Neural Ordinary Differential Equations model dynamical systems with \textit{ODE}s learned by neural networks. However, ODEs are fundamentally inadequate to model systems with long-range dependencies or discontinuities, which are common in engineering and biological systems. Broader classes of differential equations (DE) have been proposed as remedies, including delay differential equations and integro-differential equations. Furthermore, Neural ODE suffers from numerical instability when modelling stiff ODEs and ODEs with piecewise forcing functions. In this work, we propose \textit{Neural Laplace}, a unified framework for learning diverse classes of DEs including all the aforementioned ones. Instead of modelling the dynamics in the time domain, we model it in the Laplace domain, where the history-dependencies and discontinuities in time can be represented as summations of complex exponentials. To make learning more efficient, we use the geometrical stereographic map of a Riemann sphere to induce more smoothness in the Laplace domain. In the experiments, Neural Laplace shows superior performance in modelling and extrapolating the trajectories of diverse classes of DEs, including the ones with complex history dependency and abrupt changes.

提出了一种名为 Neural Laplace 的框架，它使用 Laplace 域来建模系统的动态，并能学习各种微分方程类，包括延迟微分方程和积分微分方程，能够更稳健地建模粘性微分方程和具有分段强制函数的微分方程。在实验中，证明其在建模和外推各种微分方程的轨迹方面都能比较好的工作。

神经 Laplace：在 Laplace 域中学习不同类别的微分方程