SPIKE: Sparse Koopman Regularization for Physics-Informed Neural Networks

TL;DR

SPIKE introduces a regularization framework for PINNs using continuous-time Koopman operators, leading to sparse, interpretable dynamics models that improve generalization and stability in solving complex differential equations.

Contribution

The paper proposes SPIKE, a novel regularization method combining Koopman operators with PINNs, enhancing their ability to learn parsimonious and stable representations of physical systems.

Findings

Improved extrapolation and generalization in PDE solutions.

Sparse Koopman matrices capture low-dimensional dynamics.

Enhanced stability for stiff systems using matrix exponential integration.

Abstract

Physics-Informed Neural Networks (PINNs) provide a mesh-free approach for solving differential equations by embedding physical constraints into neural network training. However, PINNs tend to overfit within the training domain, leading to poor generalization when extrapolating beyond trained spatiotemporal regions. This work presents SPIKE (Sparse Physics-Informed Koopman-Enhanced), a framework that regularizes PINNs with continuous-time Koopman operators to learn parsimonious dynamics representations. By enforcing linear dynamics $dz/dt = Az$ in a learned observable space, both PIKE (without explicit sparsity) and SPIKE (with L1 regularization on $A$) learn sparse generator matrices, embodying the parsimony principle that complex dynamics admit low-dimensional structure. Experiments across parabolic, hyperbolic, dispersive, and stiff PDEs, including fluid dynamics (Navier-Stokes) and chaotic ODEs (Lorenz), demonstrate consistent improvements in temporal extrapolation, spatial generalization, and long-term prediction accuracy. The continuous-time formulation with matrix exponential integration provides unconditional stability for stiff systems while avoiding diagonal dominance issues inherent in discrete-time Koopman operators.