A free surface wave field reconstruction method based on dynamic sub-domain conservative integration

Abstract

This invention discloses a free surface wave field reconstruction method based on dynamic subdomain conservation integrals, involving the interdisciplinary fields of fluid mechanics and artificial intelligence. The method includes the following steps: S1, dataset construction and coordinate normalization; S2, constructing an FS-S-PINN dual-network coupled architecture to approximate the free surface elevation, velocity field, and pressure field within the fluid domain using functions; S3, selecting effective subdomains, constructing irregularly shaped subdomains, and setting one-dimensional boundary integral subdomains at the free surface boundary and bottom boundary respectively; S4, constructing a total loss function, and performing global inference reconstruction after training. This invention employs the above-mentioned free surface wave field reconstruction method based on dynamic subdomain conservation integrals to achieve high-precision and stable free surface wave flow field reconstruction, while enhancing training robustness and noise resistance. It can accurately predict the spatiotemporal evolution characteristics of the wave free surface and flow field velocity and pressure under sparse observation conditions.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of fluid mechanics and artificial intelligence, and in particular to a method for reconstructing free surface wave fields based on dynamic subdomain conservation integrals. Background Technology

[0002] Wave dynamics research is of great significance to engineering fields such as coastal protection, port safety assessment, and wave energy development and utilization. Currently, the mainstream methods for obtaining free surface wave field data are mainly divided into three categories: field observation and physical model experiments rely on equipment such as buoys, sensors, PIVs, and LDVs to collect data. These methods generally suffer from sparse distribution of measurement points and insufficient spatial resolution, making it difficult to achieve high-precision reconstruction of the entire flow field. Traditional CFD numerical simulation relies on methods such as VOF, boundary element method, and smoothed particle fluid dynamics to simulate liquid surface evolution. However, this method depends on high-density grids and fine spatiotemporal discretization, resulting in large computational costs and complex modeling processes, which cannot meet the needs of rapid wave field inversion.

[0003] Physical Information Neural Networks (PINNs), combining data-driven approaches with physical constraints, offer significant advantages in sparse observation scenarios and are increasingly being applied to wave problem solving. S-PINN, in particular, improves training stability of the flow field in fixed regions through local subdomain integral constraints, while the WaveNets dual-network architecture is specifically adapted to free-floating wave scenarios. However, existing technologies still have significant shortcomings. Traditional PINNs employ point-by-point residual constraints, which are prone to collocation conflicts and gradient oscillations in highly nonlinear wave regions, resulting in slow training convergence and insufficient robustness to noisy experimental data. S-PINNs, on the other hand, only support fixed computational domains and use pre-divided static subdomain structures, making them unable to dynamically update with the free surface and prone to problems such as subdomain out-of-bounds errors and missed calculations in fluid regions. Summary of the Invention

[0004] The purpose of this invention is to provide a free surface wave field reconstruction method based on dynamic subdomain conservation integrals, thereby solving the problems mentioned in the background art.

[0005] To achieve the above objectives, this invention provides a method for reconstructing free surface wave fields based on dynamic subdomain conservation integrals, comprising the following steps: S1. Dataset Construction and Coordinate Normalization: Construct a wave field observation dataset and normalize the spatial and temporal coordinates to a preset standard range; S2. Construction of Dual-Network Coupling Architecture: Construct the Free Liquid Surface Adaptive Stable Physical Information Neural Network FS-S-PINN. FS-S-PINN adopts a dual-network coupling architecture to approximate the free liquid surface elevation, velocity field and pressure field in the fluid domain using functions. S3. Dynamic Subdomain Construction and Update: Construct a regular subdomain grid and filter effective subdomains by predicting the instantaneous free liquid surface position; construct irregular subdomains and set one-dimensional boundary integral subdomains at the free surface boundary and bottom boundary respectively; S4. Local conservation integral constraint training: Construct the total loss function and iteratively update the network parameters through backpropagation to optimize the dual-network coupling architecture. After training, perform global inference to realize the reconstruction of the free surface wave field.

