Energetic Variational Neural Network Discretizations to Gradient Flows

We propose structure-preserving neural-network-based numerical schemes to solve both L2-gradient flows and generalized diffusions. In more detail, by using neural networks as tools for spatial discretization, we introduce a structure-preserving Eulerian algorithm to solve L2-gradient flows and a structure-preserving Lagrangian algorithm to solve generalized diffusions. The Lagrangian algorithm for a generalized diffusion evolves the "flow map" which determines the dynamics of the generalized diffusion. This avoids the non-trivial task of computing the Wasserstein distance between two probability functions. Unlike most existing methods that construct numerical discretizations based on the strong or weak form of the underlying PDE, the proposed schemes are constructed using variational formulations of these PDEs for preserving their variational structures. Our schemes first perform temporal discretization on these variational systems. By doing so, they are very computer-memory-efficient. Moreover, instead of directly solving the obtained nonlinear systems after temporal and spatial discretization, a minimizing movement scheme is utilized to evolve the solutions. This guarantees the monotonic decay of the energy of the system and is crucial for the long-term stability of numerical computation. Lastly, the proposed neural-network-based scheme is mesh-free and enables us to solve gradient flows in high dimensions. Various numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed numerical approaches.