Building low-dimensional representations of reaction-diffusion dynamics

Abstract

This seminar presents ongoing research on the qualitative dynamics of reaction-diffusion processes, cornerstone complex systems in physics and biology. These equations, fundamental to modeling phenomena ranging from chemical reactions to ecological and cellular processes, encapsulate the intricate balance between transport mechanisms and local interactions. A central aim in the study of such systems is to decipher the macroscopic or qualitative behaviors that emerge from these complex interactions, seeking to understand how patterns, waves, and structures develop on larger scales. However, despite the valuable analytical insights offered by perturbation or renormalization techniques, these approaches can struggle in highly nonlinear or multi-scale regimes and do not easily generalize to new parameters. Addressing these challenges, this work leverages recent advancements in data-driven dynamical systems theory to uncover the low-dimensional dynamics governing macroscopic features of interest. By employing machine learning techniques to derive low-dimensional representations, this approach clarifies the emergence of qualitative structures which are often obscured in the high-dimensional data of the original systems. This method not only facilitates a deeper understanding of the system's dynamics but also opens new avenues for control and parameter identification. Preliminary results demonstrate the efficacy of this methodology in shedding light on the behavior of the Gray Scott model and Min protein dynamics, both examples of reaction-diffusion systems with significant theoretical and biological implications. The seminar will detail the theoretical underpinnings, methodological developments, and early outcomes of this research, highlighting its potential to advance our understanding of reaction-diffusion systems as they arise in biology