In this work, we reformulate the relativistic Bemfica-Disconzi-Noronha-Kovtun (BDNK) diffusion equation in flux-conservative form, and solve the resulting partial differential equations in $(1+1)\mathrm{D}$ 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 (SA-PINN) technique with an exact enforcement of auxiliary conditions through a transform to the output (ACTO), 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.

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• Astrophysical fluid dynamics
• Quark-gluon plasma
• Shock waves
• Artificial neural networks
• Numerical techniques
• Relativistic hydrodynamics