This paper proposes the use of spectral element methods \citep{canuto_spectral_1988} for fast and accurate training of Neural Ordinary Differential Equations (ODE-Nets; \citealp{Chen2018NeuralOD}) for system identification. This is achieved by expressing their dynamics as a truncated series of Legendre polynomials. The series coefficients, as well as the network weights, are computed by minimizing the weighted sum of the loss function and the violation of the ODE-Net dynamics. The problem is solved by coordinate descent that alternately minimizes, with respect to the coefficients and the weights, two unconstrained sub-problems using standard backpropagation and gradient methods. The resulting optimization scheme is fully time-parallel and results in a low memory footprint. Experimental comparison to standard methods, such as backpropagation through explicit solvers and the adjoint technique \citep{Chen2018NeuralOD}, on training surrogate models of small and medium-scale dynamical systems shows that it is at least one order of magnitude faster at reaching a comparable value of the loss function. The corresponding testing MSE is one order of magnitude smaller as well, suggesting generalization capabilities increase. Neural Ordinary Differential Equations (ODE-Nets; Chen et al., 2018) can learn latent models from observations that are sparse in time. This property has the potential to enhance the performance of neural network predictive models in applications where information is sparse in time and it is important to account for exact arrival times and delays. In complex control systems and model-based reinforcement learning, planning over a long horizon is often needed, while high frequency feedback is necessary for maintaining stability (Franklin et al., 2014) . Discrete-time models, including RNNs (Jain & Medsker, 1999) , often struggle to fully meet the needs of such applications due to the fixed time resolution. ODE-Nets have been shown to provide superior performance with respect to classic RNNs on time series forecasting with sparse training data. However, learning their parameters can be computationally intensive. In particular, ODE-Nets are memory efficient but time inefficient. In this paper, we address this bottleneck and propose a novel alternative strategy for system identification.
This paper proposes the use of spectral element methods for fast and accurate training of Neural Ordinary Differential Equations for system identification.
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