Open AccessLagrangian dual framework for conservative neural network solutions of kinetic equations
Author(s) -
Hyung Ju Hwang,
Hwijae Son
Publication year - 2022
Publication title -
kinetic and related models
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.987
H-Index - 28
eISSN - 1937-5093
pISSN - 1937-5077
DOI - 10.3934/krm.2021046
Subject(s) - conservation law , duality (order theory) , artificial neural network , lagrangian , mathematics , lagrangian relaxation , mathematical optimization , dual (grammatical number) , kinetic energy , homogeneous , boltzmann equation , fokker–planck equation , computer science , statistical physics , physics , classical mechanics , partial differential equation , mathematical analysis , artificial intelligence , quantum mechanics , pure mathematics , art , literature
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.