Multi-component reactive solute transport simulation method based on physical information neural network

Abstract

The invention discloses a multi-component reaction solute transport simulation method based on a physical information neural network. Establishing a multi-component convection-dispersion-reaction (ADR) equation model; constructing a double sub-network, and loading a Runge-Kutta coefficient so as to complete implicit time advance; calculating a solute concentration gradient change rate and a reaction intensity change rate which are respectively used as two types of self-adaptive control indexes, and dynamically adjusting a time step length; and training iteration is carried out by adopting an Adam and L-BFGS combined optimization algorithm, so that recursive prediction of the spatial and temporal distribution of the pollutant concentration is realized. According to the method, the time resolution can be automatically adjusted, the calculation efficiency and the numerical stability are both considered, compared with a traditional PINN and time marching PINN (TM-PINN) method, the long-term prediction precision and the convergence performance are remarkably improved, and the method is suitable for underground water pollution migration simulation and multi-component reaction-diffusion process modeling.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation technology for multi-component reaction-diffusion, specifically to a method for simulating solute transport in multi-component reactions based on a physical information neural network. Background Technology

[0002] Groundwater pollution is a major environmental issue of global concern, and its remediation and restoration are directly related to drinking water safety and ecosystem stability. Pollution sources include both natural geological processes, such as bedrock weathering, mineral dissolution, and adsorption / desorption, and human activities, such as industrial wastewater discharge and agricultural non-point source pollution. To effectively assess pollutant migration risks and formulate water resource management policies, researchers typically use mathematical models to describe and predict the transport and reaction processes of pollutants in groundwater aquifers.

[0003] Currently, solute transport models are typically based on the Advance-Dispersion-Reaction (ADR) equation. This equation comprehensively describes the concentration changes of pollutants under the combined effects of fluid convection, mechanical dispersion, and chemical reaction, and is the fundamental equation for studying the migration patterns of groundwater pollution. Traditional numerical simulations of solute transport often employ discrete numerical methods such as the Finite Difference Method (FDM), Finite Element Method (FEM), or Finite Volume Method (FVM) to solve the ADR equation. These methods have clear structures and mature theories, and have been widely used in hydrogeological and environmental engineering simulations, resulting in several mature software platforms, such as PFLOTRAN and Groundwater Modeling System (GMS). However, as research objects expand from experimental scales to site scales, high-dimensional, and multi-component systems, traditional methods have gradually revealed the following shortcomings in practical applications: (1) High computational cost and difficulty in balancing accuracy and stability. In high-dimensional heterogeneous aquifers, in order to ensure numerical stability, the model needs to use a fine grid in space and set a small time step, which increases the computational cost exponentially, resulting in huge computational costs for long-term simulations. At the same time, the accumulated error is easily amplified during long-term evolution, affecting the reliability of the results.

[0004] (2) The parameterization assumptions are too idealistic. To reduce computational complexity, traditional models usually assume that physical parameters such as dispersion coefficient, flow velocity, and reaction rate are constant or uniformly distributed in space. This simplified setting makes it difficult to accurately reflect the spatial variability of actual aquifers, thus limiting the model's ability to characterize the details of pollutant migration.

[0005] (3) Insufficient stability in solving nonlinear reaction systems. In multi-component coupled reaction systems, reaction terms often exhibit strong nonlinear or rigid characteristics. Traditional algorithms usually require repeated linearization or multiple iterations to solve the problem. Their convergence is sensitive to initial values ​​and parameters, and they are prone to problems such as oscillation or non-convergence.

[0006] (4) Fixed time step strategy limits simulation efficiency. Most existing solvers use fixed time step for time progression. When pollutant concentration or reaction rate changes rapidly, too large a step will lead to increased error, while too small a step will significantly increase computation time, making it difficult to achieve adaptive time resolution adjustment.

