cs.LGApr 4, 2026

A Theory-guided Weighted L2L^2 Loss for solving the BGK model via Physics-informed neural networks

Authors: Gyounghun Ko, Sung-Jun Son, Seung Yeon Cho, Myeong-Su Lee

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Abstract

While Physics-Informed Neural Networks offer a promising framework for solving partial differential equations, the standard L2L^2 loss formulation is fundamentally insufficient when applied to the Bhatnagar-Gross-Krook (BGK) model. Specifically, simply minimizing the standard loss does not guarantee accurate predictions of the macroscopic moments, causing the approximate solutions to fail in capturing the true physical solution. To overcome this limitation, we introduce a velocity-weighted L2L^2 loss function designed to effectively penalize errors in the high-velocity regions. By establishing a stability estimate for the proposed approach, we shows that minimizing the proposed weighted loss guarantees the convergence of the approximate solution. Also, numerical experiments demonstrate that employing this weighted PINN loss leads to superior accuracy and robustness across various benchmarks compared to the standard approach.

Figures & tables

Figure 1. Visualization of the function Kε(v)K_{\varepsilon}(v) for ε=0.01\varepsilon=0.01 .
Figure 2. Loss-accuracy curves for the explicit counterexamples: (a) Example 1 and (b) Example 2.
Figure 3. Relative error curves of the distribution function ff and macroscopic moments ( ρ,ux,T\rho,u_{x},T ) according to the polynomial growth rate β\beta at Kn=0.01\text{Kn}=0.01 .
Loss functionf\boldsymbol{f} (Relative L2L^{2} )f\boldsymbol{f} (Relative L1L^{1} )ρ\boldsymbol{\rho} (Relative L1L^{1} )ux\boldsymbol{u_{x}} (Relative L1L^{1} )T\boldsymbol{T} (Relative L1L^{1} )
Standard L2L^{2} Loss2.87×10−22.87\times 10^{-2}1.53×10−21.53\times 10^{-2}1.31×10−31.31\times 10^{-3}6.78×10−36.78\times 10^{-3}1.33×10−21.33\times 10^{-2}
Relative Loss [ 29 ]1.25×10−31.25\times 10^{-3}5.63×10−45.63\times 10^{-4}3.09×10−43.09\times 10^{-4}2.42×10−42.42\times 10^{-4}3.44×10−43.44\times 10^{-4}
α=0.1,β=2.0\alpha=0.1,\beta=2.04.71×10−34.71\times 10^{-3}2.33×10−32.33\times 10^{-3}1.78×10−41.78\times 10^{-4}9.97×10−49.97\times 10^{-4}2.08×10−32.08\times 10^{-3}
α=0.1,β=3.0\alpha=0.1,\beta=3.04.17×10−44.17\times 10^{-4}2.68×10−42.68\times 10^{-4}1.07×10−41.07\times 10^{-4}1.11×10−41.11\times 10^{-4}1.96×10−41.96\times 10^{-4}
α=0.1,β=3.5\alpha=0.1,\beta=3.51.88×10−41.88\times 10^{-4}1.30×10−41.30\times 10^{-4}5.55×10−55.55\times 10^{-5}5.02×10−55.02\times 10^{-5}8.06×10−58.06\times 10^{-5}
α=0.1,β=4.0\alpha=0.1,\beta=4.01.54×10−4\mathbf{1.54\times 10^{-4}}1.10×10−4\mathbf{1.10\times 10^{-4}}4.46×10−5\mathbf{4.46\times 10^{-5}}3.98×10−5\mathbf{3.98\times 10^{-5}}4.54×10−5\mathbf{4.54\times 10^{-5}}
Table 1. Ablation study results of relative errors at Kn=0.01\text{Kn}=0.01 for the standard L2L^{2} Loss, relative loss [ 29 ] , and the proposed loss with specific choices of α\alpha and β\beta .
Figure 4. Distribution function ff (top row) and macroscopic moments ( ρ,ux,T\rho,u_{x},T ) (bottom row) at t=0.1t=0.1 for the 1D Smooth problem at Kn=0.01\text{Kn}=0.01 .
Figure 5. Relative error curves of the distribution function ff and macroscopic moments ( ρ,ux,T\rho,u_{x},T ) according to the polynomial growth rate β\beta at Kn=0.1\text{Kn}=0.1 .
