Physics-Informed Neural Networks for coupled stiff transport systems
Purpose: Physics-Informed Neural Networks (PINNs) struggle with stiff, regime-changing transport equations due to instability, loss imbalance, and violations of physical consistency. This paper investigates these failures through the Marshak wave equations - a canonical benchmark from radiative transport - where initial and boundary conditions differ by up to 12 orders of magnitude, and proposes targeted modifications to the standard PINN framework to overcome them. Design/methodology/approach: Three modifications are introduced: (1) a ScaledSigmoid final activation enforcing physical bounds and positivity of the unknowns; (2) a logarithmic MSE loss replacing the standard quadratic loss for initial and boundary conditions, enabling training across extreme scale disparities; and (3) explicit enforcement of global conservation laws derived from the governing equations as an additional physics loss term. Monte Carlo sampling with exponential time weighting is used throughout. Findings: The proposed framework successfully recovers the Marshak wave dynamics - including the hot, cold, and wave-front regions - in agreement with a reference Implicit Monte Carlo solution, with run times under 30 minutes. Ablation studies confirm that each ingredient is essential: linear activation, absence of the logarithmic loss, or removal of the PDE term each independently cause the method to fail qualitatively. Originality/value: This work identifies and resolves three concrete failure modes of standard PINNs on stiff hyperbolic systems with nonlinear coupling. The combination of bounded activations, scale-aware loss functions, and conservation law enforcement constitutes a novel and practically validated framework, with applicability to radiative transport and other coupled stiff PDE systems in engineering.