[0007] To address the aforementioned issues, the Physics-Informed Neural Network (PINN) method, based on machine learning, has emerged in recent years. This method directly embeds the physical constraints of partial differential equations into the loss function of the neural network, achieving meshless solutions through data-driven end-to-end training, providing a new approach to complex multiphysics problems. Research on PINN in fluid mechanics, reaction-diffusion systems, and other fields has shown that it can balance computational efficiency and solution accuracy to a certain extent. However, standard PINN still suffers from optimization difficulties, gradient imbalances, and convergence instability when handling high-dimensional, strongly nonlinear, and long-duration simulation problems. To improve this limitation, researchers have proposed Time-Marching PINN (TM-PINN), which effectively improves the stability and accuracy of long-duration problems by dividing the time domain into several sub-intervals and training them step by step. TM-PINN has been applied to structural dynamics and groundwater flow, but it still suffers from fixed time steps, error accumulation, and insufficient computational efficiency in reactive solute transport problems. Another approach is Runge–Kutta PINN (RK-PINN), which embeds an implicit Runge–Kutta time-discrete scheme within the PINN framework, improving the numerical stability and accuracy of rigid systems. However, the original RK-PINN can only perform single-step predictions, making it difficult to achieve long-term, multi-step recursive simulations. Furthermore, its time step remains a fixed value and cannot be automatically adjusted according to dynamic changes in the system.

[0008] In summary, while existing PINN, TM-PINN, and RK-PINN methods have overcome some of the limitations of traditional numerical methods, they still suffer from problems such as fixed time steps, error accumulation, unstable training, and difficulty in balancing accuracy and efficiency. Summary of the Invention

[0009] The purpose of this invention is to provide a method for simulating solute transport in multi-component reactions based on a physical information neural network (PINN). This method is based on an adaptive step-size Runge-Kutta PINN architecture, introducing a recursive prediction mechanism on the basis of an implicit Runge-Kutta discrete structure. It achieves dynamic adjustment of the time step through two types of adaptive controllers: gradient rate of change and nonlinear reaction intensity. This significantly improves computational efficiency and prediction accuracy while ensuring numerical stability. It addresses the problems of fixed time step, error accumulation, optimization instability, and low computational efficiency inherent in existing PINNs and their improved forms when solving multi-component reaction solute transport problems.

[0010] To solve the above-mentioned technical problems, the present invention provides the following technical solution: The multi-component reaction solute transport simulation method based on physical information neural networks includes the following steps: Establish a convection-diffusion-reaction mathematical model and determine the initial and boundary conditions.

[0011] Dimensionless convection-diffusion-reaction mathematical model, initial conditions, and boundary conditions. Specifically, the convection-diffusion-reaction mathematical model is dimensionless, and the characteristic scale is defined as follows: , in, The horizontal feature length, For characteristic time, For concentration characteristic values, Characteristic flow velocity, For longitudinal dispersion scale, is the characteristic dispersion coefficient.

[0012] Dimensionless equations , in The initial and boundary conditions, after being dimensionless, are identical in form to the original expression.

[0013] Constructing an adaptive step-size Runge-Kutta physical information neural network model; the adaptive step-size Runge-Kutta physical information neural network model includes: The parallel prediction network module is used to predict the properties of each solute component at each stage of the Runge-Kutta time step. The parallel prediction network module has a shared backbone and multiple output branches structure. The backbone extracts general features, and the branches output the intermediate values ​​of each component in q stages. It is suitable for two-component or multi-component cases.

[0014] The implicit Runge-Kutta module couples the output data of the prediction network module with the residuals of each stage at the current time step through the Butcher table coefficients to obtain the concentration at the next time step.

[0015] The adaptive step size control module uses dynamic indicators. The next time step is dynamically adjusted.

[0016] The adaptive step-size Runge-Kutta physical information neural network model is trained with the loss function as the optimization objective. The model parameters are updated through a phased optimization strategy. When the network training converges at the nth time step, the output concentration field is... The training is performed using the initial conditions for the (n+1)th time step, and this process is repeated until a set termination time or error threshold is reached, at which point the spatiotemporal concentration distribution of the multi-components within the study domain is output. The training point set is fixed or updated according to rules at each time step, including: points inside the equation, initial condition points, and boundary condition points; the boundary conditions are Dirichlet boundaries and / or Neumann boundaries, constrained by corresponding loss terms. Simultaneously, an anomaly detection and training restart mechanism is implemented during the recursion process. When an abnormal increase in loss or a non-zero gradient (NaN) occurs, the process reverts to the most recent stable checkpoint and reduces the time step size to retrain the current time step.

