Physical field prediction method based on deep fuzzy physical information neural network

Abstract

The invention discloses a physical field prediction method based on a deep fuzzy physical information neural network, and the method comprises the following steps: S1, collecting physical field data, carrying out the preprocessing of the data with noise, and obtaining a processed training data set Dtrain and a test data set Dtest; s2, constructing a deep fuzzy physical information neural network model; s3, inputting a physical field training data set with Gaussian noise and a corresponding partial differential equation, and training the neural network model to obtain a trained neural network model; s4, physical quantities in different time-space coordinates in the test data set are calculated through the trained neural network model, and physical field prediction is achieved; the method solves the problems that the existing physical information neural network method does not consider the condition that noise exists in a training data set, the training effect of a deep neural network model is poor under the interference of a noise data set, and physical information in a partial differential equation cannot be accurately learned, so that the prediction performance of a physical field is seriously reduced.

Description

Technical Field

[0001] The present invention relates to physical field prediction technology, and in particular to a physical field prediction method based on a deep fuzzy physical information neural network. Background Art

[0002] With the rapid development of deep learning technology, remarkable progress has been made in various fields. In recent years, physically-informed neural networks (PINNs) have been widely used to solve partial differential equations (PDEs), with the goal of incorporating physical information into the loss function of the neural network using automatic differentiation. The effectiveness of PINNs has been demonstrated in various disciplines, including optics, fluid dynamics, systems biology, and biomedical research. Specifically, PINNs have a certain generalization property 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 excellent results in physical field prediction, PINNs always assume that the data is completely accurate. However, considering that the data is calculated through commercial software simulation, it is inherently deviated from the true value of the physical world. This difference can easily invalidate the above assumptions, resulting in ambiguity or uncertainty in the representation of learning data. This leads to poor generalization of PINNs models and insufficient accuracy. That is, the existing PINNs method faces this problem: there is a heavy reliance on the accuracy of software simulation data. This means that the prediction performance of existing methods depends on the closeness of the simulated data to the true value of the physical world, and it is difficult to ensure the robustness of the method. Summary of the invention

[0004] In view of the above-mentioned deficiencies in the prior art, the present invention provides a physical field prediction method based on a deep fuzzy physical information neural network, which solves the problem that the existing physical information neural network method does not take into account 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, and the physical information in the partial differential equations cannot be accurately learned, which leads to a serious decline in the physical field prediction performance.

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

[0006] S1. Collect physical field data and preprocess the noisy data to obtain the processed training data set D train And the test dataset D test ;

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

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

[0009] S4. Calculate the physical quantities at different time and space coordinates in the test data set through the trained neural network model to achieve physical field prediction;

[0010] The calculation formula for 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 under the physical field extracted by the model after training. f is the deep fuzzy physical information neural network that has completed training;

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

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

[0015] 2. Introducing fuzzy representation into physical information neural network can largely avoid the interference of data noise on training. At the same time, the obtained fuzzy features can also deal with the uncertainty of the data itself, solve the problem of mismatch between data and partial differential equations caused by data noise, and thus obtain better physical field prediction effects; in addition, fusing fuzzy features with deep features can achieve more comprehensive feature representation from different angles, reducing the problem of decreased prediction performance caused by single feature representation;

[0016] 3. This 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] Furthermore, the step 1 comprises the following steps:

[0018] S11. Divide the noisy data set into training data set D according to a certain ratio train And the test dataset D test ;

[0019] S12, training data set D train And the test data set D test All time and space coordinate data are normalized to the maximum and minimum values ​​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 value in the physical field, x is the original spatial coordinate value, t is the original time coordinate value, u is the original physical value in the physical field, and 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 It is the minimum value of the physical quantity in the physical field.

[0023] S13, using the processed image dataset as a new training dataset and test dataset;

[0024] Furthermore, 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 to extract 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 to extract 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 to predict physical quantities in the physical field;

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

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

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

[0031] DF=f0(x,t)

[0032] FF=f1(x,t)

[0033] Among them, f0 and f1 are the deep feature extractor and 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 deep feature DF extracted by the deep 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 updating formula is as follows:

[0035] TF=concat(DF,FF)

[0036] Among them, concat means concatenating the deep feature DF and the fuzzy feature FF, and TF means the fused feature.

[0037] S33. Input the fused features into the physical field predictor f2 of the model to obtain the final predicted value of the physical quantity.

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

[0039] u pred =f2(TF)

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

[0041] S34. Calculate data and physical losses to update model parameters and perform training.

