Lagrangian dual framework for conservative neural network solutions of kinetic equations
This addresses the challenge of ensuring physical consistency in neural network solutions for kinetic equations, which is crucial for computational physics applications.
The authors tackled the problem of solving kinetic equations with neural networks by formulating a constrained optimization approach that enforces physical conservation laws through Lagrangian duality. They demonstrated significantly more accurate approximations for the kinetic Fokker-Planck and homogeneous Boltzmann equations, with concrete error reductions.
In this paper, we propose a novel conservative formulation for solving kinetic equations via neural networks. More precisely, we formulate the learning problem as a constrained optimization problem with constraints that represent the physical conservation laws. The constraints are relaxed toward the residual loss function by the Lagrangian duality. By imposing physical conservation properties of the solution as constraints of the learning problem, we demonstrate far more accurate approximations of the solutions in terms of errors and the conservation laws, for the kinetic Fokker-Planck equation and the homogeneous Boltzmann equation.