Chaotic dynamical systems (DS) are ubiquitous in nature and society. Often we
are interested in reconstructing such systems from observed time series for
prediction or mechanistic insight, where by reconstruction we mean learning
geometrical and invariant temporal properties of the system in question (like
attractors). However, training reconstruction algorithms like recurrent neural
networks (RNNs) on such systems by gradient-descent based techniques faces
severe challenges. This is mainly due to exploding gradients caused by the
exponential divergence of trajectories in chaotic systems. Moreover, for
(scientific) interpretability we wish to have as low dimensional
reconstructions as possible, preferably in a model which is mathematically
tractable. Here we report that a surprisingly simple modification of teacher
forcing leads to provably strictly all-time bounded gradients in training on
chaotic systems, and, when paired with a simple architectural rearrangement of
a tractable RNN design, piecewise-linear RNNs (PLRNNs), allows for faithful
reconstruction in spaces of at most the dimensionality of the observed system.
We show on several DS that with these amendments we can reconstruct DS better
than current SOTA algorithms, in much lower dimensions. Performance differences
were particularly compelling on real world data with which most other methods
severely struggled. This work thus led to a simple yet powerful DS
reconstruction algorithm which is highly interpretable at the same time.

通过对混沌动力系统进行简单的修改以及与可处理的递归神经网络设计的简单架构重排，可以在低维空间中准确还原动力系统，并在真实世界数据上表现出较高的性能。