Efficient and precise determination of manifold coordinates for systems with complex spatiotemporal dynamics

Nonlinear dissipative partial differential equations (PDEs) are ubiquitous in describing phenomena in physics and engineering that display out-of-equilibrium dynamics, complex nonlinear behaviors, and even spatiotemporal chaos. The long-time dynamics of these formally infinite-dimensional systems are known to collapse onto finite-dimensional invariant manifolds. The task of identifying the manifold dimension/coordinates of these systems from simulation or experimental data is generally a nontrivial task. We address this challenge by combining a novel autoencoder architecture with implicit and weight regularization. Compared to existing methods, our framework does not require combinatorically expensive calculations of distance between data points and yields a sharp distinction between nonlinear degrees of freedom that are important and those that are not. Using our method, we successfully estimate the manifold dimension for a zoo of datasets including human handwriting data, an embedded Lorenz attractor, the Kuramoto-Sivashinsky Equation at domain sizes of L=22, 44, 66, and 88, 2D Kolmogorov flow and turbulent plane Couette flow. Finally, we demonstrate that our architecture provides a natural workflow for downstream reduced-order modeling and forecasting tasks.