(226a) Learning Partial Differential Equations in Emergent Coordinates

Coarse-grained descriptions of the emergent dynamics of large systems of interacting units still remain elusive in modeling contexts (from statistical mechanics to socioeconomic agent-based modeling) [1]. In particular, tasks such as bifurcation analysis become increasingly difficult when the number of agents increases. It has thus historically been an important undertaking to find effective evolution equations for agent-based or particle systems. Prominent examples include Fick's law for random walkers/diffusion or the Navier-Stokes equations for molecular dynamics/flows. However, for general networks of agents, such as for systems of coupled oscillators, there does not even exist an obvious spatial coordinate in which to find such partial differential evolution equations.

Here, we propose a methodology to learn effective partial differential equations for general multi-agent systems. Our two-step approach is based on (a) learning a so-called emergent space embedding [2] of the agents based on their observed time series/behavior, and (b) learning the law of the partial differential equation in this emergent space. The former step can be realized using manifold learning, identifying low-dimensional parametrizations of the agents, whereas the law of the partial differential equation can, as we show, be learned using artificial neural networks. This ansatz transforms a system of coupled agents into a problem with local interactions only, allowing well established simulation and bifurcation analysis tools to analyze the dynamics. Our approach is illustrated on two systems of coupled oscillators -the second one from computational neuroscience-, through which advantages and open problems of our methodology are highlighted [3].

[1] Tamás Vicsek, Anna Zafeiris, Collective motion, Physics Reports, Volume 517, Issues 3–4, 2012

[2] Felix P. Kemeth, Sindre W. Haugland, Felix Dietrich, Tom Bertalan, Kevin Höhlein, Qianxiao Li, Erik M. Bollt, Ronen Talmon, Katharina Krischer, and Ioannis G. Kevrekidis. An emergent space for distributed data with hidden internal order through manifold learning.IEEE Access, 6:77402–77413, 2018

[3] Felix P. Kemeth, Tom Bertalan, Thomas Thiem, Felix Dietrich, Sung Joon Moon, Carlo R. Laing, Ioannis G. Kevrekidis, Learning emergent PDEs in a learned emergent space, arXiv:2012.12738, 2020

Figure Caption: On a collection of time series, we first learn (a) an embedding φ_i of the time series that organizes the observed data and (b) learn a PDE in this emergent space, accelerating our understanding of multi-agent systems.