# SPIKE: Sparse Koopman Regularization for Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2601.10282 · HTML: https://mlsanity.com/paper/2601.10282/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 27 references that are in the map.
- First posted on arXiv 15 Jan 2026.

## Builds on (most cited first, 13 of 13)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [On the Principles of Parsimony and Self-Consistency for the Emergence of Intelligence](https://mlsanity.com/paper/2207.04630/) (2022)
- [Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methods](https://mlsanity.com/paper/2412.01181/) (2024)
- [Mesh-free sparse identification of nonlinear dynamics](https://mlsanity.com/paper/2505.16058/) (2025)
- [Physics-Informed Koopman Network](https://mlsanity.com/paper/2211.09419/) (2022)
- [Performance Evaluation of Single-step Explicit Exponential Integration Methods on Stiff Ordinary Differential Equations](https://mlsanity.com/paper/2411.19374/) (2024)
- [Improved Physics-informed neural networks loss function regularization with a variance-based term](https://mlsanity.com/paper/2412.13993/) (2024)
- [AC-PKAN: Attention-Enhanced and Chebyshev Polynomial-Based Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2505.08687/) (2025)

## Abstract (arXiv)

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.