Solving Euler equations with Multiple Discontinuities via Separation-Transfer Physics-Informed Neural Networks

TL;DR

This work tackles solving the Euler equations with multiple discontinuities using physics-informed neural networks (PINNs), where standard PINNs suffer from gradient pathology near shocks and interfaces. The authors introduce Separation-Transfer PINNs (ST-PINNs), which identify the strongest discontinuity via a discontinuity intensity metric, solve for it with a PINN, partition the domain at that discontinuity, and then apply transfer learning to the subdomains to capture remaining discontinuities, iterating until all are resolved. The approach is validated on 1D shock-interface, quasi-1D planar shock-interface, and 2D unsteady planar shock refraction problems, showing sharper discontinuities and markedly lower relative errors than baseline PINN variants, all without relying on external data. ST-PINNs thus offer a scalable, data-free framework to extend PINN applicability to complex shock–interface interactions and potentially to three-dimensional and boundary-conditioned problems in fluid dynamics.

Abstract

Despite the remarkable progress of physics-informed neural networks (PINNs) in scientific computing, they continue to face challenges when solving hydrodynamic problems with multiple discontinuities. In this work, we propose Separation-Transfer Physics Informed Neural Networks (ST-PINNs) to address such problems. By sequentially resolving discontinuities from strong to weak and leveraging transfer learning during training, ST-PINNs significantly reduce the problem complexity and enhance solution accuracy. To the best of our knowledge, this is the first study to apply a PINNs-based approach to the two-dimensional unsteady planar shock refraction problem, offering new insights into the application of PINNs to complex shock-interface interactions. Numerical experiments demonstrate that ST-PINNs more accurately capture sharp discontinuities and substantially reduce solution errors in hydrodynamic problems involving multiple discontinuities.

Figures (7)

• Figure 1: Schematic of ST-PINNs. (a) Solve the entire solution domain by variants of PINNs, generating Model 1, with only the strongest discontinuity D1 well-captured. (b) Identify the location of discontinuity D1. The solution domain is divided into subdomains S1 and S2 along D1, with D2 becoming the primary discontinuity in S2. Transfer learning is performed based on Model 1 to generate Model 2, with which D2 is well-captured. (c) Identify the location of discontinuity D2. The subdomain S2 is further divided into S3 and S4 at D2, with D3 becoming the primary discontinuity in S4. Transfer learning is performed based on Model 2 to generate Model 3, with which D3 is well-captured. (d) Models 1, 2, and 3, which predicts different subdomains well, are combined to obtain the predictions for the entire solution domain.
• Figure 2: The position of the transmitted shock front: blue circles is the transmitted shock positions obtained according to Model 1.1, the blue solid line is the fitted positions, and the red solid line represents the theoretical solution.
• Figure 3: The comparison between the neural network predictions (blue circles) and the theoretical solution (black solid line) for density, velocity, and pressure at $t=0.1$ in one-dimensional shock-interface interaction problem. R, C, and T correspond to the reflected shock, contact discontinuity, and transmitted shock, respectively. (a) Predictions of Model 1.1 trained by PINNs-WE method across the entire solution domain, which accurately capture the transmitted shock front. (b) Transfer learning is applied in the subdomain post of the transmitted shock based on Model 1.1 with Model 1.2 generating and the reflected shock and contact discontinuity accurately captured. (c) Combine the predictions from Models 1.1 and 1.2 at the transmitted shock to obtain the prediction across the entire solution domain.
• Figure 4: Initial conditions of the mass density $\rho$ for the quasi one-dimensional planar shock-interface interaction problem. $\rm{Region \ I}$ is the light medium compressed by the incident shock, and $\rm{region \ II}$ is the heavy medium.
• Figure 5: Results of the quasi-one-dimensional planar shock-interface interaction problem at $t=0.1$ solved by ST-PINNs based on GA-PINNs. R, C, and T correspond to the reflected shock, contact discontinuity, and transmitted shock, respectively. (a) Results solved by the GA-PINNs method, generating Model 2.1, with the transmitted shock well-captured. (b) Transfer learning based on Model 2.1 is applied to the subdomain post the transmitted shock, generating Neural Network Model 2.2, with the reflected shock and contact discontinuity well captured. (c) Predictions at both sides of the transmitted shock front provided by Models 2.1 and 2.2 respectively are combined to obtain the results across the entire solution domain. (d) Theoretical solutions.