[0006] Preferably, the specific formula for FS-S-PINN in S2 is as follows: ; ; in, , Representing neural networks and Trainable parameters, ( x , t () represents the normalized two-dimensional spatiotemporal coordinates; x , y , t () represents the normalized three-dimensional spatiotemporal coordinates; or Indicates the predicted free surface elevation, ( û , v̂ , p̂ ) represents the predicted velocity and pressure fields.

[0007] Preferably, the dual-network coupling architecture in S2 includes a first neural network and a second neural network; The first neural network takes two-dimensional spatiotemporal coordinates as input, performs a function approximation of the free liquid surface elevation, and outputs the free liquid surface elevation. The second neural network takes three-dimensional spatiotemporal coordinates as input, performs function approximation on the velocity and pressure fields in the fluid domain, and outputs the velocity and pressure fields in the fluid domain.

[0008] Preferably, the specific steps of S3 are as follows: S31. Discretize the time axis into several time slices, and construct a regular sub-domain grid in the spatial domain of each time slice. The regular sub-domain grid covers the fluid region and the external gas region. S32. Predict the instantaneous free liquid surface position through the first neural network, and filter the effective rule subdomains below the liquid surface accordingly, and eliminate the gas region subdomains. S33. Based on the local first-order geometric information of the free liquid surface, a heterogeneous subdomain with boundary matching the local slope of the free liquid surface is constructed, and Gaussian integral calculation is performed using four-node isoparametric element coordinate transformation. S34. One-dimensional boundary integral subdomains are set up at the free surface boundary and the fixed bottom boundary respectively. The free surface boundary subdomain is updated in real time with the wave evolution, while the bottom boundary subdomain remains unchanged.

[0009] Preferably, the specific steps of S4 are as follows: S41. The point-by-point residuals of the continuity equation, momentum equation, bottom impermeable boundary, free surface kinematics and dynamic boundary conditions of the incompressible Navier-Stokes equations are converted into the local integral residuals corresponding to each subdomain. S42. Using the Gauss-Legendre quadrature formula, calculate the integral residual through Gaussian integration points; S43, combined with data loss, control equation loss and various boundary losses, constructs the total loss function of the dual-network coupled architecture; S44. Use backpropagation to iteratively update network parameters and train the dual-network coupled architecture; S45. Using the trained neural network, reasoning is performed on the complete spatiotemporal domain to obtain the velocity field, pressure field, and free liquid surface elevation distribution of the free surface wave, thereby realizing the full-field reconstruction of the free surface wave field.

[0010] Preferably, S41 uses a local conserved integral form to physically constrain each control equation and boundary condition, transforming the point-by-point residual into an integral residual within the subdomain. By constructing a two-dimensional local integral subdomain that strictly fits the instantaneous flow domain, it ensures that all subdomains fall within the fluid region.

[0011] Preferably, the specific formula for the total loss function in S43 is as follows: in, For data loss function, For physical loss function, The bottom boundary loss function is... Let the free surface kinematic boundary loss function be... Let be the loss function of the free surface dynamic boundary, and let their corresponding weights be denoted as . , , , .

[0012] Preferably, the specific steps of S44 are as follows: S441. Train using the Adam optimizer for 50,000 steps, setting the initial learning rate to 1×10⁻⁶. -3 The learning rate is reduced by a decay factor of 0.9 every 2000 epochs; the subdomain is updated every 2000 iterations based on the real-time predicted free surface, and the training weights are updated in combination with the gradients of each loss term. S442. The L-BFGS optimization algorithm is used for refined solution, and the iteration termination is automatically controlled according to the built-in incremental tolerance criterion of the optimizer.

[0013] Therefore, the present invention employs the above-mentioned free surface wave field reconstruction method based on dynamic subdomain conservation integral, which has the following beneficial effects: (1) Adapting to dynamic free surface evolution: This invention constructs a dynamic subdomain update mechanism based on predicted free surface, which can adaptively identify instantaneous fluid regions and avoid subdomain boundary crossing or fluid region omission problems that occur in the fixed subdomain method, thus realizing continuous and effective constraints on dynamic free surface problems.

