In dynamical systems reconstruction (DSR) we aim to recover the dynamical system (DS) underlying observed time series. Specifically, we aim to learn a generative surrogate model which approximates the underlying, data-generating DS, and recreates its long-term properties (`climate statistics'). In scientific and medical areas, in particular, these models need to be mechanistically tractable -- through their mathematical analysis we would like to obtain insight into the recovered system's workings. Piecewise-linear (PL), ReLU-based RNNs (PLRNNs) have a strong track-record in this regard, representing SOTA DSR models while allowing mathematical insight by virtue of their PL design. However, all current PLRNN variants are discrete-time maps. This is in disaccord with the assumed continuous-time nature of most physical and biological processes, and makes it hard to accommodate data arriving at irregular temporal intervals. Neural ODEs are one solution, but they do not reach the DSR performance of PLRNNs and often lack their tractability. Here we develop theory for continuous-time PLRNNs (cPLRNNs): We present a novel algorithm for training and simulating such models, bypassing numerical integration by efficiently exploiting their PL structure. We further demonstrate how important topological objects like equilibria or limit cycles can be determined semi-analytically in trained models. We compare cPLRNNs to both their discrete-time cousins as well as Neural ODEs on DSR benchmarks, including systems with discontinuities which come with hard thresholds.

翻译：在动态系统重构（DSR）中，我们的目标是从观测到的时间序列中恢复其背后的动态系统（DS）。具体而言，我们旨在学习一个生成式代理模型，该模型能够近似底层的数据生成动态系统，并重现其长期特性（“气候统计”）。特别是在科学和医学领域，这些模型需要具备机制上的可追溯性——通过数学分析，我们希望获得对恢复系统运作机制的深入理解。基于ReLU的分段线性（PL）循环神经网络（PLRNN）在这方面有着良好的记录，它们代表了最先进的DSR模型，同时凭借其PL设计允许进行数学分析。然而，目前所有的PLRNN变体都是离散时间映射。这与大多数物理和生物过程所假设的连续时间性质不符，并且难以处理在非规则时间间隔到达的数据。神经常微分方程是一种解决方案，但它们无法达到PLRNN的DSR性能，并且通常缺乏其可追溯性。本文发展了连续时间PLRNN（cPLRNN）的理论：我们提出了一种新颖的算法来训练和模拟此类模型，通过有效利用其PL结构来绕过数值积分。我们进一步展示了如何在训练后的模型中半解析地确定重要的拓扑对象，如平衡点或极限环。我们将cPLRNN与其离散时间对应模型以及神经常微分方程在DSR基准测试上进行比较，包括包含硬阈值的具有不连续性的系统。