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August  2022, 15(4): 551-568. doi: 10.3934/krm.2021046

## Lagrangian dual framework for conservative neural network solutions of kinetic equations

 1 Department of Mathematics, Pohang University of Science and Technology, Pohang, Republic of Korea 2 Stochastic Analysis and Application Research Center, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea

* Corresponding author: Hyung Ju Hwang

Received  June 2021 Published  August 2022 Early access  December 2021

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.

Citation: Hyung Ju Hwang, Hwijae Son. Lagrangian dual framework for conservative neural network solutions of kinetic equations. Kinetic and Related Models, 2022, 15 (4) : 551-568. doi: 10.3934/krm.2021046
##### References:

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##### References:
Left: Relaxation to Maxwellian of f(t, x = 1, v) from an initial condition $f_0^{(1)}$. Right: Relaxation to Maxwellian of f(t, x = 1, v) from an initial condition $f_0^{(2)}$
Numerical solutions of Test 1 at $t = 0, \frac{1}{2},$ and $1$
Left: Loss value in training epoch at log scale. Right: Actual error in training epoch at log scale
Left: Time averaged mass in training epoch. Right: Total mass in time after training is finished
Numerical solutions of Test 1 at $t = 0, \frac{3}{2},$ and $3$
Left: Loss value in training epoch at log scale. Right: Actual error in training epoch at log scale
Left: Time averaged mass in training epoch. Right: Total mass in time after training is finished
Left: Value of the loss $\widehat{Loss}_B$ in training epoch. Right: $L^{\infty}((0,1);L^2_{v_x,v_y})$ error in training epoch
Left: Total mass in time after the training is finished. Right: Kinetic energy in time after the training is finished
Momentum in time after the training is finished
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