PINN-based substance transport prediction and model parameter inversion method

Abstract

The invention provides a PINN-based substance transport prediction and model parameter inversion method, which comprises the following steps of: constructing a partial differential equation set for solving substance transport; determining a network structure, and constructing a PINN-based multi-component substance transport prediction model by using a residual connection network; constructing a loss function; acquiring data of observation points; model input including solute initial concentration, boundary conditions, space domain, time domain and observation point concentration; model training: optimizing a loss function of the multi-component substance transportation prediction model through an optimization algorithm; in the loss function optimization process, a successive approximation method is adopted for coefficients to be solved of a partial differential equation set of the model; and model output: after the model training is finished, calculating the concentration of any unknown time and the coefficient to be solved of the partial differential equation set in the time-space domain through the model output.

Description

Technical Field

[0001] The present invention relates to the technical field of numerical simulation based on the physics-informed neural network (PINN), and particularly to a method for predicting mass transport and inverse model parameter inversion based on PINN. This method is widely applied in the fields of environmental science, water resource management, petroleum engineering, chemical engineering, etc. Background Art

[0002] In recent years, with the sharp increase in the production of deep-sea oil extraction, the leakage of pollutants and the damage to the water environment have become a major global concern. Predicting pollution based on water pollutant data is essentially an inverse problem in the field of multi-reactant transport, that is, given the governing equation, initial conditions, boundary conditions, and partial observation data, the pollutant diffusion process is reconstructed in reverse. In existing research, the convection-diffusion equation is used to describe the pollutant diffusion process [1]. However, due to the uncertainty of the relevant parameters in the transport model and certain errors in the measured data, accurate results cannot be obtained using traditional numerical methods and experimental methods.

[0003] With the development of deep learning technology, the physics-informed neural network (PINN), as an emerging method, has shown great potential in solving complex physical problems. The physics-informed neural network (PINN) method is an algorithm for solving partial differential equations based on the fitting ability of neural networks. It was proposed by Raissi and Karniadakis [2, 3]. Raissi [4] first used the PINNs model to solve the forward and inverse convection-diffusion problems, opening up a new research method for traditional fluid mechanics research. The main difference between this algorithm model and the ordinary artificial neural network ANN is that PINN can combine the physical properties of the problem and obtain the target result by controlling the differential equation of the observation data. When He et al. [5] used the deep learning model PINN method to solve the inverse convection-diffusion problem, it was found by comparison that the accuracy of the solution obtained by PINN is better than some traditional discretization-based methods. Compared with the reference concentration field, the relative error in most cases remains below 5%. In previous PINN research, two simple residual point sampling methods were mainly used for collocation points, namely equidistant uniform grids and uniform random sampling. Such sampling methods are very effective for solving some simple partial differential equations, but may not be efficient for those more complex equations. In addition, in principle, PINNs regard physical laws as part of the loss function, which means that the training process needs to optimize the network simultaneously to satisfy the data-driven loss and the physics-driven loss [6], resulting in the network being often difficult to train.

[0004] Patent CN202411079622.8 designed a PINN-based method for inverse calculation of magneto-fluid system coefficients. Patent CN202110334092.7 designed a PINN-based method for correcting turbulence model coefficients. Patent CN202311765731.0 optimized the PINN network structure based on the particle swarm optimization algorithm.

[0005] To further improve the prediction accuracy, application range, and model interpretability of the convection-diffusion inverse problem, a residual network structure is integrated into the original PINN model, and probability-based point selection is used for the selection of control points. The present invention proposes a new method for material transport prediction and model parameter inversion based on the physics-informed neural network, and the model is verified through a numerical example of a multi-reactant two-dimensional convection-diffusion model.

[0006] References:

[0007] [1] P. Jing, Z. Yang, W. Zhou, W. Huai, X. Lu, Inverse estimation of finite-duration source release mass in river pollution accidents based on adjoint equation method, Environ Sci Pollut Res Int 27(13)(2020)14679 - 14689.

[0008] [2] G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics-informed machine learning, Nature Reviews Physics 3(6)(2021)422 - 440.

[0009] [3] M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational physics 378(2019)686 - 707.

