Physics-guided critical region fluid nonlinear wave motion prediction method, system, device and medium

Abstract

The present invention discloses a physically guided prediction method, system, device and medium for nonlinear wave motion of critical region fluid, wherein the method comprises: establishing a critical region fluid calculation domain and a nonlinear time-varying equation; constructing a sample data set; constructing a physical information neural network model; constructing a loss function including residual loss of the nonlinear time-varying equation, initial point error loss and boundary point error loss; training the physical information neural network model to obtain a prediction model for the free water surface displacement wave envelope of the critical region fluid; using a gradient-free nonlinear attention masking method based on residual accumulation to weight the internal collocation point samples in each iteration round during the training process; inputting the space-time coordinates of the critical region fluid to be predicted point into the prediction model to obtain the free water surface displacement wave envelope of the fluid to be predicted point. The present invention effectively reduces error accumulation, accelerates the model convergence speed, and improves the prediction accuracy of nonlinear wave motion of critical region fluid.

Description

Technical Field

[0001] The present invention relates to the field of fluid mechanics technology, and in particular to a physically guided critical region fluid nonlinear wave motion prediction method, system, equipment and medium. Background Art

[0002] More and more studies are using methods that combine artificial intelligence with physical models to deal with complex phenomena. In the ocean field, identifying abnormal wave flow characteristics to predict subsequent changes has become an important means to achieve abnormal sudden warning. The non-autonomous Schrödinger equation can be used to derive the nonlinear development system of fluids in critical areas such as deep water and shallow water transition, thereby studying flow characteristics. In practical applications, due to the difficulty in obtaining effective data, it is an effective method to use physical information neural networks to solve nonlinear development systems and predict the development and evolution of nonlinear waves. In view of the characteristics of strong nonlinearity and high-order differential operators of the non-autonomous Schrödinger equation, the standard physical information neural network has the disadvantages of large error accumulation and long training time in long-term prediction. Therefore, the present invention designs an improved physical information neural network for the above-mentioned problems to improve prediction accuracy and training efficiency. Summary of the invention

[0003] The present invention provides a physically guided critical region fluid nonlinear wave motion prediction method, system, device and medium to solve the technical problems of large long-time domain error accumulation and slow convergence speed in the prior art in the prediction of critical region fluid nonlinear wave dynamic characteristics.

[0004] In order to achieve the above object, the present invention adopts the following technical solution.

[0005] In a first aspect, a physically guided method for predicting nonlinear wave motion of a critical region fluid is provided, comprising the following steps:

[0006] According to the spatiotemporal characteristics of the fluid flow in the critical region, a computational domain is established, and a nonlinear time-varying equation for describing the nonlinear wave dynamics characteristics of the fluid in the critical region is determined; the solution of the nonlinear time-varying equation is the free water surface displacement wave envelope of the fluid, which is expressed in complex form, i.e., includes a real part solution and an imaginary part solution;

[0007] Selecting initial points, boundary points and internal matching points for model training in the computational domain to construct a sample data set;

[0008] Construct a physical information neural network model with the space-time coordinates of the fluid in the critical region as input and the free water surface displacement wave envelope of the fluid at the space-time coordinates as output;

[0009] Construct the residual loss, initial point error loss and boundary point error loss of the nonlinear time-varying equation, and add the three together to get the loss function; each loss includes the corresponding real part loss and imaginary part loss;

[0010] The physical information neural network model is trained with sample data sets to obtain the displacement wave envelope prediction model of the free water surface of the critical region fluid. During the training process, the gradient-free nonlinear attention mask method based on residual accumulation is used to weight the internal collocation point samples in each iteration round.

[0011] The space-time coordinates of the critical region fluid point to be predicted are input into the critical region fluid free water surface displacement wave envelope prediction model to obtain the free water surface displacement wave envelope of the fluid at the point to be predicted.

[0012] According to the first aspect, in some possible implementations, during the training process, a gradient-free nonlinear attention mask method based on residual accumulation is used to weight the internal collocation point samples in each iteration round, specifically including:

[0013] Initialize the weights of the internal collocation point samples in the first round of iteration;

[0014] Starting from the second round of iteration, the weights of the internal collocation point samples in each iteration round are determined by the following formula:

[0015] ;

[0016] ;

[0017] In the formula, and denote the real weight of the kth internal collocation point sample in the mth and m+1th iterations respectively; and Respectively represent the imaginary weight of the kth internal collocation point sample in the mth and m+1th iterations; represents the attenuation coefficient; represents the learning rate; and They represent the real residual and imaginary residual of the nonlinear time-varying equation of the kth internal collocation point sample in the mth iteration respectively; and They represent the real residual mean and imaginary residual mean of the nonlinear time-varying equation of all internal collocation point samples in the mth iteration respectively; Represents a minimum constant that prevents division by 0; Represents the number of internal collocation point samples in each round of iteration;

[0018] Then the residual loss of the nonlinear time-varying equation in the mth iteration is Calculated by the following formula:

[0019] ;

[0020] In the formula, represents the real residual of the nonlinear time-varying equation of the kth internal collocation point sample, represents the imaginary residual of the nonlinear time-varying equation of the kth internal collocation point sample, Represents the spatiotemporal coordinates of the kth internal collocation point sample.

[0021] According to the first aspect, in some possible implementations, in the process of training a physical information neural network model using a sample data set, a gradient-free nonlinear attention mask method based on residual accumulation is used to weight the initial point samples and / or boundary point samples in each iteration round.

