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.