[0010] [4]M.Raissi,A.Yazdani,G.E.Karniadakis,Hidden fluid mechanics:Learning velocity and pressure fields from flow visualizations,Science 367(6481)(2020)1026-1030.

[0012] [5]Q.He,A.M.Tartakovsky,Physics-Informed neural network method for forward and backward advection-dispersion equations,Water Resources Research 57(7)(2021)e2020WR029479.

[0013] [6]S.Wang,X.Yu,P.Perdikaris,When and why PINNs fail to train:A neural tangent kernel perspective,Journal of Computational Physics 449(2022)110768. Summary of the Invention

[0014] The purpose of the present invention is to provide a transport prediction and model parameter inversion method based on a PINN network to overcome the deficiencies existing in the prior art. This method utilizes the powerful fitting ability and physical constraints of PINNs to efficiently and accurately predict the transport process of multi-component solutes and achieve the inversion of model parameters. The technical solution of the present invention is as follows:

[0015] A method for predicting the transport of substances and inverting model parameters based on PINN, comprising the following steps:

[0016] (1) Construct a partial differential equation system for solving substance transport;

[0017] (2) Determine the network structure and construct a multi-component substance transport prediction model based on PINN using a residual connection network: x [d] represents the state vector of the nodes in the d-th layer, and H [i] is both the output of the i-th residual block and the input of the (i + 1)-th residual block; use N h to represent the number of hidden layers contained within each residual block. The i-th residual block consists of the network from the ((i - 1)N h + 1)-th layer to the iN h -th layer; H[i] is equal to the state vector X[iN h +1] of the node in the (iN h +1)-th layer;

[0018] (3) Construct the loss function. The loss function in the multi-component mass transport prediction model based on PINN consists of four parts. One part is the structural loss l c controlled by the collocation points X s selected within the computational domain. This part represents the constraint of physical information, and |X c | represents the number of collocation points; l ic and l bc are the losses on the initial condition and the boundary condition respectively. X ic and X bc represent the initial condition and boundary condition points. This part represents the constraints on the initial and boundary conditions, and |X ic | and |X bc | represent the corresponding numbers; The fourth part is the loss l between the predicted concentration of the model at the observation points ob , and the observed concentration. Θ represents the number of observation points; Θ represents the model parameters, c Θ is the predicted value of the concentration when the parameter is Θ, D s represents the predicted value of the coefficient to be solved in the partial differential equation system when the parameter is Θ, and ω ic , ω bc , ω ob represent the weights of each item:

[0019]

[0020] The collocation points are selected by the method of probability-based sampling, that is, each residual point is sampled according to the probability density function p(x) proportional to the residual of the partial differential equation;

[0021] (4) Obtain the observation point data;

[0022] (5) Model input: including the initial concentration of the solute, boundary conditions, spatial domain, time domain, and the concentration at the observation points;

[0023] (6) Model training: Optimize the loss function of the multi-component mass transport prediction model through an optimization algorithm; During the optimization of the loss function, for the coefficient D x to be solved in the partial differential equation system of the model, the method of gradual approximation is adopted. First, assume an initial value for D x , and then use the optimization algorithm to perform gradient descent on the loss function to update the network parameters. Through multiple iterations and error feedback, ensure that the model can accurately predict each parameter in the solute transport process;

[0024] (7) Model output: After the model is trained, the concentration at any unknown time in the spatio-temporal domain and the coefficient D to be solved in the partial differential equation system are calculated through the model output. x 。

[0025] Furthermore, the partial differential equation system for solving mass transport established in step (1):

[0026]

[0027] where c represents the mass concentration, x represents the position, t represents the time, u is the velocity term which is default as a known value in the solution of the convection-diffusion problem, and D x represents the coefficient to be solved in the partial differential equation system, S(x, t) is the source term. When t = 0, it represents the initial concentration distribution condition. When x = 0 or x = 1, it represents the model boundary condition.

[0028] Furthermore, the network model established in step (2) is set to have seven hidden layers, with 100 neurons in each layer.

[0029] Furthermore, in step (3), ω s , ω ic , ω bc , ω ob are all 1.

