  We propose a systematic method for learning stable and physically
interpretable dynamical models using sampled trajectory data from physical
processes based on a generalized Onsager principle. The learned dynamics are
autonomous ordinary differential equations parameterized by neural networks
that retain clear physical structure information, such as free energy,
diffusion, conservative motion and external forces. For high dimensional
problems with a low dimensional slow manifold, an autoencoder with metric
preserving regularization is introduced to find the low dimensional generalized
coordinates on which we learn the generalized Onsager dynamics. Our method
exhibits clear advantages over existing methods on benchmark problems for
learning ordinary differential equations. We further apply this method to study
Rayleigh-Benard convection and learn Lorenz-like low dimensional autonomous
reduced order models that capture both qualitative and quantitative properties
of the underlying dynamics. This forms a general approach to building reduced
order models for forced dissipative systems.

Dynamical Systems;Machine Learning;Computational Physics