[0014] (2) Improved training stability and convergence speed: This invention transforms the pointwise residual constraint of traditional PINN into a local integral residual constraint, which effectively alleviates the training instability problem caused by the lack of spatial correlation between collocation points in the strongly nonlinear region. During training, FS-S-PINN exhibits excellent convergence speed and training stability.

[0015] (3) Improve the accuracy of flow field reconstruction: Compared with the traditional PINN and S-PINN (stabilized physical information neural network) methods, the FS-S-PINN of this invention shows more uniform and lower error in the prediction of velocity field, pressure field and free liquid surface, avoiding the local error concentration phenomenon that occurs near the boundary in traditional methods.

[0016] (4) Excellent noise resistance and robustness: The local integral form of the present invention is essentially equivalent to a spatial filtering process, which can effectively smooth high-frequency noise components and reduce gradient oscillations caused by local anomalies or noise disturbances.

[0017] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0018] Figure 1 This is a schematic flowchart of an embodiment of a free surface wave field reconstruction method based on dynamic subdomain conservation integrals according to the present invention. Figure 2 This is a schematic diagram of the neural network structure of FS-S-PINN, an embodiment of the free surface wave field reconstruction method based on dynamic subdomain conservation integral of the present invention. Figure 3 This is a schematic diagram illustrating the process of constructing a subdomain of a fluid region according to an embodiment of a free surface wave field reconstruction method based on dynamic subdomain conservation integrals of the present invention. Figure 4 This is a schematic diagram illustrating the calculation of Gaussian integrals in a non-circular subdomain using isoparametric mapping, according to an embodiment of the free surface wave field reconstruction method based on dynamic subdomain conservation integrals of the present invention. Figure 5 This is a comparative schematic diagram of the construction residual points of three methods in an embodiment of the free surface wave field reconstruction method based on dynamic subdomain conservation integral of the present invention. (a) is PINN, (b) is S-PINN, and (c) is FS-S-PINN. Figure 6 This invention presents three methods for reconstructing free surface wave fields based on dynamic subdomain conservation integrals during the training process of a free surface wave field reconstruction method. L 2. The process of error change is shown in the graph, (a) is u relative L 2. Error, (b) is v relative L 2. Error, (c) is p relative L 2. Error, (d) is or relative L 2. Error; Figure 7 The horizontal velocity is an embodiment of a free surface wave field reconstruction method based on dynamic subdomain conservation integrals according to the present invention. u The reconstruction results and their error distribution diagram are shown in (a1), which is the speed based on PINN. u The reconstruction result, (a2) is the speed based on S-PINN. u The reconstruction result, (a3) ​​is the speed based on FS-S-PINN. u Reconstruction result; (b1) is the speed corresponding to PINN. u Error distribution, (b2) is the velocity corresponding to S-PINN u Error distribution, (b3) is the velocity corresponding to FS-S-PINN u Error distribution; Figure 8 The vertical velocity is an embodiment of the free surface wave field reconstruction method based on dynamic subdomain conservation integral of the present invention. v The reconstruction results and their error distribution diagram are shown in (a1), which is the speed based on PINN. v The reconstruction result, (a2) is the speed based on S-PINN. v The reconstruction result, (a3) ​​is the speed based on FS-S-PINN. v Reconstruction result; (b1) is the speed corresponding to PINN. v Error distribution, (b2) is the velocity corresponding to S-PINN v Error distribution, (b3) is the velocity corresponding to FS-S-PINN. v Error distribution; Figure 9 Pressure in the embodiments of the present invention pThe reconstruction results and their error distribution diagram, (a1) is the pressure based on PINN. p The reconstruction result, (a2) is the pressure based on S-PINN. p The reconstruction result, (a3) ​​is the pressure based on FS-S-PINN. p Reconstruction result; (b1) is the pressure corresponding to PINN. p Error distribution, (b2) is the pressure corresponding to S-PINN. p Error distribution, (b3) represents the pressure corresponding to FS-S-PINN. p Error distribution; Figure 10 The free liquid surface elevation in this embodiment of the invention or The reconstruction results and their error distribution diagram are shown in (a1), which is the free surface elevation based on PINN. or The reconstruction result, (a2) is the free surface elevation based on S-PINN. or The reconstruction result, (a3) ​​is the free surface elevation based on FS-S-PINN. or Reconstruction result; (b1) is the free surface elevation corresponding to PINN. or Error distribution, (b2) is the free surface elevation corresponding to S-PINN. or Error distribution, (b3) is the free surface elevation corresponding to FS-S-PINN. or Error distribution. Detailed Implementation

