Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as the geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerating the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE(MP-NODE), is applied to chaotic systems such as the Kuramoto-Sivashinsky equation and the two-dimensional Kolmogorov flow. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.

提出了一种新颖的多步惩罚 NODE（MP-NODE）算法，可以强化学习混沌动力学系统，并显示在短期轨迹预测和代表这些动力学混沌特性的不变统计方面具有可行的性能。

分而治之: 学习具有多步惩罚的混沌动力系统的神经常微分方程