[0030] Furthermore, in step (3), the collocation points are selected by probability in the following way:

[0031] For any point in the computational domain, calculate the equation residual:

[0032]

[0033] where N r is the total number of points participating in the calculation in the computational domain, is the actual concentration of the point;

[0034] Calculate the probability:

[0035] A = ∫ Ω ∈(x)dx

[0036] p(x) ∝ ∈(x)

[0037]

[0038] m is a small positive number.

[0039] Furthermore, in step (4), the concentration of the observation point is either calculated by the MTVDLF-Superbee method, or the actual observation data or the calculation results of other numerical algorithms are selected.

[0040] Further, step (5) includes: when t = 0, given the initial concentration distribution condition:

[0041] c(x, t = 0) = c ic

[0042] The boundary conditions are given in the following form:

[0043] c(x, t) = c bc , x = 0 or x = 1

[0044] After normalization, the time domain and space domain are [0, 1].

[0045] Further, in step (8), the optimization algorithm is the gradient descent method.

[0046] The innovation of the present invention lies in combining physical information with neural networks, enabling the model to not only follow physical laws but also improve prediction accuracy through data-driven methods. The PINN model can not only handle transport problems but also efficiently invert unknown model parameters, making the prediction results more reliable. Through the method provided by the present invention, accurate prediction of multi-component solute transport in the environment can be achieved, providing a scientific basis for environmental protection and resource management. At the same time, this method also has broad application prospects in industrial fields such as chemical engineering and petroleum. Brief Description of the Drawings

[0047] In order to more detailedly and intuitively illustrate the reliability and accuracy of the technical solutions in the embodiments of the present invention, the drawings required in the embodiments will be briefly introduced below. The following described drawings are only partial embodiments of the present invention. Through some simple modifications, the present invention can be applied to other embodiments without creative efforts.

[0048] Figure 1 Physics-informed neural network model

[0049] Figure 2 Local residual connection network structure

[0050] Figure 3 Comparison of the results of the predicted concentration change of convective reaction with the exact solution

[0051] Figure 4 Results of the predicted concentration change of convective-diffusion reaction Detailed Embodiments

[0052] Next, the specific solution process of this method will be described by taking the variable coefficient multi-component problem as an example.

[0054] Implementation step 1: Construction of the PINN model

[0055] (1) Construct a partial differential equation system for solving mass transport.

[0056]

[0057] Among them, c represents the substance concentration, x represents the position, t represents the time, u is the velocity term which is default as a known value in the solution of the convection-diffusion problem, and D x represents the coefficient to be solved, S(x, t) is the source term. When t = 0, it represents the initial concentration distribution condition, and when x = 0 or x = 1, it represents the model boundary condition.

[0058] (2) Construct a physics-informed neural network model. The schematic diagram of the PINN algorithm is shown in the figure. The input training points (, ) consist of three parts: the initial sampling points X ic , the boundary sampling points X bc , and the collocation points X c . Use the fully connected feedforward DNN corresponding to the input points to calculate the predicted values. The parameter is the parameter set of the network, including weights and biases, and is the activation function. Use automatic differentiation to calculate the partial derivatives of c with respect to and Θ . The loss function evaluates the governing equation using the contributions of the initial boundary conditions and the residuals of the partial differential equations. Then, find the optimal values of and to minimize the loss function below a specific specified parameter or until a specified maximum number of iterations.

[0059] Determine the network structure. The PINN model contains multiple input layers, hidden layers, and output layers through which the network can learn complex non-linear relationships. Different from the fully connected network of the traditional PINN, the optimized model uses a residual connection network. x [d] represents the state vector of the nodes in the d-th layer, while H [i] is both the output of the i-th residual block and the input of the (i + 1)-th residual block. Use N h to represent how many hidden layers are inside each residual block. The i-th residual block consists of the network from the ((i - 1)N h + 1)-th layer to the iN h -th layer. H [i] is equal to the state vector X[iN h + 1] of the nodes in the ((iN h + 1)-th layer. The network model is set to have seven hidden layers with 100 neurons in each layer.

