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

arXiv:2605.23391v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter-equation coupling strengthens. We provide a theoretical explanation for this phenomenon through neural tangent kernel (NTK) analysis: for linearly coupled systems, we prove that the standard NTK's spectral radius grows as Omega(gamma^2) with coupling strength gamma, shrinking the stable learning rate, while block-diagonal Gauss--Newton (GN) preconditioning yields a preconditioned NTK K_P = J H^{+} J^op (where H is the block-diagonal GN Hessian) whose spectral radius is bounded by S (S = number of networks), independent of gamma. We verify the Omega(gamma^2) growth numerically across symmetric, asymmetric, and nonlinear coupled PDE systems, and confirm lambda_{max}(K_P) = S with equality in all cases. 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 1.1imes (ratio of strong- to weak-coupling error) even as coupling parameters vary over one to two orders of magnitude, compared with > 10^2imes for Adam+GN. 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).

Source: arXiv cs.LG | 2026-05-25