Coupling-Robust Accuracy in Multiphysics Physics Informed Neural Networks via Kronecker-Preconditioned Optimization

Abstract

Physics-informed neural networks (PINNs) offer a mesh-free route to solving coupled multiphysics systems, but their accuracy degrades systematically as inter-equation coupling strengthens, and inverse-gradient-norm loss balancing alone does not reliably prevent this failure. This study explains why coupling degrades PINN training and identifies an optimizer structure that removes the dependence, replacing case-by-case tuning with a principled remedy. Through a neural tangent kernel analysis, we prove that the standard kernel's spectral radius grows as $\Omega ({\gamma }^2)$ with coupling strength $\gamma$, whereas block-diagonal Gauss-Newton (GN) preconditioning bounds it by the number of networks $S$, independent of $\gamma$; no diagonal preconditioner recovers this bound for any coupling type or loss weighting. We realize block-diagonal GN preconditioning through the Kronecker-preconditioned optimizer SOAP combined with inverse-gradient-norm loss balancing (SOAP+GradNorm) and evaluate it across 222 experiments on four benchmarks of increasing difficulty. Across all systems, SOAP+GradNorm is the only configuration whose degradation remains bounded in every regime tested: it preserves weak-coupling accuracy in linear problems and limits degradation to $2.3\times$ in the nonlinear Nernst-Planck-Poisson system, whereas Adam-based training leaves the $L_2$ error above the 0.1 failure threshold. The same effect applies to a six-residual, four-network, 2D electro-osmotic flow where the electric double layer is resolved down to a Debye length of $\epsilon =0.01$ on an $x$-invariant reference solution. These results recast coupling-induced accuracy loss as a problem of the preconditioner's structure rather than loss weighting and identify Kronecker preconditioning as a structural lever for training PINNs on strongly coupled, stiff multiphysics systems.

Extended version of AI4Physics Workshop accepted paper (ICML 2026)