Coupling-Robust Accuracy in Multiphysics PINNs via Kronecker-Preconditioned Optimization

Physics-informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter-equation coupling strengthens. We systematically evaluate how PINN accuracy scales with coupling strength and show that combining the Kronecker-preconditioned optimizer SOAP with inverse-gradient-norm loss balancing (SOAP+GN) yields coupling-robust accuracy. Across 234 experiments spanning three 1D systems of increasing nonlinearity and a 2D electroosmotic flow benchmark, SOAP+GN maintains final-epoch $L_2$ degradation $\leq 1.1\times$ (ratio of strong- to weak-coupling error) even as coupling parameters vary over one to two orders of magnitude, compared with $> 10^2\times$ for Adam+GN. In the nonlinear Nernst--Planck--Poisson system, $L_\infty$-norm balancing (LRA) catastrophically fails due to weight explosion to $O(10^9)$, whereas GN remains stable---revealing a structural limitation of LRA for nonlinearly coupled PDEs. SOAP+GN further scales to a 2D, 6-PDE electroosmotic flow system at EDL-resolved conditions---a regime that all prior PINN electrokinetics studies have avoided through simplified physics---where Adam+GN fails entirely ($L_2 > 0.9$).