We focus on learning hidden 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网络从数据中学习隐藏动力学。首先，我们采用展开的隐式格式对ODE网络进行逆修正误差分析，以提升其可解释性。结果表明，使用展开隐式格式训练ODE网络能够获得对逆修正微分方程（IMDE）的精确近似。此外，我们为训练此类ODE网络时的超参数选择奠定了理论基础，而当前策略通常将ODE网络的数值积分视为黑箱操作。据此，我们提出一种自适应算法，在训练过程中监测误差水平并动态调整（展开的）隐式求解迭代次数，使得展开近似的误差低于当前学习损失函数值。该方法在保持精度的同时加速了训练。通过多项数值实验，验证了所提算法相比非自适应展开策略的优势，并证实了理论分析的正确性。我们进一步指出，该方法可自然地将方程中的部分已知物理项纳入学习过程，从而形成所谓的“灰箱”辨识框架。