[0019] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0020] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Terms such as "comprising" or "including" mean that the element or object preceding the word encompasses the elements or objects listed following the word and their equivalents, without excluding other elements or objects. Terms such as "connected" or "linked" are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. Terms such as "upper," "lower," "left," and "right" are used only to indicate relative positional relationships; when the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0021] Example Please see Figure 1-Figure 10This invention provides a free surface wave field reconstruction method based on dynamic subdomain conservation integrals, used to achieve high-precision reconstruction of the velocity field, pressure field, and free surface elevation of free surface waves under sparse observation conditions. This method combines a dual-network coupling architecture with dynamic subdomain conservation integral constraints, embedding the Navier-Stokes equations and free surface boundary conditions into a physical information neural network framework. It uses a data-efficient and training-stable neural network method, overcoming the drawbacks of traditional computational fluid dynamics methods, such as reliance on high-density grids, high computational costs, and complex numerical implementation. It also overcomes the limitations of existing physical information neural networks, such as training instability and insufficient prediction robustness caused by the lack of spatial correlation in point-by-point residual calculations. Figure 1 As shown, the method includes the following steps: S1. Dataset Construction and Coordinate Normalization: Construct a dataset based on wave field observation data to provide sparse data of wave field velocity field, pressure field and free surface elevation for neural network training, and normalize spatial and temporal coordinates to a predefined standard range [0, 1].

[0022] In this example, a two-dimensional numerical wave flume was constructed using FLOW-3D computational fluid dynamics software to obtain Stokes wave field data. The wave conditions were as follows: water depth 4 m, wave height 0.04 m, and period 2 s. The flume boundary conditions were set as follows: a second-order Stokes wave generation condition was applied at the inlet boundary; the bottom was set as a no-slip solid wall boundary; and an outflow boundary was set at the outlet, with a sponge layer arranged to absorb the incident wave.

[0023] S2. Construction of a Dual-Network Coupling Architecture: FS-S-PINN, or Free-Surface Stabilized Physics-Informed Neural Network, approximates the free surface elevation and the internal velocity and pressure fields of the fluid using functions. FS-S-PINN employs a dual-network coupling architecture composed of a first neural network and a second neural network. The first neural network uses two-dimensional spatiotemporal coordinates (… x , t () is the input, and the output is the free surface elevation. or The second neural network uses three-dimensional spatiotemporal coordinates ( x , y , t The input is the velocity field within the fluid domain, and the output is the velocity field within the fluid domain. u , v and pressure field p This network structure aims to learn the spatiotemporal evolution of free surface wave fields by introducing dynamic subdomain conservation integral constraints.

[0024] The network structure example for this embodiment is as follows: First neural network ( NN 1): Two neurons in the input layer (corresponding spatial coordinates) x and time coordinates t The system has four hidden layers, each containing 64 neurons; the activation function is the hyperbolic tangent function tanh; and the output layer has one neuron, representing the predicted free surface elevation. or The value of .

[0025] Second neural network ( NN 2): 3 neurons in the input layer (corresponding spatial coordinates) x , y and time coordinates t The hidden layer has 6 hidden layers, each containing 64 neurons; the activation function is the hyperbolic tangent function tanh; the output layer has 3 neurons, representing the predicted horizontal velocity. u Vertical velocity v and pressure p The value of .

