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.

翻译：神经网络所学的具有\ textit{ODE} 神经网络所学的神经等式模型动态系统。 然而, 数字基本不足以模拟具有远距离依赖性或不连续性(在工程和生物系统中是常见的)的模型系统。 提出了更广泛的差异方程(DE)作为补救措施, 包括延迟差异方程和内分异方程。 此外, 神经极值在模拟硬性ODE和带有片段驱动功能的 ODE 时会受到数字不稳定的影响。 在这项工作中, 我们提出\ textit{ Neural Laplace}, 是一个用于学习不同类别的DE(包括所有上述)的统一框架。 我们没有模拟时间域的动态, 而是在 Laplace 域建模它, 在那里历史- 依赖性和时间不连续性可以代表复杂指数的对比。 为了提高学习效率, 我们使用里曼域的大地测量地形图在 Laplace 域中带来更大的光滑。 在实验中, Neuralplace 显示在建模和复杂历史依赖性等级中的高级性表现, 包括急剧依赖性和外等。