[0017] Eliminating the differences in physical dimensions can improve the numerical stability of neural network training. After dimensionless processing, each physical quantity in the model is expressed based on the feature scale, thereby eliminating the differences in dimensions between different physical quantities and enabling the network to maintain good stability and comparability under different spatial scales and flow velocities.

[0018] According to the above technical solution, the convection-dispersion-reaction mathematical model uses a two-dimensional or three-dimensional aquifer as the research region to construct a partial differential equation system describing the migration and reaction of pollutants: for Each solute component, research domain ( or ), time interval ,Establish ; in, express A set of concentrations of each solute component. This indicates the concentration of the first solute component. This represents the concentration of the Nth solute component. This represents the concentration of the i-th solute component. Given a velocity field, It is the diffusion coefficient tensor (which can be in anisotropic diagonal form). For the first The reaction terms of the components (supporting first-order, binary, or more general nonlinear reaction forms).

[0019] The initial condition is the spatial distribution of pollutants at the initial time. ; Boundary conditions include: The upstream boundary is a given (Dirichlet) concentration boundary: ;in Components In position and time concentration, The concentration function is given in advance at the upstream boundary. This indicates the upstream Dirichlet boundary.

[0020] The downstream and bottom boundaries are zero-flux (Neumann) boundaries: ;in , For the diffusion tensor, Represents the outer normal vector of the boundary. This is the Neumann boundary.

[0021] The upper boundary can be set as the Neumann boundary or the constant flux input boundary.

[0022] According to the above technical solution, the implicit Runge-Kutta module couples the residuals of each stage to obtain the concentration at the next time step. : ; In the formula, Indicates solute components At time step n, the concentration Indicates the time step. Represented as the Butcher coefficient, For the Runge–Kutta order, For the residual operator of the ADR equation, Indicates the current time step Runge–Kutta The intermediate concentration of the stage. Among them, the order q of Runge–Kutta can be adaptively adjusted with the time step or switched according to preset rules to balance computational cost and numerical accuracy.

[0023] By embedding this discrete structure into the network loss function, physical constraints at multiple time stages can be explicitly coupled, thereby improving the system's rigidity and stability.

[0024] According to the above technical solution, the adaptive step size control module adjusts the time step size by individually or jointly controlling the gradient change rate controller and the reaction intensity controller.

[0025] The gradient rate of change controller has the following control index: ; when If the gradient increase is significant (potentially indicating frontal / rigid enhancement), then reduce the step size; when If the solution is stable, then increase the step size.

[0026] in, This represents the rate of change of the gradient at the nth iteration. This represents the gradient vector of the i-th component in the n-th iteration. This represents the gradient vector of the i-th component in the (n-1)-th iteration. Small positive numbers (used to avoid the denominator being 0, ensuring the numerical stability of the formula). Represents norm operations.

[0027] The reaction intensity controller has the following control parameters: ; When the relative increase in reaction intensity is significant ( If the step size is less than 0, then decrease the step size; otherwise, increase the step size.

[0028] In the formula, This represents the rate of change of reaction intensity at the nth iteration. This represents the i-th reaction intensity component in the n-th iteration. It represents the i-th reaction intensity component in the (n-1)-th iteration.

[0029] According to the above technical solution, a multiplicative update is used, and upper and lower bounds and a smoothing coefficient are set. The time step update formula is as follows: ; in, Indicates the first The "adaptive index" obtained from the step calculation can be taken as follows: or ; This represents the next time step size obtained after adaptive adjustment; These represent the lower and upper bounds of the step size hard constraint, respectively, used to prevent the time step from diverging or growing indefinitely during training; lip( () indicates a range operation, which limits the updated step size to within a certain range. Within the range; This indicates that no adaptive adjustment is performed at the first time step to ensure the stability of the training start.

[0030] , and Preferably, the first time step is taken as... and to Use exponential / moving average smoothing to avoid drastic step size fluctuations.

[0031] In the formula, Indicates the updated time step; Indicates the current time step; This represents a step-size monotonic adjustment function, and when Decrease step size when Increase the step size as needed.