[0026] The FS-S-PINN form can be represented as: ; ; in, i 1 , i 2 Representing neural networks NN 1 and NN 2 Trainable parameters, ( x , t () represents the normalized two-dimensional spatiotemporal coordinates; x , y , t () represents the normalized three-dimensional spatiotemporal coordinates; or Indicates the predicted free surface elevation, ( û , v̂ , p̂ ) represents the predicted velocity and pressure fields.

[0027] S3. Dynamic Subdomain Construction and Update: The time axis is discretized into several time slices. A regular subdomain grid is constructed within the spatial domain of each slice. The instantaneous free surface position is predicted using a first neural network. Based on this, effective subdomains below the liquid surface are selected, and gas region subdomains are eliminated. Simultaneously, irregularly shaped subdomains with boundaries consistent with the local slope of the free surface are constructed near the free surface. One-dimensional boundary integral subdomains are generated at the free surface boundary and the bottom boundary, respectively. The free surface boundary subdomain is updated in real time with wave evolution, while the bottom boundary subdomain remains fixed. This includes the following steps: S31. Dynamic Subdomain Construction and Update: Discretize the time axis and divide it into regular subdomains within the spatial domain of each time slice, so that it covers the fluid region and the gas region outside the boundary.

[0028] S32. Predict the instantaneous free surface position using the first neural network. or ( x , t After obtaining the free surface prediction results, then based on... or ( x , t ) Filter out the regular subdomains below the liquid surface.

[0029] S33. To further preserve local information near the free surface and accurately apply free surface boundary conditions, this invention, based on the predicted free surface... or ( x , t Based on the local first-order geometric information of the free liquid surface, a non-uniform subdomain is constructed, ensuring that its boundary maintains the same local slope as the free liquid surface. Gaussian integrals are then calculated for this type of non-uniform subdomain using coordinate transformation techniques based on four-node isoparametric elements.

[0030] S34. Generate one-dimensional boundary integral subdomains on the trimmed free surface boundary and fixed bottom boundary, and configure corresponding Gaussian integration points. The free surface boundary subdomain is updated in real time with wave evolution, while the bottom boundary subdomain remains fixed. This ultimately forms a dynamic spatiotemporal integral structure composed of the internal integral subdomain, the free surface boundary subdomain, and the bottom boundary subdomain, thereby achieving a unified integral solution for the governing equations and boundary conditions. During training, the model periodically reconstructs the subdomain topology based on the updated free surface position.

[0031] In this embodiment, the Stokes wave propagation problem is used as an example, within a range of one wavelength ( x Flow field reconstruction is carried out within the range of [3.7 m, 7.4 m], reconstructing a complete wave cycle. t(∈ [3 s, 5 s]). In the time domain, 11 time slices are divided at 0.2 s intervals, and a 37 × 5 regular subdomain grid is constructed in the spatial domain of each slice, with a subdomain size of 0.1 m × 0.1 m. For the boundary subdomains, 37 boundary subdomains are divided along the x-direction at the predicted free surface and bottom boundary of each time slice. Local integration is performed in each internal subdomain using 4 Gaussian integration points (2 × 2), and boundary integrals are calculated for each boundary subdomain using 2 Gaussian integration points.

[0032] S4. Local Conservation Integral Constraint Training: The pointwise residuals of the continuity equation, momentum equation, bottom impermeable boundary conditions, and kinematic and dynamic boundary conditions of the incompressible Navier-Stokes equations are transformed into local integral residuals within the aforementioned subdomains. The Gauss-Legendre quadrature formula is used to calculate the local integral residuals through Gaussian integration points. A total loss function is constructed by combining data loss, equation loss, and boundary losses. The network parameters are iteratively updated through backpropagation to train the dual-network coupled architecture. After training, the trained neural network is used to infer the entire spatiotemporal domain to obtain the velocity field, pressure field, and free surface elevation of the free surface waves, achieving full-field reconstruction of the free surface wave field.

