Boundary self-learning PINN solving method under high-speed flow
By introducing flow conservation and wall treatment functions to replace wall data in high-speed flow scenarios, the problem of insufficient shock wave capture ability of PINN in high-speed flows is solved, and efficient and accurate prediction is achieved under inaccurate boundary conditions or scarce data.
Patent Information
- Application Number
- CN202510780065.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-06-12
- Publication Date
- 2025-09-16
AI Technical Summary
In high-speed flow scenarios, physical information neural networks (PINNs) have difficulty accurately capturing strong nonlinear phenomena such as shock waves, and the accuracy and consistency of boundary conditions have a significant impact on model performance. Traditional methods rely on large amounts of observational data and gradient information, which is costly to obtain and difficult to implement universally.
By introducing the conservation of flow in multiple regions in the computational domain as a physical constraint, combining the wall treatment function to replace the wall data, adopting the boundary self-learning strategy, using hypercube sampling and smoothed particle hydrodynamics to determine the adaptive smoothing length, decomposing the computational domain and introducing conservation constraints, the boundary self-learning PINN model is trained.
In the case of scarce wall data, the applicability and accuracy of the PINN model in high-speed flow prediction are improved, the dependence on high-quality data is reduced, and the training efficiency and accuracy of the model are improved.
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Abstract
Description
Technical Field
[0001] The present invention belongs to the intersection of computational fluid dynamics and artificial intelligence, and specifically relates to a boundary self-learning PINN solution method in high-speed flow scenarios. Background Art
[0002] Deep learning, with its multi-layered network models and ability to learn from large amounts of data, has achieved remarkable results in various fields, particularly in solving partial differential equations (PDEs). Traditional PDE-solving methods suffer from high computational costs and technical difficulties, while deep learning solutions offer significant advantages in dealing with nonlinear and high-dimensional complex equations, and also provide new approaches for modeling physical processes. The physical information neural network (PINN), proposed by Raissi, Maziar, Paris Perdikaris, and George E. Karniadakis in the paper "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, has been widely used in fields such as fluid mechanics and material mechanics. Compared to traditional numerical solutions, the meshless nature of PINN gives it a unique advantage in addressing the curse of dimensionality and solving inverse problems, significantly improving the flexibility of problem solving.
[0003] However, PINN suffers from slow or non-convergence when handling high-frequency and multi-scale phenomena. Currently, researchers are improving training efficiency and accuracy through methods such as dynamic adjustment of loss function weights and adaptive point resampling. They are also continuously exploring adaptive activation functions to enhance model performance. As an emerging PDE solving method, PINN holds great potential and is expected to revolutionize traditional numerical solution techniques.
[0004] Although physical information neural networks (PINNs) have demonstrated significant effectiveness in many fields, especially in dealing with complex partial differential equations, they have made remarkable progress. However, in compressible and high-speed flow applications, PINNs are still insufficient in capturing shock waves. Specifically, when faced with strong nonlinear phenomena such as shock waves, PINNs often find it difficult to accurately approximate the solutions to partial differential equations, resulting in a decrease in solution accuracy. To overcome this limitation, researchers have proposed a variety of improvement strategies. First, some scholars enhance the capture of key flow features such as shock waves by adding a small number of high-quality data points. Reference Zhu, Yinhao, et al. "Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data." Journal of Computational Physics 394(2019):56-81. In their study, a small number of shock wave data points were introduced to help the network identify the shock wave position in the flow field, thereby improving the solution accuracy. Secondly, adding gradient information has also been widely used in the study of such problems. The literature Yu, Jeremy, et al. "Gradient-enhancedphysics-informed neural networks for forward and inverse PDE problems." Computer Methods in Applied Mechanics and Engineering 393(2022):114823. proposed to improve the network's solving ability by introducing additional gradient constraints, thereby improving the training effect of PINN. Through these methods, the researchers not only improved PINN's ability to capture flow characteristics such as shock waves, but also broadened its application prospects in high-speed flow problems, demonstrating the potential for further development in complex flow fields.
[0005] However, in practical engineering applications, relying on a small number of data points and introducing additional gradient information often presents numerous challenges. Especially under high-speed flow conditions, efficiently obtaining a small number of observation points or gradient information in the aircraft flow field is a pressing problem. Even if such data can be obtained, the cost is often enormous and difficult to implement universally in practical engineering environments. To address this issue, our team has initially explored a novel solution by introducing the conservation of fluid flow quantities in multiple regions of the computational domain as additional information to guide the training of a physical information neural network (PINN). This approach significantly reduces the reliance on high-quality training data by incorporating physical constraints into the training process, thereby avoiding the need for large amounts of observational data and gradient information in traditional methods. This approach not only significantly reduces data acquisition costs but also effectively improves the accuracy and training efficiency of the PINN model for complex flow problems. By leveraging the physical guidance of flow quantity conservation, the network can obtain relatively accurate flow field solutions even without the support of extensive observational data. This is particularly effective in complex phenomena such as high-speed flows and shock waves, effectively supplementing the data-scarce nature of accurate calculations. This physical constraint-based training strategy not only improves the network's adaptability to flow field solutions, but also provides more feasible and economical solutions for complex problems such as high-speed flows.