[0042] Further, step S31 is specifically as follows:

[0043] 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, the i-th fuzzy neuron u(i):R→[0,1] maps the k-th input to the fuzzy membership degree;

[0044] In step S311, the calculation formula for mapping the kth input to the fuzzy membership by the i-th fuzzy neuron is as follows:

[0045]

[0046] where u i is a Gaussian membership function with a mean of μ and a variance of δ 2 , represents the i-th neuron in 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 for fusing the affiliation relationship and obtaining the final fuzzy feature in step S312 is as follows:

[0049]

[0050] Among them, Ω i (∑Ω i =N·d) represents the neuron set of the fuzzy membership function layer connected to the fuzzy rule neuron;

[0051] The beneficial effects of the above further scheme are:

[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 impact of noise on the accuracy of physical field prediction.

[0054] S313, inputting the normalized spatiotemporal 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] Among them, Tanh represents the tanh activation function, represents the weight of the i-th neuron in the l-th layer, represents the bias of the i-th neuron in the l-th layer.

[0058] Further, step S33 is specifically as follows:

[0059] S331. After obtaining the fused overall features, they are mapped to the output through a fully connected layer.

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

[0061]

[0062] Among them d and f denote the outputs of deep neural representation and fuzzy logic representation respectively. d and w f Represents the fusion weight.

[0063] The beneficial effects of the above further scheme are:

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

[0065] Further, step S34 is specifically as follows:

[0066] S341. Calculate the data loss L based on the physical value in the real physical field and the physical value predicted by the network. Data , the physical loss L is calculated based on the physical value predicted by the network and the corresponding partial differential equation PDE , the total loss L is obtained by summing the two losses Total ;

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

[0068]

[0069] Among them, N d and N p denote the number of outputs and the number of control equations, respectively. 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, according to the training data loss L Data and physical loss L PDE Optimize the model parameters, 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 step S342 is as follows:

[0072]

[0073] Among them, θ is the parameter of the encoder, AdamOptimizer is the Adam reverse gradient optimizer, and L-BFGSOptimizer is the L-BFGS reverse gradient optimizer. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 The present invention is a flow chart of the method. DETAILED DESCRIPTION

[0075] The technical solutions in the embodiments of the present invention will be described below in conjunction with the accompanying drawings in the embodiments of the present invention.

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

[0077] S1. Collect physical field data and preprocess the noisy data to obtain the processed training data set D train And the test dataset Dtest ; Further, the method comprises the following steps:

[0078] S11. Divide the noisy data set into training data set D according to a certain ratio train And the test dataset D test ;

[0079] S12, training data set D train And the test data set D test All time and space coordinate data are normalized to the maximum and minimum values ​​to obtain the processed physical field data set. The normalization formula is as follows:

[0080]

[0081] Where x' is the normalized spatial coordinate value, t' is the normalized time coordinate value, u' is the normalized physical value in the physical field, x is the original spatial coordinate value, t is the original time coordinate value, u is the original physical value in the physical field, and 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 It is the minimum value of the physical quantity in the physical field.

[0082] S13, using the processed image dataset as a new training dataset and test dataset;

[0083] S2. Constructing a deep fuzzy physical information neural network model; further comprising 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 to extract 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 to extract 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 to predict physical quantities in the physical field;

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

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

[0089] Furthermore, 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 features and fuzzy features DF and FF respectively, and the update formula is as follows:

[0091] DF=f0(x,t)

[0092] FF=f1(x,t)

[0093] Among them, f0 and f1 are the deep feature extractor and 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, step S31 is specifically as follows:

[0095] 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, the i-th fuzzy neuron u(i):R→[0,1] maps the k-th input to the fuzzy membership degree;

[0096] In step S311, the calculation formula for mapping the kth input to the fuzzy membership by the i-th fuzzy neuron is as follows:

[0097]

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

[0099] 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;

[0100] The calculation formula for fusing the affiliation relationship and obtaining the final fuzzy feature in step S312 is as follows:

[0101]

[0102] Among them, 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 spatiotemporal coordinate data into the deep neural network layer and convert it into a high-dimensional representation. The calculation formula is as follows:

[0104]

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

[0106] S32, the deep feature DF extracted by the deep 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 updating formula is as follows:

[0107] TF=concat(DF,TF)

[0108] Among them, concat means concatenating the deep feature DF and the fuzzy feature FF, and TF means the fused feature.

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

[0110] u pred =f2(TF)

[0111] where u pred represents the final predicted value of the physical quantity, f2 represents the physical field predictor, and TF represents the fused feature. Further, step S33 is specifically as follows:

[0112] S331. After obtaining the fused overall features, they are mapped to the output through a fully connected layer. The calculation formula is as follows:

[0113]

[0114] Among them d and f denote the outputs of deep neural representation and fuzzy logic representation respectively. d and w f Represents the fusion weight.

[0115] S34. Calculate data and physical losses to update model parameters and perform training.

