| Abstract: |
| While Physics-Informed Neural Networks offer a promising framework for partial differential equations, the standard L2 residual loss is insufficient for solving the Bhatnagar-Gross-Krook model of the Boltzmann equation. Specifically, small L2 residuals do not guarantee small errors in macroscopic moments, causing the standard PINN to fail in capturing the true solution. To resolve this, we propose a weighted L2 loss function based on a stability analysis. The stability analysis implies that minimizing this weighted residual strictly controls the distance between the exact and approximate solutions. Finally, numerical experiments shows that employing this theoretical guided PINN loss leads to superior accuracy across various benchmarks compared to the standard PINN approach. |
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