[Paper Review] Gradient-Annihilated PINNs for Solving Riemann Problems: Application to Relativistic Hydrodynamics

Antonio Ferrer-Sánchez, José D. Martín‐Guerrero+3 more|arXiv (Cornell University)|May 15, 2023

Model Reduction and Neural Networks4 citations

TL;DR

This paper introduces Gradient-Annihilated Physics-Informed Neural Networks (GA-PINNs), a novel PINN framework that enhances the solution of hyperbolic PDEs with discontinuous solutions—particularly Riemann problems in special relativistic hydrodynamics—by modifying the loss function to suppress learning in high-gradient regions. The method achieves accuracy comparable to a second-order shock-capturing finite-volume scheme and significantly outperforms standard PINNs, while eliminating the need for primitive variable recovery.

ABSTRACT

We present a novel methodology based on Physics-Informed Neural Networks (PINNs) for solving systems of partial differential equations admitting discontinuous solutions. Our method, called Gradient-Annihilated PINNs (GA-PINNs), introduces a modified loss function that requires the model to partially ignore high-gradients in the physical variables, achieved by introducing a suitable weighting function. The method relies on a set of hyperparameters that control how gradients are treated in the physical loss and how the activation functions of the neural model are dynamically accounted for. The performance of our GA-PINN model is demonstrated by solving Riemann problems in special relativistic hydrodynamics, extending earlier studies with PINNs in the context of the classical Euler equations. The solutions obtained with our GA-PINN model correctly describe the propagation speeds of discontinuities and sharply capture the associated jumps. We use the relative $l^{2}$ error to compare our results with the exact solution of special relativistic Riemann problems, used as the reference ``ground truth'', and with the error obtained with a second-order, central, shock-capturing scheme. In all problems investigated, the accuracy reached by our GA-PINN model is comparable to that obtained with a shock-capturing scheme and significantly higher than that achieved by a baseline PINN algorithm. An additional benefit worth stressing is that our PINN-based approach sidesteps the costly recovery of the primitive variables from the state vector of conserved ones, a well-known drawback of grid-based solutions of the relativistic hydrodynamics equations. Due to its inherent generality and its ability to handle steep gradients, the GA-PINN method discussed could be a valuable tool to model relativistic flows in astrophysics and particle physics, characterized by the prevalence of discontinuous solutions.

Motivation & Objective

• To address the challenge of solving hyperbolic PDEs with discontinuous solutions using PINNs, particularly in relativistic hydrodynamics.
• To overcome the poor performance of standard PINNs in capturing shocks and discontinuities due to uncontrolled gradient backpropagation.
• To eliminate the computationally expensive primitive variable recovery step inherent in traditional grid-based relativistic hydrodynamics solvers.
• To develop a generalizable, differentiable framework that maintains accuracy in steep gradient regions without relying on heuristic modifications.
• To demonstrate the viability of deep learning for modeling relativistic flows with shocks, especially in astrophysical contexts.

Proposed method

• The method introduces a modified loss function that incorporates a gradient-weighting function to suppress learning in regions of high gradients, effectively 'annihilating' the gradient signal in discontinuous zones.
• A set of hyperparameters controls the degree of gradient suppression and dynamically adjusts the influence of activation functions during training.
• The neural network is trained to minimize a physics-informed loss that includes the relativistic Euler equations, initial conditions, and the modified gradient-weighted loss term.
• The architecture allows simultaneous learning of conserved and primitive variables, avoiding the need for iterative root-finding during inference.
• The approach is validated on standard Riemann problems in special relativistic hydrodynamics, comparing solutions against exact Riemann solvers and finite-volume schemes.
• The method is designed to be extensible to ultra-relativistic flows through potential architectural modifications, such as variable-specific subnetworks.

Experimental results

Research questions

• RQ1 Can a modified PINN framework effectively capture discontinuous solutions in relativistic hydrodynamics without relying on traditional shock-capturing schemes?
• RQ2 How does gradient suppression in the loss function improve PINN performance on hyperbolic PDEs with shocks?
• RQ3 To what extent can GA-PINNs match the accuracy of high-order finite-volume schemes in solving Riemann problems?
• RQ4 Does the elimination of primitive variable recovery in GA-PINNs lead to computational and numerical advantages?
• RQ5 Can the GA-PINN framework be extended to handle ultra-relativistic flows with extreme velocity gradients?

Key findings

• The GA-PINN model achieves relative l² errors comparable to a second-order, central, shock-capturing finite-volume scheme for all tested relativistic Riemann problems.
• The method significantly outperforms standard PINNs, which fail to accurately capture shock locations and jump discontinuities due to uncontrolled gradient backpropagation.
• The model successfully captures the propagation speeds of shocks and contact discontinuities with high resolution, even in complex wave configurations.
• The framework naturally avoids the primitive variable recovery step, which is a major computational bottleneck in traditional relativistic hydrodynamics solvers.
• The method demonstrates robustness across diverse initial conditions and Riemann problem types in special relativistic hydrodynamics.
• Preliminary analysis suggests that architectural extensions—such as variable-specific subnetworks or hard enforcement of initial conditions—may further improve performance in ultra-relativistic regimes.