OnsagerNet: Learning stable and interpretable dynamics using a generalized Onsager principle

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 parametrized 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-Bénard 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.