[0022] According to the first aspect, in some possible implementations, an adaptive activation function with scalable parameters and a fixed scaling factor is introduced into the physical information neural network model, and the loss function is reconstructed by adding a slope recovery term, specifically including:

[0023] The depth of the physical information neural network model is D, which includes the input layer, D-1 hidden layers and the output layer. The physical information neural network model is expressed as follows:

[0024] ;

[0025] In the formula, Represents the output of the physical information neural network model when the input is z; represents a composite operator; represents the linear transformation of the Dth layer; represents a nonlinear activation function; represents the scaling factor; represents the expandable parameters of the D-1th hidden layer; Represents trainable parameters in physical information neural network models, including weights, biases, and scalable parameters;

[0026] Adding the slope recovery term to the loss function is expressed as follows:

[0027] ;

[0028] In the formula, represents the loss function, , , and They represent the initial point error loss, boundary point error loss, nonlinear time-varying equation residual loss and slope recovery term respectively;

[0029] The slope recovery term is calculated as follows:

[0030] ;

[0031] In the formula, , represents the expandable parameter of the ith neuron in the dth hidden layer, Represents the number of neurons in the dth hidden layer.

[0032] According to the first aspect, in some possible implementations, a Dropout regularization mechanism is used in the process of training the physical information neural network model using a sample data set. The Dropout regularization mechanism is: during the training of the physical information neural network model, a certain probability is used to Randomly set some neurons in the physical information neural network model to 0, so that some neurons and all their incoming and outgoing connections are temporarily removed from the physical information neural network model, and then conduct this round of iterative training; at this time, the physical information neural network model is expressed as follows:

[0033] ;

[0034] In the formula, express Regularization is inserted between each layer of activation functions so that a portion of the activation function output of each layer is randomly discarded during training.

[0035] According to the first aspect, in some possible implementations, the computational domain includes a spatial domain and a time domain, the boundary point samples and the internal collocation point samples in the sample data set are divided into M parts according to the time domain, and the physical information neural network model is trained in a staged training manner;

[0036] The original initial point samples in the sample data set are only used for the first stage of physical information neural network model training. The initial point samples in each subsequent stage are pseudo initial point samples, which are predicted by the physical information neural network model trained in the previous stage; and the error loss of the physical information neural network model of the current stage fitting the prediction output of all previous stages is introduced into the loss function of the qth stage. , q≥2.

[0037] According to the first aspect, in some possible implementations, the computational domain includes a spatial domain and a time domain, the boundary point samples and the internal collocation point samples in the sample data set are divided into M parts according to the time domain, and the physical information neural network model is trained in a staged training manner;

[0038] The physical information neural network model adopts its reduced-order model, whose input is the space-time coordinates of the fluid in the critical region, and whose output is the real and imaginary parts of the free water surface displacement wave envelope of the fluid at the corresponding space-time coordinates, as well as the derivatives of the real and imaginary parts of the free water surface displacement wave envelope of the fluid with respect to the space domain coordinates;

[0039] The initial point error loss is constructed based on the prediction errors of the four model outputs corresponding to the initial point samples, and the boundary point error loss is constructed based on the prediction errors of the four model outputs corresponding to the boundary point samples;

[0040] The original initial point samples in the sample data set are only used for the first stage of physical information neural network model training. The initial point samples in each subsequent stage are pseudo initial point samples, which are predicted by the physical information neural network model trained in the previous stage; and the error loss of the physical information neural network model of the current stage fitting the prediction output of all previous stages is introduced into the loss function of the qth stage. , q≥2.

[0041] In a second aspect, a physically guided critical region fluid nonlinear wave prediction system is provided, comprising:

[0042] An equation building module is used to establish a computational domain according to the spatiotemporal characteristics of the fluid flow in the critical region, and determine a nonlinear time-varying equation for describing the nonlinear wave dynamics characteristics of the fluid in the critical region; the solution of the nonlinear time-varying equation is the free water surface displacement wave envelope of the fluid, which is expressed in complex form, i.e., includes a real part solution and an imaginary part solution;

[0043] The data set construction module is used to select the initial points, boundary points and internal matching points for model training in the computational domain to construct a sample data set;

[0044] A neural network model building module is used to build a physical information neural network model with the space-time coordinates of the fluid in the critical region as input and the displacement wave envelope of the fluid free water surface at the space-time coordinates as output;

[0045] The loss function determination module is used to construct the residual loss, initial point error loss and boundary point error loss of the nonlinear time-varying equation, and add the three to obtain the loss function; each loss includes the corresponding real part loss and imaginary part loss;

[0046] The training module is used to train the physical information neural network model using a sample data set to obtain a prediction model for the displacement wave envelope of the free water surface of the fluid in the critical region; during the training process, a gradient-free nonlinear attention mask method based on residual accumulation is used to weight the internal collocation point samples in each iteration round;

[0047] The prediction module is used to input the time and space coordinates of the critical region fluid to be predicted point into the critical region fluid free water surface displacement wave envelope prediction model to obtain the free water surface displacement wave envelope of the fluid to be predicted point.

[0048] In a third aspect, an electronic terminal is provided, comprising:

[0049] a memory having a computer program stored thereon;

[0050] A processor is used to load and execute the computer program to implement the physical-guided critical region fluid nonlinear wave motion prediction method as described above.

[0051] In a fourth aspect, a computer-readable storage medium is provided, on which a computer program is stored, and when the computer program is executed by a processor, the critical region fluid nonlinear wave motion prediction method guided by physics as described above is implemented.

[0052] The present invention proposes a physically guided critical region fluid nonlinear wave motion prediction method, system, equipment and medium. By training, a critical region fluid free water surface displacement wave envelope prediction model based on a physical information neural network model is obtained to replace the nonlinear time-varying equation, and the critical region fluid free water surface displacement wave envelope is quickly and accurately predicted, thereby realizing the prediction of the critical region fluid nonlinear wave evolution, intensity change and possible critical state, which is very useful for early warning of extreme wave events, assessing marine risks and formulating safety strategies. In the process of training the physical information neural network model, the present invention adopts a gradient-free attention mask method based on residual accumulation to weight the internal matching point samples in each iteration round, that is, the nonlinear time-varying equation residual and the historical weight are used to calculate the adaptive weight of the internal matching point samples. Because the nonlinear time-varying equation residual contains information about the difficult-to-solve area, the obtained weight can be used as an attention mask to help the physical information neural network model to concentrate on capturing specific sample features, and the historical weight information is taken into account, which means that the historical training dynamics of the physical information neural network model are included in the weight calculation range. This method increases the attention to the difficult-to-solve area, and combines the influence of the historical residual evolution, thereby helping the physical information neural network model to get rid of the local optimal solution and find other better local (or global) optimal solutions, effectively reducing the training error accumulation, and accelerating the model convergence speed, and improving the accuracy of the model in predicting the nonlinear wave motion of the fluid in the critical area.

