Connecting the geometry and dynamics of many-body complex systems with message passing neural operators
This work addresses the challenge of modeling large-scale, noisy complex systems like Kuramoto oscillators and social dynamics, offering a novel AI framework that could enhance predictive capabilities in physics and social sciences, though it appears incremental in combining existing techniques.
The authors tackled the problem of learning multiscale evolution operators for many-body complex systems by introducing the ROMA framework, which integrates renormalization group methods with neural operators and attention mechanisms, demonstrating improved scalability and positive transfer in forecasting and effective dynamics tasks for systems with over 1M nodes.
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.