Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations
This work addresses the challenge of efficiently simulating chaotic systems for researchers in computational physics and applied mathematics, though it is incremental as it builds on existing neural ODE and autoencoder frameworks.
The authors tackled the problem of modeling spatiotemporal chaos in dissipative PDEs by developing a data-driven reduced-order method that discovers low-dimensional manifold coordinates via an autoencoder and approximates the dynamics with neural ODEs, achieving excellent short- and long-time statistical recreation of true dynamics with widely spaced data (spacing of ~0.7 Lyapunov times).
Dissipative partial differential equations that exhibit chaotic dynamics tend to evolve to attractors that exist on finite-dimensional manifolds. We present a data-driven reduced order modeling method that capitalizes on this fact by finding the coordinates of this manifold and finding an ordinary differential equation (ODE) describing the dynamics in this coordinate system. The manifold coordinates are discovered using an undercomplete autoencoder -- a neural network (NN) that reduces then expands dimension. Then the ODE, in these coordinates, is approximated by a NN using the neural ODE framework. Both of these methods only require snapshots of data to learn a model, and the data can be widely and/or unevenly spaced. We apply this framework to the Kuramoto-Sivashinsky for different domain sizes that exhibit chaotic dynamics. With this system, we find that dimension reduction improves performance relative to predictions in the ambient space, where artifacts arise. Then, with the low-dimensional model, we vary the training data spacing and find excellent short- and long-time statistical recreation of the true dynamics for widely spaced data (spacing of ~0.7 Lyapunov times). We end by comparing performance with various degrees of dimension reduction, and find a "sweet spot" in terms of performance vs. dimension.