OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle

We propose a systematic method for learning stable and 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, dissipation, conservative interaction and external force. The neural network representations for the hidden dynamics are trained by minimizing the empirical risk based on an embedded Runge-Kutta method. For high dimensional problems with a low dimensional slow manifold, an autoencoder with isometric regularization is introduced to find generalized coordinates on which we learn the Onsager dynamics. We apply the method to learn reduced order models for the Rayleigh-B\'{e}nard convection problem, where we obtain low dimensional autonomous equations that capture both qualitative and quantitative properties of the underlying dynamics. In particular, this validates the basic approach of Lorenz, although we also discover that the dimension of the learned autonomous model required for faithful representation increases with the Rayleigh number.