Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEs
This addresses solving complex PDEs with particle interactions for physics and ML applications, but appears incremental as an extension of existing self-consistency methods to more general equations.
The authors tackled solving McKean-Vlasov equations (MVEs) with singular interactions like Coulomb and 2D Navier-Stokes by extending self-consistency from Fokker-Planck equations to control KL-divergence via entropy dissipation, and proposed minimizing this potential with neural networks, achieving validation against state-of-the-art NN-based PDE solvers on example problems.
We extend the concept of self-consistency for the Fokker-Planck equation (FPE) to the more general McKean-Vlasov equation (MVE). While FPE describes the macroscopic behavior of particles under drift and diffusion, MVE accounts for the additional inter-particle interactions, which are often highly singular in physical systems. Two important examples considered in this paper are the MVE with Coulomb interactions and the vorticity formulation of the 2D Navier-Stokes equation. We show that a generalized self-consistency potential controls the KL-divergence between a hypothesis solution to the ground truth, through entropy dissipation. Built on this result, we propose to solve the MVEs by minimizing this potential function, while utilizing the neural networks for function approximation. We validate the empirical performance of our approach by comparing with state-of-the-art NN-based PDE solvers on several example problems.