[0053] In addition to the above-described purposes, features and advantages, the present invention has other purposes, features and advantages. The present invention will be further described in detail in the following specific embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0055] Figure 1 It is a flow chart of a physically guided critical region fluid nonlinear wave motion prediction method provided by an embodiment of the present invention;

[0056] Figure 2 is a comparison diagram of a model prediction solution and an exact solution in a long time domain provided by an embodiment of the present invention, wherein (a) is an exact solution and (b) is a model prediction solution;

[0057] Figure 3 3. The figures are comparison diagrams of the exact solution and the model prediction solution at three moments provided by the embodiments of the present invention, wherein (a) is a comparison diagram of the exact solution and the model prediction solution at t=2.69, (b) is a comparison diagram of the exact solution and the model prediction solution at t=3.33, and (c) is a comparison diagram of the exact solution and the model prediction solution at t=3.87. DETAILED DESCRIPTION

[0058] To make the purpose, technical solution and advantages of the present invention clearer, the technical solution of the present invention will be described in detail below. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other implementation methods obtained by ordinary technicians in this field without creative work belong to the scope of protection of the present invention.

[0059] Example 1

[0060] This embodiment provides a physically guided critical region fluid nonlinear wave prediction method, such as Figure 1 As shown, the following steps are included:

[0061] S1: A computational domain is established according to the spatiotemporal characteristics of the fluid flow in the critical region, and a nonlinear time-varying equation is determined to describe the nonlinear wave dynamics characteristics of the fluid in the critical region; the solution of the nonlinear time-varying equation is the free water surface displacement wave envelope of the fluid, which is expressed in complex form, i.e., it includes real and imaginary solutions.

[0062] The computational domain includes the space domain and the time domain. In this embodiment, a 1+1 dimensional nonlinear time-varying equation in the complex space is used as the nonlinear time-varying equation to describe the nonlinear wave dynamics characteristics of the fluid in the critical region. Considering the general form of the 1+1 dimensional nonlinear time-varying equation, it is expressed as follows:

[0063] (1);

[0064] In the formula, and denote the time domain coordinates and the spatial coordinates as the wave group propagates, respectively. Constitute space-time coordinates; The equation depends on The complex-valued solution of represents the free-surface displacement wave envelope of the fluid, which varies with space and time; Representation solution right The partial derivative of Respectively represent the solution right The first, second, and third order partial derivatives of ; Yes and about Nonlinear functionals of arbitrary order derivatives of ; Represents the spatial domain; represents the time domain, and Respectively represent the initial time and the end time of the time domain. is plural, so Decomposition into real solution and the imaginary part solution ,Right now Therefore, equation (1) can be rewritten into equation (2) and equation (3) as:

[0065] (2);

[0066] (3);

[0067] same, and Denote the real part solution and the imaginary part solution right The partial derivative of and is the real solution and the imaginary part solution and about Nonlinear functionals of arbitrary order derivatives of .

[0068] S2: Select the initial points, boundary points and internal matching points for model training in the computational domain to construct a sample data set.

[0069] Among them, the initial point is the sample point corresponding to the initial moment, and the boundary point is the sample point corresponding to the boundary of the space domain.

[0070] In this embodiment, the nonlinear time-varying equation is described by taking the non-autonomous fifth-order derivative Schrödinger equation as an example. The non-autonomous fifth-order derivative Schrödinger equation is expressed as follows:

[0071] (4);

[0072] In the formula, and denote the coefficients of the cubic and quintic nonlinear terms, respectively. and are the coefficients of self-steepening effect and dispersion effect, respectively.

[0073] In this embodiment, the breather solution (analytical solution) of the non-autonomous quintic derivative Schrödinger equation is used to construct a data set, and initial point data, boundary point data, and internal collocation point data are extracted from the data to form a sample data set using at least one of a uniform sampling method, a random sampling method, and a Latin hypercube sampling method.

[0074] S3: Construct a physical information neural network model with the space-time coordinates of the fluid in the critical region as input and the free water surface displacement wave envelope of the fluid at the space-time coordinates as output.

[0075] In this embodiment, the input of the physical information neural network model is the space-time coordinates , its output includes two, corresponding to the solution The real part solution of and the imaginary part solution , that is, the real and imaginary parts of the free water surface displacement wave envelope of the fluid.

[0076] S4: Construct the nonlinear time-varying equation residual loss, initial point error loss and boundary point error loss, and add the three together to get the loss function; each loss includes the corresponding real part loss and imaginary part loss.

[0077] In this embodiment, the damage function It is expressed as follows:

[0078] (5);

[0079] (6);

[0080] (7);

[0081] (8);

[0082] In the formula, The trainable parameters of the physical information neural network model, including weights and biases; , and They represent the initial point error loss, boundary point error loss and nonlinear time-varying equation residual loss respectively; , represents the spatial domain, represents the space-time coordinates of the kth initial point; , Representation space domain The borders of represents the space-time coordinates of the kth boundary point; , represents the space-time coordinates of the kth internal collocation point; and They represent the real part model prediction value and real part analytical value of the free water surface displacement wave envelope of the fluid, and They represent the imaginary part model prediction value and imaginary part analytical value of the free water surface displacement wave envelope of the fluid respectively; and They represent the real part residual loss of the nonlinear time-varying equation and the imaginary part residual loss of the nonlinear time-varying equation respectively; , and are the number of initial point samples, boundary point samples and internal collocation point samples in each round of iterative training.

[0083] S5: The physical information neural network model is trained using a sample data set to obtain a prediction model for the displacement wave envelope of the free water surface of the fluid in the critical region. During the training process, a gradient-free nonlinear attention mask method based on residual accumulation is used to weight the internal matching point samples in each iteration round.

