Unverified 2026

Capacitary Boundary Regularizer

Implementation & benchmark of arXiv:2608.26663 — Capacitary-Distance Hardy Inequality

Usefulness5/10

Difficulty6/10

Novelty8/10

Regularization Stability Accuracy Generalization Mlp Loss Regularization Functional analysis Geometry Measure theory Optimization

Source paper: Capacitary-Distance Hardy Inequality arXiv:2608.26663 ⓘ · analyzed Aug 29, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Add an inverse-capacitary-distance penalty to coordinate-network outputs near complex forbidden sets, rather than using only Euclidean distance-to-boundary weighting. The penalty is theoretically compatible with the network's spatial Dirichlet energy: it suppresses large values near obstacles while the gradient penalty controls the weighted singularity, even when the obstacle is thin, perforated, or fractal-like.

Formulas

$$\operatorname{cap}(K):=\inf\left\{\int_{\mathbb{R}^{n}}|\nabla\varphi|^{2}\,dx:\ \varphi\in C_{c}^{\infty}(\mathbb{R}^{n}),\ \varphi\geq 1\ \text{on a neighborhood of }K\right\}.$$

$$d_{\alpha}(x):=\inf\left\{r>0:\operatorname{cap}\!\left(\overline{F\cap B(x,r)}\right)\geq\alpha\operatorname{cap}(B(\mathbf{0},r))\right\},\qquad F=\mathbb{R}^{n}\setminus\Omega.$$

$$\int_{\Omega}\frac{|u(x)|^{2}}{d_{\alpha}(x)^{2}}\,dx\leq\frac{C_{n}}{\alpha^{2}}\int_{\Omega}|\nabla u(x)|^{2}\,dx,$$

$$\mathcal{L}(\theta)=\mathcal{L}_{\mathrm{task}}(\theta)+\lambda\,\mathbb{E}_{x\sim\Omega}\left[\frac{|u_{\theta}(x)|^{2}}{(d_{\alpha}(x)+\varepsilon)^{2}}+\frac{\alpha^{2}}{C_n}|\nabla_xu_{\theta}(x)|^{2}\right].$$

Mathematical statement

For n>=3, let Omega subset R^n be open, F=R^n\Omega its forbidden set, and cap(K)=inf{int_R^n |grad phi|^2 dx: phi in C_c^infty(R^n), phi>=1 on a neighborhood of K} the Newtonian capacity. For alpha in (0,1], define d_alpha(x)=inf{r>0: cap(cl(F intersection B(x,r))) >= alpha cap(B(0,r))}, where B(x,r) is the radius-r Euclidean ball and x is in Omega. The paper proves the Hardy inequality int_Omega |u(x)|^2/d_alpha(x)^2 dx <= (C_n/alpha^2) int_Omega |grad u(x)|^2 dx for every compactly supported smooth u, with C_n depending only on n. It also constructs domains for which the best constant is at least c_n/alpha^2, so the alpha^-2 scaling cannot generally be improved. We use d_alpha as a geometry-dependent weight for a neural field u_theta and retain the right-hand Dirichlet energy as a stabilizing loss. The theorem strictly applies to smooth compactly supported functions; for an MLP, enforce approximate compact support with a smooth domain window and validate the discretized inequality empirically.

Implementation notes

Integrate this into a coordinate MLP u_theta(x) used for a neural field, implicit geometry model, or PDE surrogate on a 3D voxel domain. Represent the forbidden set F by occupied or invalid voxels and Omega by valid voxels. Precompute d_alpha on valid grid points. For each query point x and candidate radius r, estimate cap(F intersect B(x,r)) by solving a local discrete capacity problem: on a grid graph inside B(x,r), minimize the sum over edges (phi_i-phi_j)^2 subject to phi_i=1 on forbidden nodes and phi_i=0 on the outer shell. Estimate cap(B(0,r)) with the same graph solver on a centered full ball. Use logarithmically spaced radii and choose the smallest r whose estimated ratio is at least alpha; trilinearly interpolate d_alpha during training. Pseudocode: sample x from Omega; evaluate u=u_theta(x) and g=grad_x(u); retrieve d=d_alpha(x); compute L_cap=(u^2/(d+eps)^2)+(alpha^2/C_est)*||g||^2; optimize L_task+lambda*mean(L_cap) with AdamW. C_est is not assumed known: estimate it from the maximum observed ratio mean(u^2/(d+eps)^2)/mean(||g||^2) on random smooth test fields, while alpha and eps are hyperparameters. Use a smooth window w(x) that vanishes at the outer domain boundary and train u_theta=w(x)f_theta(x) to approximate compact support. First experiment: fit a Fourier-feature MLP to a signed-distance or Poisson solution in a 64^3 cube containing many thin plates and small holes. Compare no geometric regularizer, Euclidean distance weighting, and capacitary weighting at equal parameter count and training steps. Measure boundary overshoot, validation error near obstacles, gradient norm variance, and the empirical Hardy ratio; success is lower near-obstacle error and fewer unstable spikes without increasing total gradient energy.

Verification

This idea has not been verified yet.

Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.

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