[0116] Further, step S34 is specifically as follows:

[0117] S341. Calculate the data loss L based on the physical value in the real physical field and the physical value predicted by the network.Data , the physical loss L is calculated based on the physical value predicted by the network 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 step S341 is as follows:

[0119]

[0120] Among them, N d and N p denote the number of outputs and the number of control equations, respectively. 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.

[0121] S342, using Adam optimizer and L-BFGS optimizer, according to the training data loss L Date and physical loss L PDE Optimize the model parameters, 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 step S342 is as follows:

[0123]

[0124] Among them, θ is the parameter of the encoder, AdamOptimizer is the Adam reverse gradient optimizer, and L-BFGSOptimizer is the L-BFGS reverse gradient optimizer.

[0125] S4. The trained neural network model is used to calculate the physical quantities at different time and space coordinates in the test data set to achieve physical field prediction. Specifically, the calculation formula is as follows:

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

[0127] 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 under the physical field extracted by the model after training. f is the deep fuzzy physical information neural network that has completed training.

[0128] In the embodiments provided by the present invention, it should be understood that the disclosed methods can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are performed in hardware or software depends on the specific application and design constraints of the technical solution. Professional and technical personnel can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of the present invention.

[0129] In the embodiments provided by the present invention, it should be understood that the disclosed methods can be implemented in other ways. For example, the method embodiments described above are merely schematic. For example, the division of the method step units is only a logical function division. There may be other division methods in actual implementation, such as more steps and details, or some steps may be ignored or not executed. The above is only a preferred embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the embodiments of the present invention may have various changes and variations. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

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 D train And the test dataset D test ; S2, build 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, training the neural network model, and obtaining a trained neural network model; S4. The trained neural network model is used 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 S3 comprises the following sub-steps: S31, input the training data into the deep feature extractor f0 and the fuzzy feature extractor f1 of the model to obtain the deep features and fuzzy features DF and FF respectively, and the update formula is as follows: DF=f0(x,t) FF=f1(x,t) Among them, f0 and f1 are the deep feature extractor and 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; S32, the deep feature DF extracted by the deep 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 updating formula is as follows: TF = concat(DF, FF) Where 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 physical field predictor f2 of the model to obtain the final physical quantity prediction value; The updating formula in step S33 is as follows: u pred =f2(TF) where u pred represents the final predicted value of the physical quantity, f2 represents the physical field predictor, and TF represents the fused feature; S34. Calculate data and physical losses to update model parameters and perform training.

3. A physical field prediction method based on fuzzy deep neural network according to claim 2, characterized in that: The 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 the Gaussian membership. Specifically, the i-th fuzzy neuron u(i): R→[0,1] maps the k-th input to the fuzzy membership; In step S311, the calculation formula for mapping the kth input to the fuzzy membership by the i-th fuzzy neuron is as follows: where u i is a Gaussian membership function with a mean of μ and a variance of δ 2 , represents the i-th neuron in the l-th layer; S312, input the obtained fuzzy membership into the fuzzy rule layer, merge 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: Among them, Ω i (∑Ω i =N·d) represents the neuron set of the fuzzy membership function layer connected to the fuzzy rule neuron; 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: Among them, Tanh represents the tanh activation function, represents the weight of the i-th neuron in the l-th layer, represents the bias of the i-th neuron in the l-th layer.

4. A physical field prediction method based on fuzzy deep neural network according to claim 2, characterized in that: The step S33 comprises the following sub-steps: S331, after obtaining the fused overall features, map them 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: Among them d and f denote the outputs of deep neural representation and fuzzy logic representation respectively; w d and w f Represents the fusion weight.

5. The physical field prediction method based on fuzzy deep neural network according to claim 2 is characterized in that: The step S34 includes the following sub-steps: S341. Calculate the data loss L based on the physical value in the real physical field and the physical value predicted by the network. Data , the physical loss L is calculated based on the physical value predicted by the network and the corresponding partial differential equation PDE , the total loss L is obtained by summing the two losses Total ; The loss function calculation formula in step S341 is as follows: Among them, N d and N p denote the number of outputs and the number of control equations, respectively, represents the value of the physical quantity predicted by the neural network, o i represents the value of the real physical quantity, ε is the physical loss, which changes with the change of the control equation; S342, using Adam optimizer and L-BFGS optimizer, according to the training data loss L Data and physical loss L PDE Optimize the model parameters, complete the training of the deep model, and obtain the trained model M0 = {f0, f1, f2}; The calculation formula of the optimized fuzzy deep neural network parameters in step S342 is as follows: Among them, θ is the parameter of the encoder, AdamOptimizer is the Adam reverse gradient optimizer, and L-BFGSOptimizer is the L-BFGS reverse gradient optimizer.

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