  Modeling complex systems using standard neural ordinary differential
equations (NODEs) often faces some essential challenges, including high
computational costs and susceptibility to local optima. To address these
challenges, we propose a simulation-free framework, called Fourier NODEs
(FNODEs), that effectively trains NODEs by directly matching the target vector
field based on Fourier analysis. Specifically, we employ the Fourier analysis
to estimate temporal and potential high-order spatial gradients from noisy
observational data. We then incorporate the estimated spatial gradients as
additional inputs to a neural network. Furthermore, we utilize the estimated
temporal gradient as the optimization objective for the output of the neural
network. Later, the trained neural network generates more data points through
an ODE solver without participating in the computational graph, facilitating
more accurate estimations of gradients based on Fourier analysis. These two
steps form a positive feedback loop, enabling accurate dynamics modeling in our
framework. Consequently, our approach outperforms state-of-the-art methods in
terms of training time, dynamics prediction, and robustness. Finally, we
demonstrate the superior performance of our framework using a number of
representative complex systems.
