Minkowski gradient barrier for neural fields

Integrate this at the loss level of a coordinate MLP u_ heta(x), with x\in\mathbb{R}^N and spatial derivatives obtained by automatic differentiation. Use it first for a PINN or implicit scalar-field task with known boundary samples, such as solving Poisson's equation on a unit square or fitting the distance to the boundary of a disk. The network loss should be: (i) the original PDE residual or supervised field loss, (ii) a boundary loss \mathbb{E}_{x_b\in\partial\Omega}|u_ heta(x_b)|^2, and (iii) the Minkowski barrier above. Pseudocode is: sample interior points x_i and boundary points b_j; compute u_i=u_ heta(x_i) and g_i=
abla_xu_ heta(x_i); set s_i=\|g_i\|_2^2; compute B_i=1-\sqrt{\max(\varepsilon,1-s_i)}; compute the overflow penalty O_i=\max(0,\|g_i\|_2-1+\delta)^2; form L=L_task+\lambda_bL_boundary+\beta\operatorname{mean}(B_i)+\gamma\operatorname{mean}(O_i); update heta\leftarrow heta-\eta
abla_ heta L. Use \varepsilon=10^{-6} and \delta=0.02 initially. The paper supplies the geometric flux and its singular behavior; the task loss, \beta, \gamma, and the finite overflow handling are engineering choices. Log the fraction of points with \|g_i\|>1, the 95th/99th percentile gradient norm, boundary error, task residual, and gradient norm of the parameter update. Start with a 4-layer width-128 MLP on a 64 imes64 unit-square grid, compare against no regularizer and the standard penalty \alpha\|
abla u\|^2, and train at equal steps and FLOPs. A successful result is fewer gradient violations, fewer NaNs or training collapses, lower test residual, and—when the target has zero boundary values—a closer match to the true distance profile. For a forcing sweep, increase the analogue of \mu and test whether the normalized field approaches the analytic disk distance 1-\|x\| without developing boundary-layer oscillations.