[0032] Based on the above technical solution, the step size monotonic adjustment function Given a piecewise monotonic function, with upper and lower limits set: , in, Indicates the lower limit of duration. Indicates the maximum duration.

[0033] According to the above technical solution, the phased optimization strategy first uses Adam for pre-training to obtain stable initial values, and then uses L-BFGS for fine-tuning to accelerate convergence and improve in-step training stability.

[0034] According to the above technical solution, the loss function adopts a weighted sum form, and the weights can be normalized or adaptively updated based on the dimensions or relative amplitudes of each item to alleviate the imbalance problem of the loss terms. The loss function... : ; In the formula, Indicates the IRK residual loss. Indicates boundary condition loss. Indicates the weight of the IRK residual loss. This represents the boundary condition loss weight.

[0035] Among them, the IRK residual loss is: ; In the formula, Indicates components In the The time step, the first Each IRK stage, position The discrete residuals indicate whether the network output satisfies the IRK time discrete formula and PDE. This represents the total number of time steps, i.e., the total number of iterations in time-marching. This indicates the order of the IRK (Gauss-Legendre) method, corresponding to the dimension of the network output layer. Indicates the first The number of training points used in each IRK stage. L2 represents the squared error, used to measure the magnitude of the residual.

[0036] Boundary condition loss (Dirichlet / Neumann / Robin errors are constructed according to boundary type).

[0037] According to another aspect of the present invention, the present invention further provides an electronic device, comprising: one or more processors; and a storage device for storing one or more programs, wherein, when the one or more programs are executed by the one or more processors, the one or more processors cause the one or more processors to perform the multi-component reaction solute transport simulation method based on physical information neural networks described in any of the above technical solutions.

[0038] According to another aspect of the present invention, the present invention further provides a storage medium storing at least one instruction, which is loaded and executed by a processor to implement the multi-component reaction solute transport simulation method based on physical information neural network as described in any of the above technical solutions.

[0039] Compared with the prior art, the beneficial effects achieved by the present invention are: 1. This invention designs two types of adaptive step size controllers (based on gradient rate of change and based on nonlinear reaction intensity) to automatically adjust the time step size at different stages. When the system state changes drastically, the step size is automatically reduced to ensure stability; when the solute concentration field changes gently, the step size is automatically increased to improve efficiency, thereby achieving a dynamic balance between time resolution and computational cost.

[0040] 2. This invention employs an implicit Runge-Kutta structure and a recursive prediction mechanism, which allows the prediction results of each time step to be directly used as the input for the next time step. By suppressing error propagation through stage coupling, it significantly improves the prediction stability and accuracy of long-term reaction-convection-diffusion processes.

[0041] 3. The present invention employs parallel multi-output prediction branches in the network structure, which can simultaneously learn multiple solute components and their mutual reaction terms, thereby achieving a unified solution for multi-component nonlinear coupled systems and avoiding the physical inconsistency problem caused by independent solution of each component in traditional methods.

[0042] 4. This invention employs a two-stage optimization strategy: first, the Adam optimizer is pre-trained to obtain stable initial values, and then the L-BFGS optimizer is used for fine-tuning iterations. This strategy can significantly accelerate the convergence speed of the loss function within the implicit Runge-Kutta framework.

[0043] 5. The network structure and control mechanism of this invention have good scalability and can be applied to two-dimensional or three-dimensional aquifers, multi-reactant systems, and different types of boundary conditions. Through dimensionless processing, the model is also applicable to other types of convection-diffusion-reaction processes (such as surface water pollution migration, chemical reaction diffusion systems, etc.), improving network training stability and model accuracy.

