Solving stiff ordinary differential equations (StODEs) requires sophisticated numerical solvers, which are often computationally expensive. In particular, StODE's often cannot be solved with traditional explicit time integration schemes and one must resort to costly implicit methods to compute solutions. On the other hand, state-of-the-art machine learning (ML) based methods such as Neural ODE (NODE) poorly handle the timescale separation of various elements of the solutions to StODEs and require expensive implicit solvers for integration at inference time. In this work, we embark on a different path which involves learning a latent dynamics for StODEs, in which one completely avoids numerical integration. To that end, we consider a constant velocity latent dynamical system whose solution is a sequence of straight lines. Given the initial condition and parameters of the ODE, the encoder networks learn the slope (i.e the constant velocity) and the initial condition for the latent dynamics. In other words, the solution of the original dynamics is encoded into a sequence of straight lines which can be decoded back to retrieve the actual solution as and when required. Another key idea in our approach is a nonlinear transformation of time, which allows for the "stretching/squeezing" of time in the latent space, thereby allowing for varying levels of attention to different temporal regions in the solution. Additionally, we provide a simple universal-approximation-type proof showing that our approach can approximate the solution of stiff nonlinear system on a compact set to any degree of accuracy, {\epsilon}. We show that the dimension of the latent dynamical system in our approach is independent of {\epsilon}. Numerical investigation on prototype StODEs suggest that our method outperforms state-of-the art machine learning approaches for handling StODEs.

翻译：求解刚性常微分方程需要复杂的数值求解器，这通常计算成本高昂。具体而言，刚性常微分方程往往无法用传统的显式时间积分方案求解，必须采用计算代价较高的隐式方法。另一方面，基于机器学习的前沿方法（如神经常微分方程）难以处理刚性常微分方程解中不同元素的时间尺度分离问题，且在推理时仍需昂贵的隐式求解器进行积分。本研究提出了一种不同的路径：通过学习刚性常微分方程的潜动力学，完全避免数值积分。为此，我们构建了一个恒定速度潜动力系统，其解为一系列直线段。给定常微分方程的初始条件和参数，编码器网络可学习潜动力学的斜率（即恒定速度）和初始条件。换言之，原始动力学的解被编码为一系列直线段，可在需要时解码还原为实际解。本方法的另一个关键创新在于时间的非线性变换，该变换允许在潜空间中对时间进行"拉伸/压缩"，从而实现对解中不同时间区域的可变关注度。此外，我们通过简单的通用逼近型证明表明：本方法能在紧集上以任意精度{\epsilon}逼近刚性非线性系统的解，且潜动力系统的维度与{\epsilon}无关。在典型刚性常微分方程上的数值实验表明，本方法在处理刚性常微分方程方面优于当前最先进的机器学习方法。