Lagrangian dual framework for conservative neural network solutions of kinetic equations

TL;DR

This paper introduces a Lagrangian dual framework for neural network solutions to kinetic equations, ensuring physical conservation laws are respected, leading to more accurate and physically consistent results.

Contribution

It presents a novel conservative formulation using Lagrangian duality to incorporate conservation laws into neural network solutions for kinetic equations.

Findings

Improved accuracy in solving kinetic Fokker-Planck and Boltzmann equations.

Enhanced conservation law adherence in neural network solutions.

Demonstrated effectiveness of the dual framework in physical law enforcement.

Abstract

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.