[0044] 6. This invention avoids fine mesh generation and complex iterative solution processes. The adaptive time step significantly reduces invalid training rounds and shortens the overall training time. By embedding an adaptive time step mechanism into the Runge-Kutta physical information neural network, this invention achieves high-precision, stable, and efficient simulation of complex multi-component reactive solute transport processes in groundwater. Attached Figure Description

[0045] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used in conjunction with embodiments of the invention to explain the invention and do not constitute a limitation thereof. In the drawings: Figure 1 This is a schematic diagram of the principle of the invention based on the Adaptive Step-Size Runge-Kutta Physical Information Neural Network Model (ASR-PINN); Figure 2 This is a schematic diagram of the study area where the simulation examples are implemented; Figure 3 The L2 error box plots of ASR-PINN-I / II and the benchmark algorithm (TM-PINN) at three training resolutions are shown. Figure 4 These are representative seeds of ASR-PINN-I / II and TM-PINN at various time steps regarding pollutants. / Comparison of prediction results; Figure 5 ASR-PINN-I / II and TM-PINN are representative of pollutant activity under representative seed conditions. Comparison of prediction results; Figure 6 This is a comparison of step size changes between ASR-PINN-I / II and TM-PINN under representative seeds. Detailed Implementation

[0046] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0047] Example 1: A multi-component reaction solute transport simulation method based on a physical information neural network, comprising the following steps: S1. Establish a convection-diffusion-reaction mathematical model and determine the initial and boundary conditions, specifically: In two-dimensional or three-dimensional aquifer research domains, establish a set of convection-dispersion-reaction equations for the transport of multi-component reactive solutes: , in, Let i be the concentration of the i-th solute component. For Darcy's velocity field, For the diffusion tensor, This is a nonlinear reaction term.

[0048] The initial condition is the spatial distribution of pollutants at the initial time. Boundary conditions include: (1) The upstream boundary is a constant (Dirichlet) concentration boundary; (2) The downstream and bottom boundaries are zero flux (Neumann) boundaries; (3) The upper boundary can be set as the Neumann boundary or the constant flux input boundary.

[0049] This equation describes the spatiotemporal evolution of multi-component pollutants in an aquifer under the combined influence of convection, dispersion, and reaction processes.

[0050] S2, Dimensionless convection-dispersion-reaction mathematical model, initial and boundary conditions, specifically: The convection-diffusion-reaction mathematical model is dimensionless, and the characteristic scale is defined as follows: , in, The horizontal feature length, For characteristic time, For concentration characteristic values, Characteristic flow velocity, For longitudinal dispersion scale, is the characteristic dispersion coefficient.

[0051] Dimensionless equations , in The initial and boundary conditions, after being dimensionless, are identical in form to the original expression.

[0052] S3. Construct a physical information neural network model based on adaptive step size Runge-Kutta; the adaptive step size Runge-Kutta physical information neural network model includes: Parallel prediction network module, for each solute component Construct an independent prediction subnetwork, with spatial coordinates as input. The output is the intermediate concentration of each stage of Runge–Kutta at this time step; the subnetwork adopts a multi-layer fully connected structure, and the activation function can be SIREN (sine activation) to enhance the ability to represent complex spatial patterns.

[0053] The implicit Runge-Kutta module couples the residuals of each stage at the current time step with the output data of the prediction network module through the Butcher table coefficients to obtain the concentration at the next time step; The adaptive step size control module uses dynamic indicators. To dynamically adjust the next time step, the gradient rate of change controller ASR-PINN-I and the reaction intensity controller ASR-PINN-II are designed for individual or combined control to adjust the time step. (1) ASR-PINN-I (based on gradient rate of change) ; in, This represents the rate of change of the gradient at the nth iteration. This represents the gradient vector of the i-th component in the n-th iteration. This represents the gradient vector of the i-th component in the (n-1)-th iteration. Small positive numbers (used to avoid the denominator being 0, ensuring the numerical stability of the formula). Represents norm operations.

[0054] (2) ASR-PINN-II (based on nonlinear reaction strength) ; In the formula, This represents the rate of change of reaction intensity at the nth iteration. This represents the i-th reaction intensity component in the n-th iteration. It represents the i-th reaction intensity component in the (n-1)-th iteration.

[0055] Use multiplicative updates, setting upper and lower bounds and a smoothing coefficient, to update the time step. The time step update formula is: , in This is the adjustment coefficient. This mechanism can automatically increase or decrease the time step according to system changes, thereby achieving a balance between computational accuracy and efficiency. Pick or , Indicates the lower limit of duration. Indicates the maximum duration.

