Implementation and (Inverse Modified) Error Analysis for implicitly-templated ODE-nets
This work addresses the challenge of efficiently training ODE-nets for dynamics learning, offering a theoretical basis for hyper-parameter selection and an adaptive approach, though it is incremental as it builds on existing implicit solver frameworks.
The paper tackles the problem of learning unknown dynamics from data using ODE-nets based on implicit numerical solvers, by developing an adaptive algorithm that monitors error and adjusts unrolled iterations during training to accelerate it while maintaining accuracy, with numerical experiments showing advantages over nonadaptive methods.
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.