[0084] Since it is necessary to train the physical information neural network model by minimizing the loss function to obtain the solution of the approximate partial differential equation, maintaining the balance between the various loss terms in the loss function is crucial to avoid the problems of slow convergence and low accuracy, especially when dealing with the non-autonomous quintic derivative Schrödinger equation with high-order differential operators and strong nonlinearity. In order to solve this imbalance problem, there are many adaptive weighting methods in the prior art that solve the loss imbalance problem by iteratively changing the contribution of specific terms during the training process. This change can be done through adaptive resampling or by directly weighting the coefficients. However, the implementation cost of these methods is very high because they require multiple neural networks to assist in the calculation, or they need to calculate and process gradients, or they do not add constraints, resulting in unlimited growth of the weighting coefficients.

[0085] In order to solve the above problems, this embodiment proposes a gradient-free nonlinear attention mask scheme based on residual accumulation, which uses the nonlinear time-varying equation residuals and historical weights in the loss function to calculate the adaptive weights of internal collocation point samples. Because the nonlinear time-varying equation residuals contain information about difficult-to-solve areas, the obtained weights can be used as attention masks to help the neural network focus on capturing the features of specific sampling points, and historical weight information is taken into account, which means that the historical training dynamics of the neural network are included in the weight calculation range. The method proposed in this embodiment aims to increase attention to difficult-to-solve areas, and combines the influence of historical residual evolution, so as to help the physical information neural network model get rid of local optimal solutions and find other better local (or global) optimal solutions.

[0086] In this embodiment, a gradient-free nonlinear attention mask method based on residual accumulation is used to weight the internal collocation point samples in each iteration round, specifically including:

[0087] Initialize the weights of the internal collocation point samples in the first round of iteration. Generally, the weights of all internal collocation point samples in the first round of iteration are set to 1 by default.

[0088] Starting from the second round of iteration, the weights of the internal collocation point samples in each iteration round are determined by the following formula:

[0089] (9);

[0090] (10);

[0091] In the formula, and denote the real weight of the kth internal collocation point sample in the mth and m+1th iterations respectively; and Respectively represent the imaginary weight of the kth internal collocation point sample in the mth and m+1th iterations; represents the attenuation coefficient; represents the learning rate; and They represent the real residual and imaginary residual of the nonlinear time-varying equation of the kth internal collocation point sample in the mth iteration respectively; and They represent the real mean of the residual of the nonlinear time-varying equation and the imaginary mean of the residual of the nonlinear time-varying equation of all internal collocation point samples in the mth iteration respectively; Represents a minimum constant that prevents division by 0; Represents the number of internal matching point samples in each iteration; the superscript 2 represents the square, which introduces nonlinearity. Compared with linear weights, nonlinear weights are more sensitive to large residual areas, especially in areas that require higher precision, while suppressing the interference of smaller noise; in addition, the square form can also enhance the numerical stability of the optimization process, avoid directional interference from positive and negative errors, and make the model more accurate in describing complex nonlinear characteristics.

[0092] Then the residual loss of the nonlinear time-varying equation in the mth iteration is From formula (8) to formula (11):

[0093] (11);

[0094] In the formula, represents the real residual of the nonlinear time-varying equation of the kth internal collocation point sample, Represents the imaginary residual of the nonlinear time-varying equation of the kth internal collocation point sample.

[0095] S6: Input the time and space coordinates of the critical region fluid point to be predicted into the critical region fluid free water surface displacement wave envelope prediction model to obtain the free water surface displacement wave envelope of the fluid at the point to be predicted.

[0096] Free surface displacement wave envelope of a fluid Represents the overall shape and outline of the fluid wave packet, describing the distribution of the wave amplitude over time and space; Fluid nonlinear wave amplitude: the vertical distance from the equilibrium position of the wave to the wave crest ; Spatial distribution of amplitude: fixed time , about Curve; Time evolution of wave packet amplitude: The wave packet amplitude at a fixed spatial position The rate of change over time .

[0097] The above embodiment proposes a physically guided method for predicting nonlinear wave motion of fluid in critical region. By training, a fluid free water surface displacement wave envelope prediction model based on a physical information neural network model is obtained to replace the nonlinear time-varying equation, so as to quickly and accurately predict the free water surface displacement of fluid in critical region, thereby realizing the prediction of nonlinear wave evolution, intensity change and possible critical state of fluid in critical region, which is very useful for early warning of extreme wave events, assessing marine risks and formulating safety strategies. In the process of training the physical information neural network model, the present invention adopts a gradient-free attention mask method based on residual accumulation to weight the internal matching point samples in each iteration round, that is, the nonlinear time-varying equation residual and the historical weight are used to calculate the adaptive weight of the internal matching point samples. Because the nonlinear time-varying equation residual contains information about the difficult-to-solve area, the obtained weight can be used as an attention mask to help the physical information neural network model to concentrate on capturing specific sample features, and the historical weight information is taken into account, which means that the historical training dynamics of the physical information neural network model are included in the weight calculation range. This method increases the attention to the difficult-to-solve area, and combines the influence of the historical residual evolution, thereby helping the physical information neural network model to get rid of the local optimal solution and find other better local (or global) optimal solutions, effectively reducing the training error accumulation, and accelerating the model convergence speed, and improving the accuracy of the model in predicting the nonlinear wave motion of the fluid in the critical area.

[0098] Example 2

[0099] This embodiment provides a physically guided critical region fluid nonlinear wave prediction method, which differs from Embodiment 1 only in that, in the process of training the physical information neural network model using the sample data set, in addition to weighting the internal matching point samples, a gradient-free nonlinear attention mask method based on residual accumulation is also used to weight the initial point samples and / or boundary point samples in each iteration round.

