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Lagrangian dual framework for conservative neural network solutions of kinetic equations

  • * Corresponding author: Hyung Ju Hwang

    * Corresponding author: Hyung Ju Hwang 
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  • 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.

    Mathematics Subject Classification: Primary: 68T07, 82B40.


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  • Figure 1.  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)} $

    Figure 2.  Numerical solutions of Test 1 at $ t = 0, \frac{1}{2}, $ and $ 1 $

    Figure 3.  Left: Loss value in training epoch at log scale. Right: Actual error in training epoch at log scale

    Figure 4.  Left: Time averaged mass in training epoch. Right: Total mass in time after training is finished

    Figure 5.  Numerical solutions of Test 1 at $ t = 0, \frac{3}{2}, $ and $ 3 $

    Figure 6.  Left: Loss value in training epoch at log scale. Right: Actual error in training epoch at log scale

    Figure 7.  Left: Time averaged mass in training epoch. Right: Total mass in time after training is finished

    Figure 8.  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

    Figure 9.  Left: Total mass in time after the training is finished. Right: Kinetic energy in time after the training is finished

    Figure 10.  Momentum in time after the training is finished

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