[0056] S4. The adaptive step-size Runge-Kutta physical information neural network model is trained with the loss function as the optimization objective. A phased optimization strategy is used to update the model parameters (first, pre-training with the Adam optimizer to obtain stable initial parameters; then switching to the L-BFGS optimizer for fine-tuning, accelerating convergence and improving accuracy). When the network training converges at the nth time step, the output concentration field is... The training is performed using the initial conditions of the (n+1)th time step, and this process is repeated until the set termination time or error threshold is reached, at which point the spatiotemporal concentration distribution of the multi-components in the study domain is output.

[0057] Among them, the loss function : ; In the formula, This represents the IRK residual loss. Indicates boundary condition loss. Indicates the weight of the IRK residual loss. This represents the boundary condition loss weight.

[0058] Through the above process, the model can adaptively adjust the time step while ensuring accuracy, significantly reducing the number of training rounds and overall computational cost.

[0059] Example 2 uses a two-dimensional reactive solute transport problem as an example, and TM-PINN as the benchmark algorithm to verify the effectiveness and superiority of the multi-component reactive solute transport simulation method based on a physical information neural network described in this invention. The study area setup is shown in [link to study area setup]. Figure 2 As shown, the governing equations for the two components are: , , in, and These represent the concentrations of two different solute components, and their units are... ; For hydrodynamic dispersion tensor; The groundwater velocity field is spatially variable; Retention factor; It is the first-order attenuation coefficient; This represents the nonlinear reaction rate constant. Coupling term. , The dimensions are This represents a class of higher-order transformation channels whose interaction rates affect... It exhibits secondary dependency. See Table 1 for specific parameter settings.

[0060] Table 1: Model parameters used in the implementation examples

[0061] This implementation example uses the dimensionless method described in the above technical solution, and the equations after characteristic transformation are as follows: , , Initial conditions: , Boundary conditions: Upstream : , Other boundaries: , , In this implementation example, under the aforementioned heterogeneous porous media background, based on the adaptive step-size Runge-Kutta physical information neural network model (ASR-PINN) framework proposed in the above technical solution, a systematic numerical experiment was conducted on two sets of convection-dispersion-reaction equation systems (ADR problems). The results were compared and evaluated with the commonly used TM-PINN model to examine accuracy, stability, and computational efficiency. To reduce the randomness introduced by weight initialization, sampling, and randomized trajectory optimization, each method was evaluated under six independent random seeds. The training hyperparameters are shown in Table 2.

[0062] Table 2. Training hyperparameters used by ASR-PINN-I / II and TM-PINN at three training resolutions in the implementation examples.

[0063] After training, the ASR-PINN model was used for predictions at different resolutions and with random seeds, and compared with the benchmark method TM-PINN. The results show ( Figure 3 The L2 relative errors of ASR-PINN-I and ASR-PINN-II were significantly lower than those of TM-PINN across all configurations. In the 300 / 100 configuration, the average error of TM-PINN was approximately... The error can be reduced by orders of magnitude, while ASR-PINN's error can be reduced by 1–2 orders of magnitude. – Furthermore, it exhibits low cross-seed dispersion. As the number of training points increases, ASR-PINN maintains high accuracy, while TM-PINN shows limited error reduction. Overall, the ASR-PINN method achieves an average error reduction of at least 65% compared to TM-PINN, exceeding 95% at coarse resolution.

[0064] In terms of training efficiency, ASR-PINN, due to its adaptive step size control, significantly reduces the training iterations per step and the total training time while maintaining prediction accuracy. As shown in Table 3, ASR-PINN improves the training efficiency of TM-PINN by approximately 2–5 times, and up to 8 times under some seeds. Figure 3 The L2 error box plots of ASR-PINN-I / II and the benchmark algorithm (TM-PINN) at three training resolutions are shown. Figure 4 These are representative seeds of ASR-PINN-I / II and TM-PINN at various time steps regarding pollutants. / Comparison of prediction results; Figure 5 ASR-PINN-I / II and TM-PINN are representative of pollutant activity under representative seed conditions. Comparison of prediction results; Figure 6 This is a comparison of step size changes between ASR-PINN-I / II and TM-PINN under representative seeds.

[0065] Table 3. Training time and speedup ratio of ASR-PINN-I / II and TM-PINN at three training resolutions in the implementation examples.