[0033] In this invention, the governing equations (Navier-Stokes equations) and free surface boundary conditions of the two-dimensional incompressible free surface wave problem are embedded into the loss function of FS-S-PINN, so that the network automatically satisfies the physical laws of fluid motion through local conservation integral constraints during the training process.

[0034] The governing equations for describing the wave problem on a free surface are given: the fluid motion is governed by the incompressible Navier-Stokes equations, and the continuity equation in two dimensions can be expressed as: ; x The equation for directional momentum can be expressed as: ; y The equation for directional momentum can be expressed as: ; in, u , v These are the horizontal velocity and the vertical velocity, respectively. p For pressure, r For fluid density, g This is the acceleration due to gravity.

[0035] At the free liquid surface y = or (x , t At point ), the fluid satisfies both kinematic and dynamic boundary conditions: ; ; The bottom of the computational domain is an impermeable boundary condition: ; To facilitate unified modeling and automatic differential calculation within the physical information neural network framework, this invention employs a local conserved integral form to physically constrain the aforementioned control equations and boundary conditions. Specifically, the pointwise residuals are transformed into integral residuals within subdomains. By constructing a two-dimensional local integral subdomain that strictly fits the instantaneous flow domain, it is ensured that all subdomains fall within the fluid region.

[0036] Define the integral residual of the continuity equation as: ; definition x The integral residual of the directional momentum equation is: ; definition y The integral residual of the directional momentum equation is: ; in, Oh I For the first i An internal integral subfield, û , v̂ For the velocity component predicted by the neural network, p̂ The pressure of neural network prediction, N pde This represents the total number of internal integral subdomains. v e Kinematic viscosity, x i Indicates the first i The center of each internal integral subdomain is located. The integral is calculated using the Gauss-Legendre quadrature formula, with 4 Gaussian quadrature points (2×2) for the internal subdomain and 2 Gaussian quadrature points for the boundary subdomain.

[0037] Define physical loss function L e This loss is constructed based on the integral residuals of the incompressible Navier-Stokes equations, and is expressed as: ; During neural network training, a boundary condition loss function is introduced to ensure that the model satisfies physical constraints at the boundaries. Specifically, an impermeable boundary condition is applied at the bottom boundary, and kinematic and dynamic boundary conditions are applied at the free surface.

[0038] Define the bottom boundary loss function as follows: ; in, N low This represents the total number of bottom boundary subdomains.

[0039] Define the free surface kinematic boundary loss function as: ; in, or The free surface elevation predicted by the neural network. N up This represents the total number of free surface boundary subdomains.

[0040] Define the free surface dynamic boundary loss function as: ; Define the data loss function as follows: ; Where, u = [ u , v , p , or This represents the actual values ​​of velocity, pressure, and surface profile obtained at each measurement point in the flow field. This represents the predicted value of the neural network at the corresponding sampling point.

[0041] The total loss function is obtained as follows: Among them, subscript i ∈ { o , p , b , k , d}, corresponding oh i These are data loss, equation loss, bottom boundary loss, and free surface kinematic and dynamic boundary loss, respectively.

[0042] After obtaining the total loss function, the network parameters are updated using the backpropagation algorithm. The model is trained using a two-stage optimization strategy: first, the Adam optimizer is used for 50,000 training steps, with an initial learning rate set to 1×10⁻⁶. -3The learning rate is decreased by a decay factor of 0.9 every 2000 epochs, and the subdomain is updated every 2000 steps based on the current predicted free surface position and the training weights are updated based on the gradient magnitude of each loss term. Then, the L-BFGS optimization algorithm is used for fine-tuning the solution, and the iteration termination is automatically controlled by the incremental tolerance criterion inside the optimizer.

