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

• URL: http://arxiv.org/abs/2605.23391v1
• Date: Fri, 22 May 2026 09:01:31 GMT
• Title: Coupling-Robust Accuracy in Multiphysics Physics Informed Neural Networks via Kronecker-Preconditioned Optimization
• Abstract summary: Physics-informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter-equation coupling strengthens.<n>We provide a theoretical explanation for this phenomenon through neural tangent kernel (NTK) analysis.
• Score: 12.079987397544953
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $Ω(γ^2)$ with coupling strength $γ$, shrinking the stable learning rate, while block-diagonal Gauss--Newton (GN) preconditioning yields a preconditioned NTK $K_P = J H^{+} J^\top$ (where $H$ is the block-diagonal GN Hessian) whose spectral radius is bounded by $S$ ($S$ = number of networks), independent of $γ$. We verify the $Ω(γ^2)$ growth numerically across symmetric, asymmetric, and nonlinear coupled PDE systems, and confirm $λ_{\max}(K_P) = S$ 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 $\leq 1.1\times$ (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. 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 ($L_2 > 0.9$).

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