Equivariant Graph Neural ODEs for Modeling Physical Dynamics

Abstract

Modeling 3D dynamical systems is a fundamental challenge in the physical and engineering sciences, where Equivariant Graph Neural Networks (EGNNs) have emerged as a powerful paradigm by incorporating geometric symmetries. However, these models are fundamentally constrained by their discrete-time, Markovian framework, which neglects long-range temporal correlations and inevitably leads to error accumulation in long-horizon forecasting. To address this limitation, we introduce the Equivariant Graph Neural Ordinary Differential Equation (EG-NODE), a novel framework that directly learns the continuous-time evolution laws of physical systems. Instead of predicting discrete future states, EG-NODE leverages an equivariant GNN as its core to directly model the ordinary differential equation governing the system's instantaneous rate of change the physical laws of motion thereby natively preserving SE(3) symmetry within the learning process. This continuous-time paradigm enables high-precision predictions at arbitrary time points and allows for the use of adaptive step-size solvers to dynamically balance computational efficiency and accuracy. Extensive experiments on N-body, molecular, and fluid dynamics benchmarks demonstrate that EG-NODE significantly outperforms existing discrete models in long-horizon prediction accuracy and effectively suppresses error propagation. Our work establishes a more fundamental, first-principles-based paradigm for learning continuous physical laws from data. Our code is available at https://anonymous.4open.science/r/egno_egnn-436D/.