[0100] The weight of the initial point sample (real weight , imaginary weight ) and the weights of the boundary point samples (real weights , imaginary weight ) is determined in a similar way to the weight determination of the internal matching points. The specific determination form can be referred to equations (9) and (10). The only difference is that the real residual of the nonlinear time-varying equation and the imaginary residual of the nonlinear time-varying equation in the formula are replaced by the real error and imaginary error of the initial point / boundary point, and the real mean of the residual of the nonlinear time-varying equation and the imaginary mean of the residual of the nonlinear time-varying equation are replaced by the real mean of the error and the imaginary mean of the error of the initial point / boundary point.

[0101] Similarly, the initial point error loss of the mth iteration is and boundary point error loss Equation (6) and (7) are transformed into Equation (12) and (13).

[0102] (12);

[0103] (13);

[0104] By weighting the initial point samples and / or boundary point samples, the model convergence speed can be further accelerated and the model prediction accuracy can be improved.

[0105] Example 3

[0106] This embodiment provides a physically guided critical region fluid nonlinear wave prediction method, which differs from Embodiment 1 or Embodiment 2 in that an adaptive activation function with scalable parameters and fixed scaling factors is introduced into the physical information neural network model, and the loss function is reconstructed by adding a slope recovery term.

[0107] Assume that the depth of the physical information neural network model is D, that is, it includes the input layer, D-1 hidden layer and output layer. Layer contains A neuron.

[0108] No. Output of the layer After linear transformation and nonlinear activation function After the action, it is sent to the next layer as input, that is, , so the standard physical information neural network model can be expressed as:

[0109] (14);

[0110] In the formula, Represents the output of the physical information neural network model when the input is z; represents a composite operator; Indicates Linear transformation of layer (output layer); represents a non-linear activation function.

[0111] Add scalable parameters and fixed scaling factors to the activation function, that is, . Then the updated physical information neural network model is expressed as follows:

[0112] (15);

[0113] In the formula, represents the scaling factor; Indicates Scalable parameters of hidden layers; Represents trainable parameters in physical information neural network models, including weights, biases, and scalable parameters .

[0114] Adding the slope recovery term to the loss function is expressed as follows:

[0115] (16);

[0116] In the formula, represents the slope recovery term.

[0117] The slope recovery term is calculated as follows:

[0118] (17);

[0119] In the formula, , Indicates The hidden layer Scalable parameters of neurons.

[0120] Introducing an adaptive activation function with scalable parameters and fixed scaling factors into the standard physical information neural network model can improve the performance and adaptability of the model in nonlinear systems; reconstructing the loss function by adding a slope recovery term can accelerate the training and convergence of the model.

[0121] Example 4

[0122] This embodiment provides a physically guided critical region fluid nonlinear wave prediction method, which is different from Embodiment 1, Embodiment 2 or Embodiment 3 in that a Dropout regularization mechanism is used in the process of training the physical information neural network model using a sample data set. The Dropout regularization mechanism is: during the training of the physical information neural network model, a certain probability is used to Randomly set some neurons in the physical information neural network model to 0, so that some neurons and all their incoming and outgoing connections are temporarily removed from the physical information neural network model, and then conduct this round of iterative training.

[0123] Taking the further improvement on the basis of Example 3 as an example for explanation, at this time, the physical information neural network model is expressed as follows:

[0124] (18);

[0125] In the formula, express Regularization is inserted between each layer of activation functions so that a portion of the activation function output of each layer is randomly discarded during training.

[0126] The Dropout regularization mechanism is used in physical information neural networks. By randomly discarding neurons, the Dropout regularization mechanism forces the model to learn more flexible and broader feature representations, thereby reducing overfitting to specific data and improving the model's adaptability to physical constraints and the stability of predictions.

[0127] Example 5

[0128] This embodiment provides a physically guided critical region fluid nonlinear wave prediction method, which differs from Embodiment 1 or Embodiment 2 or Embodiment 3 or Embodiment 4 in that the boundary point samples and internal matching point samples in the sample data set are divided into M parts according to the time domain, and the physical information neural network model is trained in a staged training manner.

[0129] The time domain is divided as follows:

[0130] (19);

[0131] In the formula, According to this time domain division method, the boundary point samples and internal matching point samples in the sample data set are divided into M parts according to the time domain.

[0132] The original initial point samples in the sample data set are only used for the first stage of physical information neural network model training. The initial point samples in each subsequent stage are pseudo initial point samples, which are predicted by the physical information neural network model trained in the previous stage; and The physical information of the current stage is introduced into the loss function of the stage. The neural network model fits the error loss of the predicted output of all previous stages. , , It is expressed as follows:

[0133] (20);

[0134] In the formula, Representation space domain The coordinates within; Representation time domain The coordinates inside, represents the initial time, Indicates The end time of the stage; express The number of samples in Indicates The sample set of the stage; and Respectively represent The physical information neural network model of the stage is about the input The real part prediction output and imaginary part prediction output of ; and Respectively represent The physical information neural network model of the stage is about the input The real part prediction output and imaginary part prediction output.

[0135] Taking the further improvement on the basis of Example 3 as an example for explanation, at this time, the The loss function of the stage is expressed as follows:

[0136] (twenty one);

[0137] (twenty two);

[0138] In the formula, and Respectively represent the current sub-time domain The starting point The real part model prediction value and the imaginary part model prediction value of the free water surface displacement wave envelope of the corresponding fluid; and Respectively represent the previous sub-time domain Physical information neural network model prediction Space coordinates at each time The real part model prediction value and imaginary part model prediction value of the free water surface displacement wave envelope of the fluid at the location. The boundary point error loss in the loss function , nonlinear time-varying equation residual and the slope recovery term All of them are the same as the above-mentioned embodiments and will not be further described here.

[0139] In this embodiment, in all stages except the first stage, the “pre-training-fine-tuning” strategy is used. For training at this stage, directly use The network parameters after the segment training are trained instead of the initialized network parameters. This strategy can save computing resources and improve the utilization of the limited data on the basis of improving the generalization ability of the model.

[0140] Example 6

[0141] This embodiment provides a physically guided critical region fluid nonlinear wave prediction method, which differs from Embodiment 1 or Embodiment 2 or Embodiment 3 or Embodiment 4 in that the boundary point samples and internal matching point samples in the sample data set are divided into M parts according to the time domain, and the physical information neural network model is trained in a staged training manner.

