Solving BDNK diffusion using physics-informed neural networks

TL;DR

This paper introduces a novel physics-informed neural network framework, SA-PINN-ACTO, for solving the relativistic BDNK diffusion equations, demonstrating comparable accuracy to traditional methods for smooth data and highlighting limitations near discontinuities.

Contribution

The paper presents the SA-PINN-ACTO framework that combines self-adaptive PINNs with boundary condition enforcement for solving relativistic diffusion equations.

Findings

SA-PINN-ACTO matches finite volume solutions for smooth initial data.

Errors increase for discontinuous profiles, indicating PINN limitations near sharp gradients.

The approach effectively handles complex relativistic diffusion problems.

Abstract

In this work, we reformulate the relativistic BDNK (Bemfica-Disconzi-Noronha-Kovtun) diffusion equation in flux-conservative form, and solve the resulting equations in $(1+1)$D using both a second-order Kurganov-Tadmor finite volume scheme and physics-informed neural networks (PINNs). In particular, we introduce the SA-PINN-ACTO framework, which combines the self-adaptive PINN technique with an exact enforcement of initial and periodic boundary conditions through an algebraic transform of the network's raw output, allowing the network to focus solely on minimizing the PDE residual. We test both approaches on smooth and discontinuous initial data, for both trivial and dynamically evolving velocity and temperature BDNK backgrounds, and for two characteristic speeds. The SA-PINN-ACTO method matches the converged Kurganov-Tadmor solutions for smooth profiles, while for discontinuous profiles the errors increase, reflecting an expected limitation of PINNs near sharp gradients.