[0043] In this embodiment, taking the Stokes wave propagation problem as an example, three methods were used for training, and the results were compared and analyzed: PINN, S-PINN, and FS-S-PINN. Figure 5 The diagram shows a comparison of the residual point construction methods for the three methods. FS-PINN uses a point-by-point residual distribution, SW-S-PINN only constructs a fixed subdomain below the lowest liquid level, while FS-S-PINN achieves complete coverage of the instantaneous fluid domain through dynamic subdomain updates and constructs irregularly shaped subdomains with fitted boundaries near the free liquid surface. Figure 6 As shown, the relative performance of the three methods during the training process is... L The error variation curves show that FS-S-PINN exhibits faster error reduction and smoother convergence across various physical quantities, ultimately achieving a significantly lower error level than FS-PINN and SW-S-PINN. This demonstrates that the conservation constraint in the form of local integrals effectively mitigates training oscillations caused by pointwise residuals. Figure 7-10 As shown, the horizontal velocity predicted by three methods at time t = 5 s is presented. u Vertical velocity v Pressure field p and free liquid surface elevation or The distribution of flow field details and error distribution shows that FS-S-PINN performs best in terms of flow field detail capture and overall reconstruction accuracy, especially in the region near the free surface, where it exhibits a more uniform error distribution. This further verifies the effectiveness of dynamic subdomain conservation integral constraints in improving the reconstruction accuracy of wave fields on free surfaces.

[0044] This invention fully considers the complexity of the continuous changes in the fluid domain caused by the dynamic evolution of the free surface. It combines a dual-network coupling architecture with dynamic subdomain conservation integral constraints and embeds it into a physical information neural network framework, thus enabling direct full-field reconstruction of the free surface wave field without the need for traditional numerical solvers. By incorporating local integral conservation constraints into the loss function of the neural network for training, and then using the trained neural network to predict the velocity field, pressure field, and free surface elevation, this invention replaces traditional numerical solution schemes such as the finite volume method. This significantly reduces computational cost, memory consumption, and numerical instability when dealing with dynamic free surface problems, overcoming the limitations of existing physical information neural network fixed subdomain methods that cannot adapt to free surface evolution, the instability of point-by-point residual constraints during training in strongly nonlinear regions, and insufficient robustness under noisy observation conditions.

[0045] Therefore, this invention adopts the above-mentioned free surface wave field reconstruction method based on dynamic subdomain conservation integral, which integrates the WaveNets dual network framework, local integral conservation constraints and dynamic subdomain update mechanism. By driving the dynamic subdomain partitioning and local integral conservation constraints through the prediction of the free liquid surface, it achieves high-precision and stable free surface wave flow field reconstruction, while enhancing training robustness and noise resistance. It can accurately predict the spatiotemporal evolution characteristics of wave free liquid surface and flow field velocity and pressure under sparse observation conditions.

[0046] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for reconstructing a free surface wave field based on dynamic subdomain conservation integrals, characterized in that, Includes the following steps: S1. Dataset Construction and Coordinate Normalization: Construct a wave field observation dataset and normalize the spatial and temporal coordinates to a preset standard range; S2. Construction of Dual-Network Coupling Architecture: Construct the Free Liquid Surface Adaptive Stable Physical Information Neural Network FS-S-PINN. FS-S-PINN adopts a dual-network coupling architecture to approximate the free liquid surface elevation, velocity field and pressure field in the fluid domain using functions. S3. Dynamic Subdomain Construction and Update: Construct a regular subdomain grid and filter effective subdomains by predicting the instantaneous free liquid surface position; construct irregular subdomains and set one-dimensional boundary integral subdomains at the free surface boundary and bottom boundary respectively; S4. Local conservation integral constraint training: Construct the total loss function and iteratively update the network parameters through backpropagation to optimize the dual-network coupling architecture. After training, perform global inference to realize the reconstruction of the free surface wave field.

2. The free surface wave field reconstruction method based on dynamic subdomain conservation integral as described in claim 1, characterized in that, The specific formula for FS-S-PINN in S2 is as follows: ； ； in, , Representing neural networks and Trainable parameters, ( x , t () represents the normalized two-dimensional spatiotemporal coordinates; x , y , t () represents the normalized three-dimensional spatiotemporal coordinates; η̂ Indicates the predicted free surface elevation, ( û , v̂ , p̂ ) represents the predicted velocity and pressure fields.

