Learning the Dynamics of Coupled Oscillator Systems through the Discovery of Emergent PDEs Via Artificial Neural Networks and Manifold Learning

The time evolution of heterogeneous coupled oscillators is typically described by systems of ordinary differential equations (ODEs). When the coupling is strong enough, these systems can exhibit attracting, low dimensional manifolds on which the collective dynamics depend on the model heterogeneities in a smooth fashion. By taking advantage of this smoothness, we are able to rephrase this discrete problem as a continuous one in a neighborhood of the attracting manifold. We liken our continuous approach to typical partial differential equations (PDEs), where the oscillators can be viewed as the discretization points customarily used for the numerical solution of PDEs, such as those employed by the method of lines approach, and as such refer to it as an “emergent PDE” description of the system. In order to find the emergent PDE we use spatial partial derivatives, here derivatives of the oscillator states with respect to the model heterogeneities, as input features with which to approximate the temporal partial derivatives of the system. By limiting ourselves to systems and regimes in which the coupled behavior can be accurately approximated by local effects, we are able to fit the temporal partial derivatives with a reduced set of these features via a neural network.

We discuss the challenges and benefits of such a coarse-grained emergent PDE description — for example, the selection of appropriate boundary conditions — and present an application of this approach to Hodgkin-Huxley type coupled neurons. We also consider the case in which the model heterogeneities are unknown, and demonstrate how a manifold learning tool (Diffusion Maps) can be leveraged to identify data-driven features that are suitable for use as independent variables for our PDE methodology [1-2]. Finally, we elaborate on the technical challenge of ensuring stability for the learned dynamics, outline different approaches to tackle this problem, and contrast these with methods proposed in recent literature.

[1] Thiem, Thomas N., Kooshkbaghi, Mahdi, Bertalan, Tom, Laing, Carlo R., and Kevrekidis, Ioannis G. "Emergent spaces for coupled oscillators." arXiv preprint., 2020, arXiv:2004.06053v1.

[2] Kemeth, Felix P., et al. "An Emergent Space for Distributed Data With Hidden Internal Order Through Manifold Learning." IEEE Access, vol. 6, 2018, pp. 77402-77413.