🤖 AI Summary
This work addresses the poor spatiotemporal extrapolation and generalization of physics-informed neural networks (PINNs), which often suffer from overfitting within the training domain. To mitigate this, the authors propose the SPIKE framework, which regularizes PINNs by incorporating a continuous-time Koopman operator to enforce linear dynamics in a learned observable space, yielding a compact and structured representation of the underlying system. By integrating L1 sparse regularization, SPIKE learns a sparse generator matrix that reflects the intrinsic low-dimensional simplicity of complex dynamical systems while avoiding the diagonal dominance commonly observed in discrete Koopman operators. Experiments on diverse partial differential equations—including Navier–Stokes—and chaotic ordinary differential equations such as the Lorenz system demonstrate that SPIKE substantially improves temporal extrapolation, spatial generalization, long-term prediction accuracy, and achieves unconditional stability.
📝 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.