Loss functionf\boldsymbol{f} (Relative L2L^{2} )f\boldsymbol{f} (Relative L1L^{1} )ρ\boldsymbol{\rho} (Relative L1L^{1} )ux\boldsymbol{u_{x}} (Relative L1L^{1} )T\boldsymbol{T} (Relative L1L^{1} )
Standard L2L^{2} Loss1.88×10−31.88\times 10^{-3}2.57×10−32.57\times 10^{-3}7.92×10−47.92\times 10^{-4}1.88×10−31.88\times 10^{-3}2.14×10−32.14\times 10^{-3}
Relative Loss [ 29 ]1.69×10−31.69\times 10^{-3}1.33×10−31.33\times 10^{-3}3.16×10−43.16\times 10^{-4}7.75×10−47.75\times 10^{-4}1.32×10−31.32\times 10^{-3}
α=0.01,β=4.0\alpha=0.01,\beta=4.03.25×10−43.25\times 10^{-4}5.87×10−45.87\times 10^{-4}8.85×10−58.85\times 10^{-5}2.72×10−42.72\times 10^{-4}2.55×10−42.55\times 10^{-4}
α=0.01,β=4.5\alpha=0.01,\beta=4.52.53×10−42.53\times 10^{-4}4.14×10−44.14\times 10^{-4}8.45×10−58.45\times 10^{-5}1.13×10−41.13\times 10^{-4}1.76×10−41.76\times 10^{-4}
α=0.01,β=5.0\alpha=0.01,\beta=5.02.34×10−4\mathbf{2.34\times 10^{-4}}3.10×10−4\mathbf{3.10\times 10^{-4}}4.34×10−54.34\times 10^{-5}5.58×10−55.58\times 10^{-5}8.58×10−58.58\times 10^{-5}
α=0.01,β=5.5\alpha=0.01,\beta=5.52.85×10−42.85\times 10^{-4}3.22×10−43.22\times 10^{-4}3.67×10−53.67\times 10^{-5}4.65×10−54.65\times 10^{-5}5.89×10−55.89\times 10^{-5}
Table 2. Ablation study results of relative errors at Kn=0.1\text{Kn}=0.1 for the standard L2L^{2} Loss, relative loss [ 29 ] , and the proposed loss with specific choices of α\alpha and β\beta .
Figure 6. Relative error curves of the distribution function ff and macroscopic moments ( ρ,ux,T\rho,u_{x},T ) according to the polynomial growth rate β\beta at Kn=1.0\text{Kn}=1.0 .
Loss functionf\boldsymbol{f} (Relative L2L^{2} )f\boldsymbol{f} (Relative L1L^{1} )ρ\boldsymbol{\rho} (Relative L1L^{1} )ux\boldsymbol{u_{x}} (Relative L1L^{1} )T\boldsymbol{T} (Relative L1L^{1} )
Standard L2L^{2} Loss1.49×10−31.49\times 10^{-3}2.89×10−32.89\times 10^{-3}1.03×10−31.03\times 10^{-3}1.82×10−31.82\times 10^{-3}4.59×10−34.59\times 10^{-3}
Relative Loss [ 29 ]2.64×10−32.64\times 10^{-3}2.37×10−32.37\times 10^{-3}4.54×10−44.54\times 10^{-4}1.33×10−31.33\times 10^{-3}2.45×10−32.45\times 10^{-3}
α=0.01,β=5.0\alpha=0.01,\beta=5.04.48×10−4\mathbf{4.48\times 10^{-4}}6.82×10−46.82\times 10^{-4}5.73×10−55.73\times 10^{-5}1.44×10−41.44\times 10^{-4}1.99×10−41.99\times 10^{-4}
α=0.01,β=5.5\alpha=0.01,\beta=5.55.18×10−45.18\times 10^{-4}6.51×10−4\mathbf{6.51\times 10^{-4}}4.35×10−54.35\times 10^{-5}7.62×10−5\mathbf{7.62\times 10^{-5}}1.03×10−4\mathbf{1.03\times 10^{-4}}
α=0.01,β=6.0\alpha=0.01,\beta=6.06.43×10−46.43\times 10^{-4}7.17×10−47.17\times 10^{-4}5.30×10−55.30\times 10^{-5}8.41×10−58.41\times 10^{-5}1.04×10−41.04\times 10^{-4}
α=0.01,β=6.5\alpha=0.01,\beta=6.51.22×10−31.22\times 10^{-3}1.11×10−31.11\times 10^{-3}1.38×10−41.38\times 10^{-4}8.53×10−58.53\times 10^{-5}1.60×10−41.60\times 10^{-4}
Table 3. Ablation study results of relative errors at Kn=1.0\text{Kn}=1.0 for the standard L2L^{2} Loss, relative loss [ 29 ] , and the proposed loss with specific choices of α\alpha and β\beta .
