用稀疏柯尔曼算子提升神经网络的物理外推能力
SPIKE: Sparse Koopman Regularization for Physics-Informed Neural Networks
- 引入连续时间柯尔曼算子约束，让网络学习简洁的动力学表示
- 在多种偏微分方程上实现更优的长期预测与空间泛化性能
- 特别适合需要稳定外推的流体动力学和混沌系统建模
物理信息神经网络（PINNs）通过将物理约束嵌入训练过程，提供了一种无网格求解微分方程的方法。然而，传统PINNs在训练域内容易过拟合，导致在未训练时空区域外推时表现不佳。本文提出SPIKE（Sparse Physics-Informed Koopman-Enhanced），通过连续时间柯尔曼算子对PINNs进行正则化，学习稀疏的动力学表示。通过在学习的观测量空间中强制线性动态 $dz/dt = Az$，PIKE（无显式稀疏性）与SPIKE（对矩阵 $A$ 加L1正则）均能学习稀疏生成矩阵，体现复杂系统具有低维结构的简约性原则。在抛物型、双曲型、色散型及刚性常微分方程（如洛伦兹系统）和流体动力学（纳维-斯托克斯方程）等多类问题上，实验表明其在时间外推、空间泛化和长期预测精度上均有持续提升。连续时间形式结合矩阵指数积分，对刚性系统提供无条件稳定性，避免了离散时间柯尔曼算子固有的对角占优问题。
原文摘要 · Abstract (English)
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.
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