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

**Authors:** Nicholas A. Gabriel, Neil F. Johnson, George Em Karniadakis  
**arXiv:** [2502.15913](https://arxiv.org/abs/2502.15913)  
**Date:** 2025-02-21  
**Summary page:** https://www.summarizepaper.com/en/paper/2502.15913v1/

## Key points

- Integration of scale transformations and dynamics through renormalization group techniques is fundamental in modern physical theories
- The ROMA framework is a scalable AI approach for acquiring knowledge about multiscale evolution operators of complex systems
- ROMA incorporates renormalization procedures based on neural analogs of geometric and laplacian renormalization groups
- An attention mechanism is used to model multiscale interactions by connecting geometric representations with dynamical operators
- ROMA framework has been successfully tested under challenging conditions involving large systems, long-range interactions, and noisy data
- It enhances scalability and facilitates positive transfer between forecasting and effective dynamics tasks compared to existing techniques
- Insights into multiscale interactions have been gained through the application of the ROMA framework
- Investigation into power law scaling in the number of model parameters has revealed deviations in scenarios involving hierarchical and multiscale interactions
- Development of neural operators that capture the interplay between multiscale structure and dynamics represents a significant advancement in extending renormalization methods for complex systems

## Summary

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The integration of scale transformations and dynamics through renormalization group techniques is a fundamental aspect of modern physical theories, spanning from fluid mechanics to elementary particle physics. This approach has the potential to establish a foundational inductive bias for learning effective dynamics while revealing multiscale organization. The ROMA (Renormalized Operators with Multiscale Attention) framework has been introduced as a scalable AI approach for acquiring knowledge about the multiscale evolution operators of complex systems. It incorporates renormalization procedures based on neural analogs of geometric and laplacian renormalization groups, which can be co-learned alongside neural operators. An attention mechanism is utilized to model multiscale interactions by connecting geometric representations of local subgraphs with dynamical operators. This framework has been successfully tested under challenging conditions involving large systems exceeding 1 million nodes, long-range interactions, and noisy input-output data using examples such as Kuramoto oscillators and Burgers-like social dynamics. Through experimentation, it has been demonstrated that the ROMA framework enhances scalability and facilitates positive transfer between forecasting and effective dynamics tasks compared to existing operator learning techniques. Additionally, insights into multiscale interactions have been gained through its application. Furthermore, an investigation into power law scaling in the number of model parameters has revealed deviations from typical power law exponents in scenarios involving hierarchical and multiscale interactions. This highlights the importance of considering these factors when designing neural operators for complex systems. In conclusion, the development of neural operators that effectively capture the interplay between multiscale structure and dynamics represents a significant advancement in extending renormalization methods for complex systems. The inherent flexibility and composability of neural operators offer a systematic approach to defining such methods while leveraging renormalizability as an inductive bias. This holds promise for modeling multiscale interactions and enhancing performance and explainability within complex systems.