# Graph Fourier Neural ODEs: Modeling Spatial-temporal Multi-scales in Molecular Dynamics

arXiv: https://arxiv.org/abs/2411.01600 · HTML: https://mlsanity.com/paper/2411.01600/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 41 references that are in the map.
- First posted on arXiv 3 Nov 2024.

## Builds on (most cited first, 17 of 17)

- [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)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [Timewarp: Transferable Acceleration of Molecular Dynamics by Learning Time-Coarsened Dynamics](https://mlsanity.com/paper/2302.01170/) (2023)
- [Learning Continuous System Dynamics from Irregularly-Sampled Partial Observations](https://mlsanity.com/paper/2011.03880/) (2020)
- [Neural Dynamics on Complex Networks](https://mlsanity.com/paper/1908.06491/) (2019)
- [Domain Agnostic Fourier Neural Operators](https://mlsanity.com/paper/2305.00478/) (2023)
- [Equivariant Graph Neural Operator for Modeling 3D Dynamics](https://mlsanity.com/paper/2401.11037/) (2024)
- [Graph Neural Networks Accelerated Molecular Dynamics](https://mlsanity.com/paper/2112.03383/) (2021)
- [Neural Laplace: Learning diverse classes of differential equations in the Laplace domain](https://mlsanity.com/paper/2206.04843/) (2022)
- [Generalizing Graph ODE for Learning Complex System Dynamics across Environments](https://mlsanity.com/paper/2307.04287/) (2023)
- [Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective](https://mlsanity.com/paper/2310.06379/) (2023)
- [PGODE: Towards High-quality System Dynamics Modeling](https://mlsanity.com/paper/2311.06554/) (2023)
- [TANGO: Time-Reversal Latent GraphODE for Multi-Agent Dynamical Systems](https://mlsanity.com/paper/2310.06427/) (2023)

## Built on by (most cited first, 2 of 2)

- [Permutation Equivariant Neural Controlled Differential Equations for Dynamic Graph Representation Learning](https://mlsanity.com/paper/2506.20324/) (2025)
- [PAINET: A Principled Efficient Transformer for 3D Dynamics Modeling](https://mlsanity.com/paper/2510.04233/) (2025)

## Abstract (arXiv)

Accurately predicting long-horizon molecular dynamics (MD) trajectories remains a significant challenge, as existing deep learning methods often struggle to retain fidelity over extended simulations. We hypothesize that one key factor limiting accuracy is the difficulty of capturing interactions that span distinct spatial and temporal scales, ranging from high-frequency local vibrations to low-frequency global conformational changes. To address these limitations, we propose Graph Fourier Neural ODEs (GF-NODE), integrating a graph Fourier transform for spatial frequency decomposition with a Neural ODE framework for continuous-time evolution. Specifically, GF-NODE first decomposes molecular configurations into multiple spatial frequency modes using the graph Laplacian, then evolves the frequency components in time via a learnable Neural ODE module that captures both local and global dynamics, and finally reconstructs the updated molecular geometry through an inverse graph Fourier transform. By explicitly modeling high- and low-frequency phenomena in this unified pipeline, GF-NODE captures long-range correlations and local fluctuations more effectively. We provide theoretical insight through heat equation analysis on a simplified diffusion model, demonstrating how graph Laplacian eigenvalues can determine temporal dynamics scales, and crucially validate this correspondence through comprehensive empirical analysis on real molecular dynamics trajectories showing quantitative spatial-temporal correlations across diverse molecular systems. Experimental results on challenging MD benchmarks demonstrate that GF-NODE achieves state-of-the-art accuracy while preserving essential geometrical features over extended simulations. These findings highlight the promise of bridging spectral decomposition with continuous-time modeling to improve the robustness and predictive power of MD simulations.