Learning emergent partial differential equations in a learned emergent space

Felix P. Kemeth, Tom Bertalan, Thomas Thiem, Felix Dietrich, Sung Joon Moon, Carlo R. Laing and Ioannis G. Kevrekidis ()
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Felix P. Kemeth: Whiting School of Engineering, Johns Hopkins University
Tom Bertalan: Whiting School of Engineering, Johns Hopkins University
Thomas Thiem: Princeton University
Felix Dietrich: Technical University of Munich
Sung Joon Moon: Princeton University
Carlo R. Laing: Massey University (Albany)
Ioannis G. Kevrekidis: Whiting School of Engineering, Johns Hopkins University

Nature Communications, 2022, vol. 13, issue 1, 1-13

Abstract: Abstract We propose an approach to learn effective evolution equations for large systems of interacting agents. This is demonstrated on two examples, a well-studied system of coupled normal form oscillators and a biologically motivated example of coupled Hodgkin-Huxley-like neurons. For such types of systems there is no obvious space coordinate in which to learn effective evolution laws in the form of partial differential equations. In our approach, we accomplish this by learning embedding coordinates from the time series data of the system using manifold learning as a first step. In these emergent coordinates, we then show how one can learn effective partial differential equations, using neural networks, that do not only reproduce the dynamics of the oscillator ensemble, but also capture the collective bifurcations when system parameters vary. The proposed approach thus integrates the automatic, data-driven extraction of emergent space coordinates parametrizing the agent dynamics, with machine-learning assisted identification of an emergent PDE description of the dynamics in this parametrization.

Date: 2022
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DOI: 10.1038/s41467-022-30628-6

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