Grazing-aware kinetic boundary loss

(1) Integration point: apply this to a PINN, neural operator, or graph neural network that predicts $f_\theta(x,v)$ for a transport or kinetic PDE. Modify only the phase-boundary condition loss on $(x,v)\in\partial\Omega\times V$; keep the interior PDE residual $R_\theta=v\cdot\nabla_xf_\theta-\mathcal C[f_\theta]$ unchanged. For inflow conditions, set $B[f_\theta](x,v)=f_\theta(x,v)$ on $\Gamma_-$; for reflection or outgoing constraints, substitute the relevant boundary operator. (2) Pseudocode: sample boundary positions $x_i$ from a surface sampler and velocities $v_i$ from $\mu$; compute normals $n_i=n(x_i)$, $a_i=v_i^\top n_i$, and $w_i=\min(|a_i|,|a_i|^p)$. Compute $e_i=B[f_\theta](x_i,v_i)-b(x_i,v_i)$ and optimize $L=L_{\rm interior}+\lambda_{\rm bc}(\sum_iw_ie_i^2)/(\sum_iw_i+\epsilon)$. Use $p=1$ on $C^{1,1/2}$ or smoother domains; if a boundary chart is only estimated as $C^{1,\alpha}$, choose the smallest tested $p$ satisfying $\alpha\ge1/(p+1)$, and clip $w_i$ to avoid numerical underflow. (3) Computed from the mathematics: $a_i$, $w_i$, the normalization, and the regularity-based candidate values of $p$. Estimated empirically: the normal from the mesh or signed-distance network, the best $\lambda_{\rm bc}$, and whether $p=1,2,3$ improves validation error. To avoid eliminating all near-grazing information, use a mixture sampler: with probability $1-\rho$ draw ordinary boundary points and with probability $\rho$ draw $|a|<\delta$ points, while retaining the same weighted loss. (4) First experiment: solve 2D free transport or BGK on a unit disk and on a polygonal/rounded domain using a small MLP or Fourier neural operator. Compare uniform boundary MSE, $|v\cdot n|$ weighting, and $\omega_2$ weighting at equal samples and optimizer budget. Measure maximum inflow violation, weighted trace error, interior $L^2$ error, and residual spikes near grazing. Success is lower worst-case boundary violation and smoother training, especially on curved domains or with Gaussian velocities, without increasing interior residual or requiring more boundary samples.