Learning emergent PDEs in a learned emergent space

by   Felix P. Kemeth, et al.

We extract data-driven, intrinsic spatial coordinates from observations of the dynamics of large systems of coupled heterogeneous agents. These coordinates then serve as an emergent space in which to learn predictive models in the form of partial differential equations (PDEs) for the collective description of the coupled-agent system. They play the role of the independent spatial variables in this PDE (as opposed to the dependent, possibly also data-driven, state variables). This leads to an alternative description of the dynamics, local in these emergent coordinates, thus facilitating an alternative modeling path for complex coupled-agent systems. We illustrate this approach on a system where each agent is a limit cycle oscillator (a so-called Stuart-Landau oscillator); the agents are heterogeneous (they each have a different intrinsic frequency ω) and are coupled through the ensemble average of their respective variables. After fast initial transients, we show that the collective dynamics on a slow manifold can be approximated through a learned model based on local "spatial" partial derivatives in the emergent coordinates. The model is then used for prediction in time, as well as to capture 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.


page 4

page 6

page 7

page 9

page 10


Neural Time-Dependent Partial Differential Equation

Partial differential equations (PDEs) play a crucial role in studying a ...

Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations

Dissipative partial differential equations that exhibit chaotic dynamics...

Machine Learning for Discovering Effective Interaction Kernels between Celestial Bodies from Ephemerides

Building accurate and predictive models of the underlying mechanisms of ...

Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data

We propose a machine learning framework for the data-driven discovery of...

Coarse-grained and emergent distributed parameter systems from data

We explore the derivation of distributed parameter system evolution laws...

Deep Learning Models for Global Coordinate Transformations that Linearize PDEs

We develop a deep autoencoder architecture that can be used to find a co...