[0066] As can be seen from the above implementation examples, the ASR-PINN method proposed in this invention can achieve high-precision, long-scale solute transport simulation in complex heterogeneous aquifers and multi-component nonlinear reaction systems. Compared with the benchmark algorithm, it significantly improves numerical stability and computational efficiency, and has good prospects for engineering applications.

[0067] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0068] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for simulating solute transport in multi-component reactions based on physical information neural networks, characterized in that, The steps include: Establish a convection-diffusion-reaction mathematical model and determine the initial and boundary conditions; Dimensionless convection-diffusion-reaction mathematical model, initial conditions and boundary conditions; Constructing an adaptive step-size Runge-Kutta physical information neural network model; the adaptive step-size Runge-Kutta physical information neural network model includes: The parallel prediction network module is used to predict the properties of each solute component at each stage of the Runge-Kutta time step. The amount at each stage or the intermediate concentration at each stage; The implicit Runge-Kutta module couples the residuals of each stage at the current time step with the output data of the prediction network module through Butcher table coefficients to obtain the concentration at the next time step; The adaptive step size control module uses dynamic indicators. Dynamically adjust the next time step; The adaptive step-size Runge-Kutta physical information neural network model is trained with the loss function as the optimization objective. The model parameters are updated through a phased optimization strategy. When the network training converges at the nth time step, the output concentration field is... The training is performed using the initial conditions of the (n+1)th time step, and this process is repeated until the set termination time or error threshold is reached, at which point the spatiotemporal concentration distribution of the multi-components in the study domain is output.

2. The method for simulating solute transport in multi-component reactions based on a physical information neural network according to claim 1, characterized in that, The convection-dispersion-reaction mathematical model uses two-dimensional or three-dimensional aquifers as the research area to construct a partial differential equation system describing the migration and reaction of pollutants. The initial condition is the spatial distribution of pollutants at the initial time; The boundary conditions include: upstream boundary, downstream and bottom boundary, and upper boundary; wherein, the upstream boundary is a constant concentration boundary, the downstream and bottom boundary is a zero flux boundary, and the upper boundary is a Neumann boundary or a constant flux input boundary.

3. The method for simulating solute transport in multi-component reactions based on a physical information neural network according to claim 1, characterized in that, The implicit Runge-Kutta module couples the residuals of each stage to obtain the concentration at the next time step. : ； In the formula, Indicates solute components At time step n, the concentration Indicates the time step. Represented as the Butcher coefficient, For the Runge–Kutta order, For the residual operator of the ADR equation, Indicates the current time step Runge–Kutta Intermediate concentration at the stage.

4. The method for simulating solute transport in multi-component reactions based on a physical information neural network according to claim 1, characterized in that, The adaptive step size control module adjusts the time step size by individually or jointly controlling the gradient rate of change controller and the reaction intensity controller.

5. The method for simulating solute transport in multi-component reactions based on a physical information neural network according to claim 4, characterized in that, The time step update formula is as follows: ， In the formula, Indicates the updated time step; Indicates the current time step; This represents a step-size monotonic adjustment function, and when Decrease step size when Increase the step size as needed.

6. The method for simulating solute transport in multi-component reactions based on a physical information neural network according to claim 5, characterized in that, The step size monotonic adjustment function Given a piecewise monotonic function, with upper and lower limits set: ， in, Indicates the lower limit of duration. Indicates the maximum duration.

7. The method for simulating solute transport in multi-component reactions based on a physical information neural network according to claim 1, characterized in that, The phased optimization strategy first uses Adam for pre-training to obtain stable initial values, and then uses L-BFGS for fine-tuning to accelerate convergence and improve in-step training stability.

8. The method for simulating solute transport in multi-component reactions based on a physical information neural network according to claim 1, characterized in that, The loss function : ； In the formula, Indicates the IRK residual loss. Indicates boundary condition loss. Indicates the weight of the IRK residual loss. This represents the boundary condition loss weight.

9. An electronic device, characterized in that, include: One or more processors; A storage device for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors cause the one or more processors to perform the method according to any one of claims 1-8.

10. A storage medium, characterized in that, The storage medium stores at least one instruction, which is loaded and executed by a processor to implement the method as described in any one of claims 1-8.

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