[0142] All subsequent stages use the partial predicted solution of the previous stage as the pseudo initial condition. Due to the accumulation of errors, in order to improve the overall prediction accuracy, the error of the first stage should be reduced as much as possible. In this embodiment, the physical information neural network model adopts its reduced-order model to reduce the error of automatic differentiation by increasing the output dimension of the model. Specifically, in this embodiment, the model input is the space-time coordinates of the fluid in the critical region, and its output is the real part of the free water surface displacement wave envelope of the fluid at the corresponding space-time coordinates. , imaginary part , and the real part of the free surface displacement wave envelope of the fluid , imaginary part About Space Domain Coordinates The derivative is used to calculate the residual loss of the time-invariant equation, thereby reducing the second-order derivative of the equation to the first order, avoiding the error of high-order derivative calculation using the automatic differentiation function of the deep learning framework (TensorFlow and Pytorch).

[0143] The initial point error loss is constructed based on the prediction errors of the four model outputs corresponding to the initial point samples, and the boundary point error loss is constructed based on the prediction errors of the four outputs corresponding to the boundary point samples. Taking the further improvement based on Example 3 as an example, at this time, for the loss function of the first stage, it is expressed as follows:

[0144] (twenty three);

[0145] (twenty four);

[0146] (25);

[0147] (26);

[0148] In the formula, the superscript (1) indicates the first stage, and Represents sub-time domain The real and imaginary parts of the free water surface displacement wave envelope of the fluid in the space domain coordinates The first-order partial derivative prediction value of ; and The real and imaginary parts of the free water surface displacement wave envelope of the fluid are expressed in terms of the spatial domain coordinates The analytical value of the first-order partial derivative of ; and They represent the predicted values ​​of the real and imaginary parts of the free water surface displacement wave envelope of the fluid obtained by automatic differentiation with respect to the spatial domain coordinates The first partial derivative of The real and imaginary parts of the free water surface displacement wave envelope of the fluid represented by the neural network output are expressed in terms of spatial domain coordinates The error between the predicted value of the first-order partial derivative and the automatic differentiation; Represents the parameters of the first-stage physical information neural network model. The original initial point samples in the sample data set are only used for the first-stage physical information neural network model training. The initial point samples of each subsequent stage are pseudo initial point samples, which are predicted by the physical information neural network model trained in the previous stage; and the error loss of the physical information neural network model of the current stage fitting the prediction output of all previous stages is introduced into the loss function of the qth stage. , q≥2, the loss function of the qth stage is expressed as follows:

[0149] (27);

[0150] (28);

[0151] (29);

[0152] (30);

[0153] (31);

[0154] In the formula, the superscript (q) indicates the qth stage, and as well as and Respectively represent the current sub-time domain The starting point The corresponding real and imaginary model prediction values ​​of the free water surface displacement wave envelope of the fluid and their spatial domain coordinates The first-order partial derivative prediction value of ; and as well as and Respectively represent the previous sub-time domain Physical information neural network model prediction Space coordinates at each time The real part model prediction value and imaginary part model prediction value of the free water surface displacement wave envelope of the fluid at the position and their relationship with the spatial domain coordinates The first-order partial derivative prediction value of ; and as well as and Respectively represent the current sub-time domain The boundary point The corresponding real and imaginary model prediction values ​​of the free water surface displacement wave envelope of the fluid and their spatial domain coordinates The first-order partial derivative prediction value of ; and as well as and Respectively represent the current sub-time domain The boundary point The corresponding real and imaginary model analytical values ​​of the free water surface displacement wave envelope and their spatial coordinates The analytical value of the first-order partial derivative of ; and Respectively represent the current sub-time domain Internal matching points The corresponding predicted values ​​of the real and imaginary parts of the displacement wave envelope of the free water surface of the fluid obtained by automatic differentiation are about the spatial domain coordinates The first partial derivative of and as well as and Respectively represent The physical information neural network model of the stage is about the input The real and imaginary predicted outputs and their spatial domain coordinates The first partial derivative of and as well as and Respectively represent The physical information neural network model of the stage is about the input The real and imaginary predicted outputs and their spatial domain coordinates The first-order partial derivative of ; the residual of the nonlinear time-varying equation in the loss function and the slope recovery term All of them are the same as the above-mentioned embodiments and will not be further described here.

[0155] Since the initial point of each subsequent sub-time domain except the first one uses a pseudo-exact solution instead of an exact solution, each subsequent sub-time domain will inevitably be affected by the error between the pseudo-exact solution and the exact solution, and the error will accumulate as the number of sub-time domains trained increases. Considering that the first sub-time domain uses an exact solution instead of a pseudo-exact solution, the prediction error of the first sub-time domain is the smallest. In order to effectively improve the overall prediction accuracy and reduce the computational cost, the optimal network parameters are quickly obtained by iterating the network parameter initialization adjustment only for the first sub-time domain. This method significantly reduces the time cost compared to full training on the entire time domain.

[0156] Based on the trained fluid free surface displacement prediction model based on the physical information neural network model, all sub-time domains are predicted continuously, and the prediction results of each sub-time domain are spliced ​​to generate a complete long-time domain prediction result, that is, .

[0157] The test set is used to verify the complete long-term prediction results, and the effect of the prediction results is demonstrated by combining visualization methods. Figure 2 (a) shows the exact solution, Figure 2 (b) shows the model prediction solution; Figure 3 The comparison diagrams of the exact solution and the model prediction solution at three moments are shown, among which (a) is the comparison diagram of the exact solution and the model prediction solution at t=2.69, (b) is the comparison diagram of the exact solution and the model prediction solution at t=3.33, and (c) is the comparison diagram of the exact solution and the model prediction solution at t=3.87. Figure 2 and Figure 3 It can be seen from the figure that the prediction accuracy of the model constructed in this embodiment is relatively high.