KnLoss functionf\boldsymbol{f} (Relative L2L^{2} )f\boldsymbol{f} (Relative L1L^{1} )ρ\boldsymbol{\rho} (Relative L1L^{1} )ux\boldsymbol{u_{x}} (Relative L1L^{1} )T\boldsymbol{T} (Relative L1L^{1} )
1.01.0Standard L2L^{2} Loss1.78×10−21.78\times 10^{-2}1.26×10−21.26\times 10^{-2}1.79×10−31.79\times 10^{-3}6.55×10−36.55\times 10^{-3}7.26×10−37.26\times 10^{-3}
Relative Loss [ 29 ]2.58×10−22.58\times 10^{-2}1.37×10−21.37\times 10^{-2}2.49×10−32.49\times 10^{-3}4.13×10−34.13\times 10^{-3}3.36×10−33.36\times 10^{-3}
Proposed Loss1.42×10−2\mathbf{1.42\times 10^{-2}}9.49×10−3\mathbf{9.49\times 10^{-3}}8.54×10−4\mathbf{8.54\times 10^{-4}}1.74×10−3\mathbf{1.74\times 10^{-3}}1.48×10−3\mathbf{1.48\times 10^{-3}}
0.10.1Standard L2L^{2} Loss7.63×10−37.63\times 10^{-3}6.39×10−36.39\times 10^{-3}1.47×10−31.47\times 10^{-3}5.47×10−35.47\times 10^{-3}4.76×10−34.76\times 10^{-3}
Relative Loss [ 29 ]1.10×10−21.10\times 10^{-2}6.53×10−36.53\times 10^{-3}2.06×10−32.06\times 10^{-3}2.97×10−32.97\times 10^{-3}2.65×10−32.65\times 10^{-3}
Proposed Loss6.64×10−3\mathbf{6.64\times 10^{-3}}4.86×10−3\mathbf{4.86\times 10^{-3}}7.62×10−4\mathbf{7.62\times 10^{-4}}1.12×10−3\mathbf{1.12\times 10^{-3}}1.10×10−3\mathbf{1.10\times 10^{-3}}
Table 4. Relative errors of the distribution function ff and macroscopic moments ( ρ,ux,T\rho,u_{x},T ) for the 1D Riemann problem across different Knudsen numbers ( Kn∈{1.0,0.1,0.01}\text{Kn}\in\{1.0,0.1,0.01\} ).
Figure 7. Distribution function ff (top row) and macroscopic moments ( ρ,ux,T\rho,u_{x},T ) (bottom row) at t=0.1t=0.1 for the 1D Riemann problem at Kn=1.0\text{Kn}=1.0 .
KnLoss functionMetricρ\boldsymbol{\rho}ux\boldsymbol{u_{x}}uy\boldsymbol{u_{y}}T\boldsymbol{T}
1.01.0Standard L2L^{2} LossRel L1L^{1}7.85×10−57.85\times 10^{-5}1.60×10−41.60\times 10^{-4}2.20×10−42.20\times 10^{-4}2.54×10−42.54\times 10^{-4}
Rel L2L^{2}1.01×10−41.01\times 10^{-4}2.24×10−42.24\times 10^{-4}2.90×10−42.90\times 10^{-4}3.18×10−43.18\times 10^{-4}
Relative Loss [ 29 ]Rel L1L^{1}2.46×10−5\mathbf{2.46\times 10^{-5}}2.60×10−52.60\times 10^{-5}3.59×10−53.59\times 10^{-5}1.82×10−51.82\times 10^{-5}
Rel L2L^{2}3.14×10−5\mathbf{3.14\times 10^{-5}}3.40×10−53.40\times 10^{-5}4.53×10−54.53\times 10^{-5}2.66×10−52.66\times 10^{-5}
Proposed LossRel L1L^{1}2.61×10−52.61\times 10^{-5}1.78×10−5\mathbf{1.78\times 10^{-5}}2.28×10−5\mathbf{2.28\times 10^{-5}}1.40×10−5\mathbf{1.40\times 10^{-5}}
Rel L2L^{2}3.58×10−53.58\times 10^{-5}2.34×10−5\mathbf{2.34\times 10^{-5}}3.18×10−5\mathbf{3.18\times 10^{-5}}1.82×10−5\mathbf{1.82\times 10^{-5}}
Table 5. Relative errors of the macroscopic moments ( ρ,ux,uy,T\rho,u_{x},u_{y},T ) for the 2D smooth problem across different Knudsen numbers ( Kn∈{1.0,0.1,0.01}\text{Kn}\in\{1.0,0.1,0.01\} ).