[0006] During the training process of physical information neural networks (PINNs), the accuracy and consistency of boundary condition data are crucial for model performance. This is particularly true in high-speed flow scenarios, where the difficulty of setting boundary conditions is even greater. High-speed flows are inherently complex, involving phenomena such as shock waves, flow separation, and turbulence. These factors place higher demands on the boundary condition handling. In this context, different boundary condition handling methods often differ significantly. This not only exacerbates the problem of insufficient training data but can also lead to inaccurate boundary condition settings, thus compromising the model's ability to learn the true flow characteristics. Furthermore, incomplete or inconsistent wall boundary data can directly interfere with the PINN model training process, resulting in a decrease in the accuracy of flow characteristics captured and, consequently, weakening the model's predictive power. For example, the thickness and flow behavior of the boundary layer play a crucial role in high-speed flows. If these characteristics are not effectively simulated through appropriate boundary conditions, the model's predictions in real-world flow situations may suffer significant errors. Furthermore, noisy or inconsistent boundary condition data can also cause model convergence issues, impairing the optimization process of the loss function, preventing the model from converging to a valid solution, affecting training efficiency, and even rendering the model ineffective in predicting flow characteristics in practical applications. Therefore, in order to improve the application efficiency of PINN in the field of high-speed flow, it is crucial to ensure the accuracy and consistency of boundary condition data, which may require the generation of high-quality boundary condition data through sophisticated experimental methods or numerical simulation techniques.
[0007] The core idea of the method provided by the present invention is to introduce regional conservation relations, which can compensate for the unsampled information in the flow area, obtain boundary conditions by replacing wall data with wall processing functions, and then combine boundary embedding training strategies to accurately capture the influence of walls on flow characteristics, thereby improving the applicability and accuracy of the PINN model in high-speed flow prediction. Summary of the Invention
[0008] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is as follows:
[0009] A boundary self-learning PINN solution method under high-speed flow includes the following steps:
[0010] The solution coordinate points in the computational domain are obtained by using the hypercube sampling method;
[0011] Adaptive smoothing length is determined using a probabilistic model in smoothed particle hydrodynamics;
[0012] The entire computational domain is divided into several subdomains, and additional conservation constraints are introduced for each subdomain;
[0013] The specific conditions of the wall boundary are determined by the corresponding wall processing function. These processed boundary conditions will replace the traditional wall data.
[0014] Combine the physical constraints in the control equations with the wall boundary conditions for training to obtain a boundary self-learning PINN model, and use the model to predict the data;
[0015] The present invention provides a boundary self-learning PINN solution method for high-speed flows. This method, when wall data is scarce, replaces wall data with a wall processing function to obtain boundary conditions. By introducing additional conservation constraints, it compensates for unsampled information in the flow region, improving the applicability and accuracy of the PINN model in high-speed flow prediction when wall data is sparse. BRIEF DESCRIPTION OF THE DRAWINGS
[0016] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings used in the embodiments.
[0017] Figure 1 1 is a flowchart of the steps of the boundary self-learning PINN solution method under high-speed flow in an embodiment;
[0018] Figure 2 3 is a comparison diagram between the predicted result and the actual value of the two-dimensional oblique shock wave in the embodiment;
[0019] Figure 33 is a comparison diagram of the predicted results and the actual values of the two-dimensional expansion wave in the embodiment. DETAILED DESCRIPTION
[0020] To make the technical solution of the present invention clearer, the present invention will be clearly and completely explained below with reference to the accompanying drawings of the embodiments of the invention. The embodiments described only use a part of the embodiments, not all of the embodiments. The present invention is specifically implemented according to the following steps:
[0021] (1) Obtain the solution coordinate point (X) in the computational domain by using the hypercube sampling method F , Y F ).
[0022] (2) The probabilistic model in smoothed particle hydrodynamics is used to determine the adaptive smoothing length L.
[0023] Adaptive smoothing length allows us to precisely determine the coordinates of the wall attachment's influence domain, which often significantly influences the flow characteristics at the wall. This allows us to determine the coordinates of the governing equations to be solved within the computational domain, as well as the sampling points that are closely related to the wall flow.
