We focus on learning unknown dynamics from data using ODE-nets templated on implicit numerical initial value problem solvers. First, we perform Inverse Modified error analysis of the ODE-nets using unrolled implicit schemes for ease of interpretation. It is shown that training an ODE-net using an unrolled implicit scheme returns a close approximation of an Inverse Modified Differential Equation (IMDE). In addition, we establish a theoretical basis for hyper-parameter selection when training such ODE-nets, whereas current strategies usually treat numerical integration of ODE-nets as a black box. We thus formulate an adaptive algorithm which monitors the level of error and adapts the number of (unrolled) implicit solution iterations during the training process, so that the error of the unrolled approximation is less than the current learning loss. This helps accelerate training, while maintaining accuracy. Several numerical experiments are performed to demonstrate the advantages of the proposed algorithm compared to nonadaptive unrollings, and validate the theoretical analysis. We also note that this approach naturally allows for incorporating partially known physical terms in the equations, giving rise to what is termed ``gray box" identification.

翻译：本文聚焦于利用基于隐式数值初值问题求解器模板化的ODE网络从数据中学习未知动力学。首先，我们对采用展开隐式格式的ODE网络进行逆修正误差分析，以简化解释过程。研究表明，使用展开隐式格式训练ODE网络能够逼近逆修正微分方程。此外，我们为训练此类ODE网络时的超参数选择建立了理论基础，而现有策略通常将ODE网络的数值积分视为黑箱操作。据此，我们设计了一种自适应算法，该算法在训练过程中监测误差水平并动态调整（展开）隐式求解迭代次数，使得展开近似的误差小于当前学习损失，从而在保持精度的同时加速训练。多项数值实验验证了所提算法相较于非自适应展开方法的优势，并证实了理论分析的有效性。值得注意的是，该方法可自然地将部分已知物理项融入方程中，形成所谓的"灰箱"辨识框架。