Energetic Variational Neural Network Discretizations to Gradient Flows

We propose structure-preserving neural-network-based numerical schemes to solve both $L^2$-gradient flows and generalized diffusions. In more detail, by using neural networks as tools for spatial discretizations, we introduce a structure-preserving Eulerian algorithm to solve $L^2$-gradient flows and a structure-preserving Lagrangian algorithm to solve generalized diffusions. The Lagrangian algorithm for the generalized diffusion evolves the "flow map" which determines the dynamics of the generalized diffusion. This avoids computing the Wasserstein distance between two probability functions, which is non-trivial. The key ideas behind these schemes are to construct numerical discretizations based on the variational formulations of the gradient flows, i.e., the energy-dissipation laws, directly. More precisely, we construct minimizing movement schemes for these two types of gradient flow by introducing temporal discretization first, which is more efficient and convenient in neural-network-based implementations. The variational discretizations ensure the proper energy dissipation in numerical solutions and are crucial for the long-term stability of numerical computation. The neural-network-based spatial discretization enables us to solve these gradient flows in high dimensions. Various numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed numerical approaches.