# From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systems

arXiv: https://arxiv.org/abs/2405.11542 · HTML: https://mlsanity.com/paper/2405.11542/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 19 of its 33 references that are in the map.
- First posted on arXiv 19 May 2024.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 19 of 19)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [Neural Flows: Efficient Alternative to Neural ODEs](https://mlsanity.com/paper/2110.13040/) (2021)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [Constructing Neural Network-Based Models for Simulating Dynamical Systems](https://mlsanity.com/paper/2111.01495/) (2021)
- [Neural Delay Differential Equations](https://mlsanity.com/paper/2102.10801/) (2021)
- [Neural Laplace: Learning diverse classes of differential equations in the Laplace domain](https://mlsanity.com/paper/2206.04843/) (2022)
- [Neural Integro-Differential Equations](https://mlsanity.com/paper/2206.14282/) (2022)
- [Hybrid Physical-Neural ODEs for Fast N-body Simulations](https://mlsanity.com/paper/2207.05509/) (2022)

## Built on by (most cited first, 9 of 9)

- [SPIN-ODE: Stiff Physics-Informed Neural ODE for Chemical Reaction Rate Estimation](https://mlsanity.com/paper/2505.05625/) (2025)
- [Is Flow Matching Just Trajectory Replay for Sequential Data?](https://mlsanity.com/paper/2602.08318/) (2026)
- [Towards Generalizable PDE Dynamics Forecasting via Physics-Guided Invariant Learning](https://mlsanity.com/paper/2509.24332/) (2025)
- [FNODE: Flow-Matching for data-driven simulation of constrained multibody systems](https://mlsanity.com/paper/2509.00183/) (2025)
- [Operator Flow Matching for Timeseries Forecasting](https://mlsanity.com/paper/2510.15101/) (2025)
- [Breaking Scale Anchoring: Frequency Representation Learning for Accurate High-Resolution Inference from Low-Resolution Training](https://mlsanity.com/paper/2512.05132/) (2025)
- [Breaking Free: Decoupling Forced Systems with Laplace Neural Networks](https://mlsanity.com/paper/2503.13158/) (2025)
- [MCMC Informed Neural Emulators for Uncertainty Quantification in Dynamical Systems](https://mlsanity.com/paper/2603.10987/) (2026)
- [Wavelet Flow Matching for Multi-Scale Physics Emulation](https://mlsanity.com/paper/2605.16573/) (2026)

## Abstract (arXiv)

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.