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.

翻译：物理信息神经网络（PINNs）通过将物理约束嵌入神经网络训练，为求解微分方程提供了一种无网格方法。然而，PINNs在训练域内易发生过拟合，导致在训练时空区域外进行外推时泛化性能较差。本文提出SPIKE（稀疏物理信息Koopman增强）框架，该框架利用连续时间Koopman算子对PINNs进行正则化，以学习简约的动力学表示。通过在学习的可观测量空间中强制线性动力学$dz/dt = Az$，PIKE（无显式稀疏性）和SPIKE（对$A$施加L1正则化）均能学习稀疏的生成矩阵，体现了复杂动力学具有低维结构的简约性原则。在抛物线型、双曲线型、色散型和刚性偏微分方程（包括流体动力学Navier-Stokes方程和混沌常微分方程Lorenz系统）上的实验表明，该方法在时间外推、空间泛化和长期预测精度方面均取得一致提升。采用矩阵指数积分的连续时间公式为刚性系统提供了无条件稳定性，同时避免了离散时间Koopman算子固有的对角占优问题。