[0158] At the same time, the present invention also compares the experimental errors of model prediction in four cases. The first is to use the standard physical information neural network model without any changes, and use the experimental error in this case as a reference benchmark; the second is that the improvement is only to train the standard physical information neural network model by the segmented training method described in Example 5; the third is that the improvement is only to use the gradient-free attention mask method based on residual accumulation to weight the internal collocation point samples in each iteration round, but the weight of the internal collocation point sample is a linear weight (hereinafter referred to as a linear attention mask), that is, the residual term of the nonlinear time-varying equation in formula (9) and formula (10) does not carry a square; the fourth is that the improvement includes the use of the gradient-free attention mask method based on residual accumulation in Example 1 to weight the internal collocation point samples in each iteration round, and the weight of the internal collocation point sample is a nonlinear weight (hereinafter referred to as a nonlinear attention mask), and the segmented training method described in Example 5. The experimental error results of the model prediction in the four cases are shown in Table 1.

[0159] Table 1

[0160] .

[0161] Example 7

[0162] This embodiment provides a physics-guided critical region fluid nonlinear wave prediction system, including:

[0163] An equation building module is used to establish a computational domain according to the spatiotemporal characteristics of the fluid flow in the critical region, and determine a nonlinear time-varying equation for describing the nonlinear wave dynamics characteristics of the fluid in the critical region; the solution of the nonlinear time-varying equation is the free water surface displacement wave envelope of the fluid, which is expressed in complex form, i.e., includes a real part solution and an imaginary part solution;

[0164] The data set construction module is used to select the initial points, boundary points and internal matching points for model training in the computational domain to construct a sample data set;

[0165] A neural network model building module is used to build a physical information neural network model with the space-time coordinates of the fluid in the critical region as input and the free water surface displacement wave envelope of the fluid at the space-time coordinates as output;

[0166] The loss function determination module is used to construct the residual loss, initial point error loss and boundary point error loss of the nonlinear time-varying equation, and add the three to obtain the loss function; each loss includes the corresponding real part loss and imaginary part loss;

[0167] The training module is used to train the physical information neural network model using a sample data set to obtain a prediction model for the free water surface displacement of the fluid in the critical region; during the training process, a gradient-free nonlinear attention mask method based on residual accumulation is used to weight the internal collocation point samples in each iteration round;

[0168] The prediction module is used to input the time and space coordinates of the critical region fluid point to be predicted into the critical region fluid free water surface displacement wave envelope prediction model to obtain the free water surface displacement of the fluid at the point to be predicted.

[0169] It should be understood that the functional unit modules in the embodiments of the present invention may be concentrated in one processing unit, or each unit module may exist physically separately, or two or more unit modules may be integrated in one unit module, and may be implemented in the form of hardware or software.

[0170] Example 8

[0171] This embodiment provides an electronic terminal, including:

[0172] a memory having a computer program stored thereon;

[0173] A processor is used to load and execute the computer program to implement the physical-guided critical region fluid nonlinear wave motion prediction method described in the above embodiment.

[0174] Example 9

[0175] This embodiment provides a computer-readable storage medium having a computer program stored thereon. When the computer program is executed by a processor, the method for predicting nonlinear wave motion of critical region fluid under physical guidance described in the above-mentioned embodiment is implemented.

[0176] It can be understood that the same or similar parts of the above embodiments can be referenced to each other, and the contents not described in detail in some embodiments can refer to the same or similar contents in other embodiments.

[0177] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, the present application may adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program codes.

[0178] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0179] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0180] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 A step that specifies a function in one or more boxes.

[0181] Although the embodiments of the present invention have been shown and described above, it is to be understood that the above embodiments are exemplary and are not to be construed as limitations of the present invention. A person skilled in the art may change, modify, replace and vary the above embodiments within the scope of the present invention.

Claims

1. A physically guided method for predicting nonlinear wave motion of critical region fluid, characterized in that: The steps include: According to the spatiotemporal characteristics of the fluid flow in the critical region, a computational domain is established, and a nonlinear time-varying equation for describing the nonlinear wave dynamics characteristics of the fluid in the critical region is determined; the solution of the nonlinear time-varying equation is the free water surface displacement wave envelope of the fluid, which is expressed in complex form; Selecting initial points, boundary points and internal matching points for model training in the computational domain to construct a sample data set; Construct a physical information neural network model with the space-time coordinates of the fluid in the critical region as input and the free water surface displacement wave envelope of the fluid at the space-time coordinates as output; Construct the residual loss, initial point error loss and boundary point error loss of the nonlinear time-varying equation, and add the three together to get the loss function; each loss includes the corresponding real part loss and imaginary part loss; The physical information neural network model is trained with sample data sets to obtain the displacement wave envelope prediction model of the free water surface of the critical region fluid. During the training process, the gradient-free nonlinear attention mask method based on residual accumulation is used to weight the internal collocation point samples in each iteration round. Inputting the time and space coordinates of the critical region fluid point to be predicted into the critical region fluid free water surface displacement wave envelope prediction model to obtain the free water surface displacement wave envelope of the fluid at the point to be predicted; During the training process, a gradient-free nonlinear attention mask method based on residual accumulation is used to weight the internal collocation point samples in each iteration round, specifically including: Initialize the weights of the internal collocation point samples in the first round of iteration; Starting from the second round of iteration, the weights of the internal collocation point samples in each iteration round are determined by the following formula: ； ； In the formula, and denote the real weight of the kth internal collocation point sample in the mth and m+1th iterations respectively; and Respectively represent the imaginary weight of the kth internal collocation point sample in the mth and m+1th iterations; represents the attenuation coefficient; represents the learning rate; and They represent the real residual and imaginary residual of the nonlinear time-varying equation of the kth internal collocation point sample in the mth iteration respectively; and They represent the real residual mean and imaginary residual mean of the nonlinear time-varying equation of all internal collocation point samples in the mth iteration respectively; Represents a minimum constant that prevents division by 0; Represents the number of internal collocation point samples in each iteration.

