Multi-resolution energy diffusion graph neural network discovers system conservation variables

Discovering conservation laws and assessing integrability are fundamental to understanding the evolution and long-term behavior of Hamiltonian dynamical systems. The neural deflation method provides a principled framework for iteratively learning conservation laws, but its performance depends on the representational capacity of the underlying neural network. Standard multilayer perceptrons lack structural priors and struggle to scale to high-dimensional lattice systems, leading to degraded accuracy and limited interpretability. We propose incorporating a multi-resolution diffusion graph neural network (MR-DGNN) as the approximator for conserved quantities within neural deflation. MR-DGNN uses multi-resolution graph diffusion operators to adaptively capture long-range and local particle interactions. This architecture aligns with the discrete geometric structure of lattice systems and achieves higher accuracy with significantly fewer parameters than baselines. Moreover, the learned diffusion weights reveal the spatial interaction range of each conserved quantity, supporting physical interpretation. We evaluate the method on several benchmark systems. Results show that MR-DGNN consistently outperforms baselines in test error, with advantages becoming more pronounced as the number of degrees of freedom increases. The interpretable weight patterns further indicate that different conserved quantities exhibit distinct interaction ranges. These findings suggest that the method can accurately discover conservation laws while providing physical insights into complex high-dimensional Hamiltonian systems.