[0060] (3) Construct the loss function:

[0061] The loss function in the multi-component substance transport prediction model based on the physics-informed neural network consists of four parts. One is the structural loss l c controlled by the collocation points X s selected within the computational domain. This part represents the constraint of the physical information. l ic and l bcis the loss on the initial conditions and boundary conditions, X ic and X bc represent the initial conditions and boundary condition points, and this part represents the constraints on the initial and boundary conditions. The third type is the observation points the loss l between the model predicted concentration and the observed concentration at the observation points ob c Θ is the predicted concentration value, ω s ω ic ω bc ω ob represent the weights of each item, and can be defaulted to 1 under normal circumstances without special requirements.

[0062]

[0063] In previous PINN studies, two simple residual point sampling methods were mainly used for the collocation points, namely equidistant uniform grids and uniform random sampling. Such sampling methods are very effective for solving some simple partial differential equations, but may not be efficient for those more complex equations. Therefore, probability-based point selection is proposed on the basis of uniform point selection, that is, all residual points are resampled according to the probability density function p(x) proportional to the residual of the partial differential equation. Specifically, for any point within the computational domain, first calculate the equation residual:

[0064]

[0065] where, N r is the total number of points participating in the calculation within the computational domain, which can be designed according to actual needs, is the actual concentration of the point.

[0066] Then calculate the probability:

[0067] A = ∫ Ω ∈(x)dx

[0068] p(x) ∝ ∈(x)

[0069]

[0070] All residual points are sampled according to the function p(x) to select collocation points, but in the process of training, this method still tends to focus on the positions with the largest equation residuals and ignores the smaller residuals. Therefore, a constant m is added to the residual ε(x) to ensure that the sampling points with smaller residuals can also be included in the sampling range, ensuring that there is enough randomness to explore different regions, but not completely random to ignore the importance of high-residual regions to avoid computational bottlenecks.

[0071] Meanwhile, for the partial differential equations of multiple components, the reaction conditions of each reactant are different. It is necessary to calculate the different probability density functions of each reactant according to the residuals of each reactant, and select the training points according to their different probability density functions. The newly selected points are the union of the points selected according to the probability density functions of each reactant. Taking the partial differential equation of two components as an example, the probability density function at this time is:

[0072]

[0073] (4) Observation point data acquisition: The concentration of the observation point is calculated from the MTVDLF-Superbee method, or the actual observation data or the calculation results of other high-precision numerical algorithms can also be selected. In this experiment, 2500 grid points in the space-time domain were selected as the observation values (50 in space × 50 in time).

[0074] (5) Model input: It includes the initial concentration of the solute, boundary conditions, spatial domain, time domain and the concentration of the observation point.

[0075] When t = 0, the initial concentration distribution condition is given:

[0076] c(x, t = 0) = c ic

[0077] The boundary conditions are given in the following form:

[0078] c(x, t) = c bc , x = 0 or x = 1

[0079] The time domain and the spatial domain default to [0, 1], and normalization processing is required if it exceeds the region.

[0080] (6) Model training: Optimize the loss function of the multi-component mass transport prediction model through an optimization algorithm (such as the gradient descent method). During the optimization process of the loss function, for the coefficients D to be solved in the model x Adopt the method of gradual approximation. First, assume D x An initial value, and then use the optimization algorithm to perform gradient descent on the loss function, update the network parameters, and through multiple iterations and error feedback, ensure that the model can accurately predict various parameters in the solute transport process.

[0081] The specific parameter inversion process includes the following steps:

[0082] Initial guess: Make an initial guess for the unknown parameter D x

[0083] Loss function calculation: Calculate the value of the model loss function.

[0084] Parameter adjustment: Use the optimization algorithm to minimize the loss function. ​

[0085] Iterative process: Repeat the above steps until the error converges to an acceptable range or the specified number of iteration steps.

[0086] Through the above inversion process, the unknown parameters in the model can be effectively determined, making the model more accurate. This method can converge quickly and avoid the local optimum problem in traditional inversion methods.