2. The physically guided critical region fluid nonlinear wave motion prediction method according to claim 1, characterized in that: The residual loss of the nonlinear time-varying equation in the mth iteration Calculated by the following formula: ； In the formula, represents the real residual of the nonlinear time-varying equation of the kth internal collocation point sample, represents the imaginary residual of the nonlinear time-varying equation of the kth internal collocation point sample, Represents the spatiotemporal coordinates of the kth internal collocation point sample.

3. The physically guided critical region fluid nonlinear wave motion prediction method according to claim 1 or 2, characterized in that: In the process of training the physical information neural network model using a sample data set, a gradient-free nonlinear attention mask method based on residual accumulation is used to weight the initial point samples and / or boundary point samples in each iteration round.

4. The physically guided critical region fluid nonlinear wave motion prediction method according to claim 1 or 2, characterized in that: An adaptive activation function with scalable parameters and fixed scaling factors is introduced into the physical information neural network model, and the loss function is reconstructed by adding a slope recovery term, including: The depth of the physical information neural network model is D, which includes the input layer, D-1 hidden layers and the output layer. The physical information neural network model is expressed as follows: ； In the formula, Represents the output of the physical information neural network model when the input is z; represents a composite operator; represents the linear transformation of the Dth layer; represents a nonlinear activation function; represents the scaling factor; represents the expandable parameters of the D-1th hidden layer; Represents trainable parameters in physical information neural network models, including weights, biases, and scalable parameters; Adding the slope recovery term to the loss function is expressed as follows: ； In the formula, represents the loss function, , , and They represent the initial point error loss, boundary point error loss, nonlinear time-varying equation residual loss and slope recovery term respectively; The slope recovery term is calculated as follows: ； In the formula, , represents the expandable parameter of the ith neuron in the dth hidden layer, Represents the number of neurons in the dth hidden layer.

5. The physically guided critical region fluid nonlinear wave motion prediction method according to claim 4, characterized in that: The Dropout regularization mechanism is used in the process of training the physical information neural network model using a sample data set. The Dropout regularization mechanism is: during the training of the physical information neural network model, a certain probability is used to Randomly set some neurons in the physical information neural network model to 0, so that some neurons and all their incoming and outgoing connections are temporarily removed from the physical information neural network model, and then conduct this round of iterative training; at this time, the physical information neural network model is expressed as follows: ； In the formula, express Regularization is inserted after each layer of activation function, so that a part of the activation function output of each layer is randomly discarded during training.

6. The physically guided critical region fluid nonlinear wave motion prediction method according to claim 1 or 2, characterized in that: The computational domain includes the space domain and the time domain. The boundary point samples and the internal collocation point samples in the sample data set are divided into M parts according to the time domain. The physical information neural network model is trained in a staged training method. The original initial point samples in the sample data set are only used for the first stage of physical information neural network model training. The initial point samples in each subsequent stage are pseudo initial point samples, which are predicted by the physical information neural network model trained in the previous stage; and the error loss of the physical information neural network model of the current stage fitting the prediction output of all previous stages is introduced into the loss function of the qth stage. , q≥2.

7. The physically guided critical region fluid nonlinear wave motion prediction method according to claim 1 or 2, characterized in that: The computational domain includes the space domain and the time domain. The boundary point samples and the internal collocation point samples in the sample data set are divided into M parts according to the time domain. The physical information neural network model is trained in a staged training method. The physical information neural network model adopts its reduced-order model, whose input is the space-time coordinates of the fluid in the critical region, and whose output is the real and imaginary parts of the free water surface displacement wave envelope of the fluid at the corresponding space-time coordinates, as well as the derivatives of the real and imaginary parts of the free water surface displacement wave envelope of the fluid with respect to the space domain coordinates; The initial point error loss is constructed based on the prediction errors of the four model outputs corresponding to the initial point samples, and the boundary point error loss is constructed based on the prediction errors of the four model outputs corresponding to the boundary point samples; The original initial point samples in the sample data set are only used for the first stage of physical information neural network model training. The initial point samples in each subsequent stage are pseudo initial point samples, which are predicted by the physical information neural network model trained in the previous stage; and the error loss of the physical information neural network model of the current stage fitting the prediction output of all previous stages is introduced into the loss function of the qth stage. , q≥2.

8. A physically guided critical region fluid nonlinear wave prediction system, characterized in that: The method for predicting nonlinear wave motion of critical region fluid guided by physics as claimed in claim 1 comprises: An equation building module is used to establish a computational domain according to the spatiotemporal characteristics of the fluid flow in the critical region, and determine a nonlinear time-varying equation for describing the nonlinear wave dynamics characteristics of the fluid in the critical region; the solution of the nonlinear time-varying equation is the free water surface displacement wave envelope of the fluid, which is expressed in complex form; The data set construction module is used to select the initial points, boundary points and internal matching points for model training in the computational domain to construct a sample data set; A neural network model building module is used to build a physical information neural network model with the space-time coordinates of the fluid in the critical region as input and the free water surface displacement wave envelope of the fluid at the space-time coordinates as output; The loss function determination module is used to construct the residual loss, initial point error loss and boundary point error loss of the nonlinear time-varying equation, and add the three to obtain the loss function; each loss includes the corresponding real part loss and imaginary part loss; The training module is used to train the physical information neural network model using a sample data set to obtain a prediction model for the displacement wave envelope of the free water surface of the fluid in the critical region; during the training process, a gradient-free nonlinear attention mask method based on residual accumulation is used to weight the internal collocation point samples in each iteration round; The prediction module is used to input the time and space coordinates of the critical region fluid to be predicted point into the critical region fluid free water surface displacement wave envelope prediction model to obtain the free water surface displacement wave envelope of the fluid to be predicted point.

9. An electronic terminal, characterized in that: include: a memory having a computer program stored thereon; A processor is used to load and execute the computer program to implement the physically guided critical region fluid nonlinear wave motion prediction method as described in any one of claims 1 to 7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the physically guided critical region fluid nonlinear wave motion prediction method according to any one of claims 1 to 7 is implemented.

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