SPIKE: Sparse Koopman Regularization for Physics-Informed Neural Networks
This addresses the generalization issue in PINNs for solving differential equations, which is incremental as it builds on existing PINN and Koopman operator methods.
The paper tackles the problem of Physics-Informed Neural Networks (PINNs) overfitting and generalizing poorly beyond trained regions by introducing SPIKE, a framework that regularizes PINNs with continuous-time Koopman operators to learn sparse dynamics representations, resulting in consistent improvements in temporal extrapolation, spatial generalization, and long-term prediction accuracy across various PDEs and ODEs.
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.