Connecting the geometry and dynamics of many-body complex systems with message passing neural operators

TL;DR

This paper introduces ROMA, a neural operator framework that integrates renormalization group concepts and multiscale attention to effectively learn and analyze the dynamics of large, complex many-body systems across multiple scales.

Contribution

The paper presents a novel neural operator architecture, ROMA, that incorporates renormalization techniques and attention mechanisms to improve scalability and understanding of multiscale interactions in complex systems.

Findings

ROMA outperforms existing methods in large-scale systems with over 1 million nodes.

It effectively captures long-range interactions and multiscale dynamics.

The framework provides insights into power law scaling and hierarchical interactions.

Abstract

The relationship between scale transformations and dynamics established by renormalization group techniques is a cornerstone of modern physical theories, from fluid mechanics to elementary particle physics. Integrating renormalization group methods into neural operators for many-body complex systems could provide a foundational inductive bias for learning their effective dynamics, while also uncovering multiscale organization. We introduce a scalable AI framework, ROMA (Renormalized Operators with Multiscale Attention), for learning multiscale evolution operators of many-body complex systems. In particular, we develop a renormalization procedure based on neural analogs of the geometric and laplacian renormalization groups, which can be co-learned with neural operators. An attention mechanism is used to model multiscale interactions by connecting geometric representations of local subgraphs and dynamical operators. We apply this framework in challenging conditions: large systems of more than 1M nodes, long-range interactions, and noisy input-output data for two contrasting examples: Kuramoto oscillators and Burgers-like social dynamics. We demonstrate that the ROMA framework improves scalability and positive transfer between forecasting and effective dynamics tasks compared to state-of-the-art operator learning techniques, while also giving insight into multiscale interactions. Additionally, we investigate power law scaling in the number of model parameters, and demonstrate a departure from typical power law exponents in the presence of hierarchical and multiscale interactions.