Training stiff neural ordinary differential equations with explicit rational Taylor series methods.

Abstract

We introduce a new approach for directly training stiff neural ordinary differential equations using explicit rational Taylor series methods. These explicit schemes, of second and third order, are A-stable and require only a single linear solve or matrix inversion per time step, providing both high efficiency and strong numerical stability. In contrast to traditional implicit methods, which are often computationally intensive due to the need to solve nonlinear systems at each step, the explicit rational Taylor methods deliver accurate and stable learning of stiff dynamics at a significantly reduced computational cost. Our results show that these methods effectively learn stiff systems, such as the van der Pol oscillator, at large step sizes without encountering stability issues that challenge implicit schemes. This work broadens the capabilities of data-driven simulation, supporting efficient and accurate modeling of complex dynamics in stiff neural differential equations, mesh-based simulation, and physics-informed neural networks.