modifications and changes. Any modification, equivalent replacement, improvement, etc. within the spirit and principles of the present application should be included in the protection scope of the present application.

Claims

1. A physical field prediction method based on fuzzy deep neural network, characterized in that: The following steps are involved: S1. Collect physical field data and preprocess the noisy data to obtain the processed training data set. and test dataset ; S2. Construct a deep fuzzy physical information neural network model; S3, inputting a physical field training data set with Gaussian noise and its corresponding partial differential equation to train the neural network model to obtain a trained neural network model; said step S3 includes the following sub-steps: S31. Input training data to the deep feature extractor of the model and fuzzy feature extractor The depth feature and fuzzy feature are obtained as DF and FF respectively, and the update formula is as follows: ， ， in and They are the deep feature extractor and fuzzy feature extractor in the model, DF is the deep feature extractor Extracted deep features, FF is the fuzzy feature extractor Extracted fuzzy features; S32, deep feature extractor Extracted deep features DF and fuzzy feature extractor The extracted fuzzy features FF are fused to update the fused features TF. The update formula is as follows: ， Among them, concat means concatenating the deep feature DF and the fuzzy feature FF, and TF means the fused feature. S33. Input the fused features into the model's physical field predictor The final predicted value of the physical quantity is obtained; The updating formula in step S33 is as follows: ， in represents the final predicted value of the physical quantity, represents the physical field predictor, TF represents the fused features; S34: Calculate data and physical losses to update model parameters and perform training. Step S31 includes the following sub-steps: S311, input the normalized spatiotemporal coordinate data into the fuzzy membership function layer, and capture the membership of the fuzzy set through Gaussian membership. Specifically, Fuzzy neurons The first The input is mapped to the fuzzy official script degree; In step S311, A fuzzy neuron will The calculation formula for mapping an input into fuzzy membership is as follows: ， in is the Gaussian membership function, whose mean is , the variance is , Indicates the Layer neurons; S312, input the obtained fuzzy membership into the fuzzy rule layer, fuse the membership relationship of the fuzzy membership function layer, and perform 'AND' fuzzy logic operation to obtain the final fuzzy feature; The calculation formula for fusing the affiliation relationship and obtaining the final fuzzy feature in step S312 is as follows: ， in, represents the set of neurons in the fuzzy membership function layer connected to the fuzzy rule neurons; S313, inputting the normalized spatiotemporal coordinate data into a deep neural network layer to convert it into a high-dimensional representation; The calculation formula for obtaining the high-dimensional representation in step S313 is as follows: ， in, represents the tanh activation function, Indicates the Layer The weight of a neuron, Indicates the Layer The bias of each neuron; S4. Use the trained neural network model to calculate the physical quantities at different time and space coordinates in the test data set to achieve physical field prediction.

2. A physical field prediction method based on fuzzy deep neural network according to claim 1, characterized in that: The step S33 includes the following sub-steps: S331. After obtaining the fused overall features, they are mapped to the output through a fully connected layer; The calculation formula for mapping the fusion feature to the output in step S331 is as follows: ， in and denote the outputs of deep neural representation and fuzzy logic representation respectively; and represents the fusion weight, Represents the bias term of this layer.

3. The physical field prediction method based on fuzzy deep neural network according to claim 1 is characterized in that: The step S34 includes the following sub-steps: S341. Calculate data loss based on the physical value in the real physical field and the physical value predicted by the network. , calculate the physical loss based on the physical value predicted by the network and the corresponding partial differential equation , the total loss is obtained by summing the two losses ; The loss function calculation formula in step S341 is as follows: ， in, and denote the number of outputs and the number of control equations, respectively, represents the value of the physical quantity predicted by the neural network, represents the value of a real physical quantity, is the physical loss, which changes with the change of the governing equation; S342, using Adam optimizer and L-BFGS optimizer, according to the training data loss and physical losses Optimize model parameters, complete the training of the deep model, and obtain the trained model ; The calculation formula of the optimized fuzzy deep neural network parameters in step S342 is as follows: ， in, are the parameters of the encoder, AdamOptimizer is the Adam reverse gradient optimizer, and L-BFGSOptimizer is the L-BFGS reverse gradient optimizer.

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