Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Study

TL;DR

The paper tackles the challenge of solving PDEs with discontinuities, such as shock fronts in multiphase flow through porous media, with Physics-Informed Neural Networks (PINNs). It provides a comprehensive review classifying shock-management techniques into physics modification, loss/training modification, and architecture modification, and benchmarks these approaches on Buckley-Leverett and a coupled two-phase flow system. The benchmarking shows that vanilla PINNs struggle to resolve shocks, while physics-based and training-based methods substantially improve front capture, though high-dimensional and complex loss landscapes remain challenging; architecture-based methods offer additional gains but do not fully eliminate infinite gradients at shocks. The work highlights the need for hybrid, domain-informed, and automated-parameter strategies to scale PINNs to realistic, high-dimensional multiphysics problems and suggests directions for integrating PINNs with traditional numerical methods to realize robust shock modeling and prediction.

Abstract

Solving partial differential equations (PDEs) with discontinuous solutions , such as shock waves in multiphase viscous flow in porous media , is critical for a wide range of scientific and engineering applications, as they represent sudden changes in physical quantities. Physics-Informed Neural Networks (PINNs), an approach proposed for solving PDEs, encounter significant challenges when applied to such systems. Accurately solving PDEs with discontinuities using PINNs requires specialized techniques to ensure effective solution accuracy and numerical stability. A benchmarking study was conducted on two multiphase flow problems in porous media: the classic Buckley-Leverett (BL) problem and a fully coupled system of equations involving shock waves but with varying levels of solution complexity. The findings show that PM and LM approaches can provide accurate solutions for the BL problem by effectively addressing the infinite gradients associated with shock occurrences. In contrast, AM methods failed to effectively resolve the shock waves. When applied to fully coupled PDEs (with more complex loss landscape), the generalization error in the solutions quickly increased, highlighting the need for ongoing innovation. This study provides a comprehensive review of existing techniques for managing PDE discontinuities using PINNs, offering information on their strengths and limitations. The results underscore the necessity for further research to improve PINNs ability to handle complex discontinuities, particularly in more challenging problems with complex loss landscapes. This includes problems involving higher dimensions or multiphysics systems, where current methods often struggle to maintain accuracy and efficiency.

Figures (15)

• Figure 1: Various types of PDE discontinuities, including jumps and kinks with real-world physical examples.
• Figure 2: A simplified visualization of a 1D shock-wave frame at the interface of two fields (e.g., flowing fluids), a) a continuum-scale perspective, b) a molecular-scale perspective Patel2022ThermodynamicallySystems.
• Figure 3: A graphical visualization of fractional flow $f_w$ vs. $s_w$ and tangent line (Welge method). The applied properties are as follows: $k_{ri}^{max}$=1.0, $n_{i}$=2.0, and $\mu_{i}$=1.0 cP.
• Figure 4: A classification of shock management techniques for PINNs.
• Figure 5: A visualization of two different techniques for defining adaptive artificial viscosity maps, based on Coutinho2023Physics-informedViscosity: (a) parametric artificial viscosity map and (b) residual-based artificial viscosity map.