3. The free surface wave field reconstruction method based on dynamic subdomain conservation integral as described in claim 1, characterized in that: The dual-network coupling architecture in S2 includes a first neural network and a second neural network; The first neural network takes two-dimensional spatiotemporal coordinates as input, performs a function approximation of the free liquid surface elevation, and outputs the free liquid surface elevation. The second neural network takes three-dimensional spatiotemporal coordinates as input, performs function approximation on the velocity and pressure fields in the fluid domain, and outputs the velocity and pressure fields in the fluid domain.

4. The free surface wave field reconstruction method based on dynamic subdomain conservation integral as described in claim 1, characterized in that, The specific steps of S3 are as follows: S31. Discretize the time axis into several time slices, and construct a regular sub-domain grid in the spatial domain of each time slice. The regular sub-domain grid covers the fluid region and the external gas region. S32. Predict the instantaneous free liquid surface position through the first neural network, and filter the effective rule subdomains below the liquid surface accordingly, and eliminate the gas region subdomains. S33. Based on the local first-order geometric information of the free liquid surface, a heterogeneous subdomain with boundary matching the local slope of the free liquid surface is constructed, and Gaussian integral calculation is performed using four-node isoparametric element coordinate transformation. S34. One-dimensional boundary integral subdomains are set up at the free surface boundary and the fixed bottom boundary respectively. The free surface boundary subdomain is updated in real time with the wave evolution, while the bottom boundary subdomain remains unchanged.

5. The free surface wave field reconstruction method based on dynamic subdomain conservation integral as described in claim 1, characterized in that, The specific steps of S4 are as follows: S41. The point-by-point residuals of the continuity equation, momentum equation, bottom impermeable boundary, free surface kinematics and dynamic boundary conditions of the incompressible Navier-Stokes equations are converted into the local integral residuals corresponding to each subdomain. S42. Using the Gauss-Legendre quadrature formula, calculate the integral residual through Gaussian integration points; S43, combined with data loss, control equation loss and various boundary losses, constructs the total loss function of the dual-network coupled architecture; S44. Use backpropagation to iteratively update network parameters and train the dual-network coupled architecture; S45. Using the trained neural network, reasoning is performed on the complete spatiotemporal domain to obtain the velocity field, pressure field, and free liquid surface elevation distribution of the free surface wave, thereby realizing the full-field reconstruction of the free surface wave field.

6. The free surface wave field reconstruction method based on dynamic subdomain conservation integral as described in claim 5, characterized in that: The S41 uses a local conserved integral form to physically constrain each control equation and boundary condition, transforming the point-by-point residual into an integral residual within the subdomain. By constructing a two-dimensional local integral subdomain that strictly fits the instantaneous flow domain, it ensures that all subdomains fall within the fluid region.

7. The free surface wave field reconstruction method based on dynamic subdomain conservation integral as described in claim 5, characterized in that, The specific formula for the total loss function in S43 is as follows: in, For data loss function, For physical loss function, The bottom boundary loss function is... Let the free surface kinematic boundary loss function be... Let be the loss function of the free surface dynamic boundary, and let their corresponding weights be denoted as . , , , .

8. The free surface wave field reconstruction method based on dynamic subdomain conservation integral as described in claim 5, characterized in that, The specific steps of S44 are as follows: S441. Train using the Adam optimizer for 50,000 steps, setting the initial learning rate to 1×10⁻⁶. -3 The learning rate is reduced by a decay factor of 0.9 every 2000 epochs; the subdomain is updated every 2000 iterations based on the real-time predicted free surface, and the training weights are updated in combination with the gradients of each loss term. S442. The L-BFGS optimization algorithm is used for refined solution, and the iteration termination is automatically controlled according to the built-in incremental tolerance criterion of the optimizer.

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