[0087] (6) Model output:

[0088] After the model training is completed, the concentration at any unknown time and the coefficient D to be determined in the calculated space-time domain can be output through the model. x 。

Claims

1. A PINN-based material transport prediction and model parameter inversion method, comprising the following steps: (1) Construct a system of partial differential equations to solve material transport; (2) Determine the network structure and use the residual connection network to build a multi-component material transport prediction model based on PINN: [d] represents the state vector of the d-th layer node, and H [i] It is both the output of the i-th residual block and the input of the i+1-th residual block; use N h Represents the number of hidden layers contained in each residual block. The i-th residual block consists of the (i-1)N h +1 level to iNth level h The network composition of the layer; H [i] Equal to the first (iN h +1) layer node state vector X[iN h +1]; (3) Constructing the loss function. The loss function in the multi-component material transport prediction model based on PINN consists of four parts. One part is the collocation point X selected in the calculation domain. c Controlled structural losses s , this part represents the constraints of physical information, |X c | indicates the number of collocation points; l ic and l bc are the losses on initial conditions and boundary conditions respectively, X ic and X bc Indicates the initial and boundary condition points. This part indicates the constraints on the initial and boundary conditions. ic | and |X bc | indicates the corresponding number; the fourth part is the observation point The loss between the model predicted concentration and the observed concentration is l ob , represents the number of observation points; Θ represents the model parameters, c Θ It represents the concentration prediction value when the parameter is Θ, D Θ represents the predicted value of the coefficient of the partial differential equation system when the parameter is Θ, ω s ,ω ic ,ω bc ,ω ob Represents the weights of each item: The collocation points are selected by probability, that is, each residual point is sampled according to the probability density function p(x) proportional to the residual of the partial differential equation; (4) Acquisition of observation point data; (5) Model input: including initial solute concentration, boundary conditions, spatial domain, time domain and observation point concentration; (6) Model training: The loss function of the multi-component material transport prediction model is optimized through an optimization algorithm. During the loss function optimization process, the coefficients D of the partial differential equations of the model are calculated. x Using the method of stepwise approximation, we first assume that D x An initial value is given, and then the optimization algorithm performs gradient descent on the loss function, updates the network parameters, and through multiple iterations and error feedback, ensures that the model can accurately predict various parameters in the solute transport process; (7) Model output: After the model training is completed, the model output is used to calculate the concentration of any unknown at any time in the space-time domain and the coefficients of the partial differential equation system to be calculated. x .

2. The PINN-based material transport prediction and model parameter inversion method according to claim 1, characterized in that: The partial differential equations for solving material transport established in step (1) are: Where C represents the concentration of the substance, x represents the position, t represents the time, and u is the velocity term, which is a known value by default in solving convection-diffusion problems. x represents the unknown coefficients of the partial differential equations, S(x,t) is the source term, when t=0, it represents the initial concentration distribution condition, when x=0 or x=1, it represents the model boundary condition.

3. The PINN-based material transport prediction and model parameter inversion method according to claim 1, characterized in that: The network model established in step (2) is set to seven hidden layers, with 100 neurons in each layer.

4. The PINN-based material transport prediction and model parameter inversion method according to claim 1, characterized in that: In step (3), ω s ,ω ic ,ω bc ,ω ob Both are 1.

5. The PINN-based material transport prediction and model parameter inversion method according to claim 1, characterized in that: In step (3), the collocation points are selected by probability as follows: For any point in the computational domain, calculate the residual of the equation: Among them, N r is the total number of points involved in the calculation in the computational domain, is the actual concentration of the point; Calculate the probability: A=∫ Ω ∈(x)dx p(x)∝∈(x) m is a small positive number.

6. The PINN-based material transport prediction and model parameter inversion method according to claim 1, characterized in that: In step (4), the concentration at the observation point is either calculated using the MTVDLF-Superbee method, or actual observation data or calculation results of other numerical algorithms are selected.

7. The PINN-based material transport prediction and model parameter inversion method according to claim 1, characterized in that: Step (5) includes: when t=0, giving the initial concentration distribution condition: c(x,t=0)=c ic The boundary conditions are given by: c(x,t)=c bc ,X=0orx=1 After normalization, the time domain and space domain are [0,1].

8. The PINN-based material transport prediction and model parameter inversion method according to claim 1, characterized in that: In step (8), the optimization algorithm is the gradient descent method.

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