Coupling-Robust Accuracy in Multiphysics Physics Informed Neural Networks via Kronecker-Preconditioned Optimization
For practitioners training PINNs on coupled multiphysics problems, this work provides a theoretically grounded optimizer that eliminates coupling-induced accuracy loss, enabling reliable simulation of strongly coupled systems.
The paper identifies and theoretically explains accuracy degradation in multiphysics PINNs under strong coupling, and proposes a Kronecker-preconditioned optimizer (SOAP+GN) that maintains final-epoch L2 degradation ≤1.1× across coupling strengths, compared to >10^2× for Adam+GN, and scales to a challenging 2D 6-PDE electroosmotic flow system where prior methods fail.
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 ($L_2 > 0.9$).