Lagrangian dual framework for conservative neural network solutions of kinetic equations

• URL: http://arxiv.org/abs/2106.12147v1
• Date: Wed, 23 Jun 2021 04:01:04 GMT
• Title: Lagrangian dual framework for conservative neural network solutions of kinetic equations
• Authors: Hyung Ju Hwang and Hwijae Son
• Abstract summary: We formulate the learning problem as a constrained optimization problem with constraints that represent the physical conservation laws. By imposing physical conservation properties of the solution as constraints of the learning problem, we demonstrate far more accurate approximations of the solutions.
• Score: 2.741266294612776
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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