[0024] (3) In order to simulate the wall boundary conditions more accurately, we combined common wall types in CFD, such as adiabatic walls, isothermal walls, slip walls, and no-slip walls, and adopted corresponding wall processing functions. These wall conditions are based on multiple coordinate points in the wall influence domain obtained by the above-mentioned adaptive smoothing length, and the specific conditions of the wall boundary are determined by weighted averaging. These processed boundary conditions will replace the traditional wall data and be directly used to drive the training of the PINN model. In order to improve the training accuracy, when the model performs poorly, we can add additional sampling points by adaptively adjusting the smoothing length to optimize the model performance.
[0025] (4) Finally, the wall characteristics are introduced into the neural network and combined with the incoming flow conditions, the conservation constraints of the solution domain and the partial differential equations to train the loss function. The boundary self-learning PINN loss function MSE is mainly composed of the wall condition loss Inflow condition loss Conservation Constraint Loss Loss with partial differential equations Composition. It can be described as (1) to (5):
[0026]
[0027]
[0028] (5) Use the total loss MSE to train PINN, update the neural network weights through automatic differentiation, back propagation, and stochastic gradient descent, and finally train the PINN model.
[0029] The present invention provides two embodiments, namely a two-dimensional oblique shock wave experiment and a two-dimensional expansion wave experiment. The incoming flow data used in the experiments are all obtained from COMSOL simulation.
[0030] Example 1: A PINN model is constructed for a two-dimensional oblique shock wave to predict the velocity, density, and pressure behind the wave. The details are as follows:
[0031] This embodiment uses COMSOL simulation data, and the results of COMSOL software simulation calculation are used as the incoming flow data. This embodiment is in an inviscid flow with a Mach number of 2, and the angle relative to the horizontal wall is -10°. The sampling points X and Y are used as the input of the model, and the output is velocity, density, and pressure. In order to simulate the engineering use scenario, the wall data is not used, and only the incoming flow data is used to prove the effectiveness and accuracy of the model. The method of the present invention is used to predict the result graph of the two-dimensional oblique shock wave, in which the density prediction is as follows Figure 2 As shown in (a) and (b), the pressure prediction and the actual value are as follows Figure 2 As shown in (c) and (d).
[0032] Example 2: A PINN model is constructed for a two-dimensional expansion wave to predict the velocity, density, and pressure behind the wave. The details are as follows:
[0033] The experimental data of this embodiment are the same as those of embodiment 1. In this embodiment, the inviscid flow is at an upstream Mach number of 2 and a downstream Mach number of 3, and the angle relative to the horizontal wall is -10°. The input and output are the same as those of embodiment 1. The result graph of the method of the present invention for predicting two-dimensional expansion waves, where the density prediction and the actual value are as shown in FIG. Figure 3 As shown in (a), the pressure prediction and the actual value are as follows Figure 3 (b) shown.
[0034] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention may be modified and varied in various ways.
Claims
1. A boundary self-learning PINN solution method under high-speed flow, characterized by: The steps include: The solution coordinate points (X F , Y F ) and the analytical solution (U, V, ρ, P) to the problem; The entire computational domain is divided into several subdomains, and additional conservation constraints are introduced for each subdomain; The specific conditions of the wall boundary are determined by the corresponding wall processing function; The physical constraints in the control equations are combined with the wall boundary conditions for training to obtain a boundary self-learning PINN model, which is used to predict the flow problem variables.
2. The boundary self-learning PINN solution method under high-speed flow according to claim 1 is characterized in that: The analytical solutions (I, V, ρ, P) to the problem are obtained from the oblique shock wave example and the expansion wave example in the COMSOL learning case library.
3. The boundary self-learning PINN solution method under high-speed flow according to claim 1 is characterized in that: The entire computational domain is divided into several subdomains, and additional conservation constraints are introduced for each subdomain, including: The computational domain is [0,1] 2 Divide into 5 subdomains, and sample 80 coordinate points at the boundary of each subdomain, i.e. [0, 0.2, 0.4, 0.6, 0.8, 1] Calculate the mass flow rate through each edge.
4. The boundary self-learning PINN solution method under high-speed flow according to claim 1 is characterized in that: Use the wall treatment function to determine the specific conditions of the wall boundary, including: Velocity wall function: relates wall shear stress to near-wall node velocity through a logarithmic law; Temperature wall function: Similar to velocity, using the temperature logarithm: Where T w is the wall temperature, T τ is the temperature scale.
5. The boundary self-learning PINN solution method under high-speed flow according to claim 1 is characterized in that: Construct the partial differential equations and boundary conditions for the entire solution domain, including: The state equation is used to close the equation group and solve the flow field variables (U, V, ρ, P) and so on.
6. The boundary self-learning PINN solution method under high-speed flow according to claim 1 is characterized in that The physical constraints in the control equations and the wall boundary conditions are combined for training to build a boundary self-learning PINN model, which includes: The MSE loss function of the boundary self-learning PINN is mainly composed of the wall condition loss Inflow condition loss Conservation Constraint Loss Loss with partial differential equations constitute: Construct a boundary self-learning PINN model based on the loss function.