KnLoss functionMetricρ\boldsymbol{\rho}ux\boldsymbol{u_{x}}uy\boldsymbol{u_{y}}T\boldsymbol{T}
1.01.0Standard L2L^{2} LossRel L1L^{1}8.09×10−38.09\times 10^{-3}3.32×10−23.32\times 10^{-2}3.75×10−23.75\times 10^{-2}1.35×10−21.35\times 10^{-2}
Rel L2L^{2}9.38×10−39.38\times 10^{-3}3.28×10−23.28\times 10^{-2}3.51×10−23.51\times 10^{-2}2.32×10−22.32\times 10^{-2}
Relative Loss [ 29 ]Rel L1L^{1}1.70×10−21.70\times 10^{-2}2.94×10−22.94\times 10^{-2}3.03×10−23.03\times 10^{-2}8.00×10−38.00\times 10^{-3}
Rel L2L^{2}2.18×10−22.18\times 10^{-2}2.82×10−22.82\times 10^{-2}2.82×10−22.82\times 10^{-2}1.19×10−21.19\times 10^{-2}
Proposed LossRel L1L^{1}7.10×10−3\mathbf{7.10\times 10^{-3}}1.59×10−2\mathbf{1.59\times 10^{-2}}1.59×10−2\mathbf{1.59\times 10^{-2}}4.70×10−3\mathbf{4.70\times 10^{-3}}
Rel L2L^{2}7.75×10−3\mathbf{7.75\times 10^{-3}}1.13×10−2\mathbf{1.13\times 10^{-2}}1.11×10−2\mathbf{1.11\times 10^{-2}}6.53×10−3\mathbf{6.53\times 10^{-3}}
Table 6. Relative errors of the macroscopic moments ( ρ,ux,uy,T\rho,u_{x},u_{y},T ) for the 2D Riemann problem across different Knudsen numbers ( Kn∈{1.0,0.1,0.01}\text{Kn}\in\{1.0,0.1,0.01\} ).
KnLoss functionMetricρ\boldsymbol{\rho}ux\boldsymbol{u_{x}}uy\boldsymbol{u_{y}}uz\boldsymbol{u_{z}}T\boldsymbol{T}
1.01.0Standard L2L^{2} LossRel L1L^{1}6.07×10−56.07\times 10^{-5}8.18×10−58.18\times 10^{-5}7.86×10−57.86\times 10^{-5}1.40×10−41.40\times 10^{-4}3.13×10−43.13\times 10^{-4}
Rel L2L^{2}8.25×10−58.25\times 10^{-5}1.23×10−41.23\times 10^{-4}1.19×10−41.19\times 10^{-4}1.99×10−41.99\times 10^{-4}4.55×10−44.55\times 10^{-4}
Relative Loss [ 29 ]Rel L1L^{1}1.90×10−51.90\times 10^{-5}1.38×10−5\mathbf{1.38\times 10^{-5}}1.63×10−51.63\times 10^{-5}1.37×10−5\mathbf{1.37\times 10^{-5}}2.75×10−52.75\times 10^{-5}
Rel L2L^{2}2.59×10−52.59\times 10^{-5}1.93×10−5\mathbf{1.93\times 10^{-5}}2.38×10−52.38\times 10^{-5}1.98×10−5\mathbf{1.98\times 10^{-5}}4.50×10−54.50\times 10^{-5}
Proposed LossRel L1L^{1}1.78×10−5\mathbf{1.78\times 10^{-5}}1.72×10−51.72\times 10^{-5}1.22×10−5\mathbf{1.22\times 10^{-5}}1.67×10−51.67\times 10^{-5}2.28×10−5\mathbf{2.28\times 10^{-5}}
Rel L2L^{2}2.39×10−5\mathbf{2.39\times 10^{-5}}2.38×10−52.38\times 10^{-5}1.71×10−5\mathbf{1.71\times 10^{-5}}2.30×10−52.30\times 10^{-5}3.21×10−5\mathbf{3.21\times 10^{-5}}
Table 7. Relative errors of the distribution function ff and macroscopic moments ( ρ,ux,T\rho,u_{x},T ) for the 3D smooth problem across different Knudsen numbers ( Kn∈{1.0,0.1,0.01}\text{Kn}\in\{1.0,0.1,0.01\} ).
Figure 8. Comparison of weight shapes between the relative loss and the proposed loss for the 1D smooth problem at Kn=0.01\text{Kn}=0.01 .
Figure 9. Comparison of weight shapes between the relative loss and the proposed loss for the 1D Riemann problem at Kn=1.0\text{Kn}=1.0 .
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