Learning emergent partial differential equations in a learned emergent space.

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.

Global and local reduced models for interacting, heterogeneous agents.

Large collections of coupled, heterogeneous agents can manifest complex dynamical behavior presenting difficulties for simulation and analysis. However, if the collective dynamics lie on a low-dimensional manifold, then the original agent-based model may be approximated with a simplified surrogate model on and near the low-dimensional space where the dynamics live. Analytically identifying such simplified models can be challenging or impossible, but here we present a data-driven coarse-graining methodology for discovering such reduced models.

Emergent Spaces for Coupled Oscillators.

Systems of coupled dynamical units (e.g., oscillators or neurons) are known to exhibit complex, emergent behaviors that may be simplified through coarse-graining: a process in which one discovers coarse variables and derives equations for their evolution.