{
 "artifacts": null,
 "category": "regularization",
 "description": "Add a differentiable penalty that encourages a neural implicit field to have a controlled local homogeneity degree across concentric spatial scales. The penalty compares the flux-normalized frequency at adjacent radii, optionally targeting a desired degree k, so the network is discouraged from producing scale-inconsistent or oscillatory local geometry.",
 "formulas_latex": [
  "$$\\operatorname{div}(|Du|^{p-2}Du)=0,\\qquad 1\u003cp\u003c\\infty.$$",
  "$$N_{*}(x_{0},r)=\\frac{r\\displaystyle\\int_{B_{r}(x_{0})}|Du|^{p}\\,dx}{\\displaystyle\\int_{\\partial B_{r}(x_{0})}|Du|^{p-2}(u-u(x_{0}))^{2}\\,dS}.$$",
  "$$N_{2}(x_{0},r)=\\frac{r\\int_{B_{r}(x_{0})}|Du|^{2}\\,dx}{\\int_{\\partial B_{r}(x_{0})}u^{2}\\,dS}.$$",
  "$$\\mathcal{L}_{\\mathrm{freq}}=\\frac{1}{|\\mathcal{C}|}\\sum_{x_{0}\\in\\mathcal{C}}\\frac{1}{J-1}\\sum_{j=0}^{J-2}\\left[N_{*}(x_{0},2r_j)-N_{*}(x_{0},r_j)\\right]^2+\\lambda_k\\left[N_{*}(x_{0},r_j)-k\\right]^2.$$"
 ],
 "id": 2905,
 "implementation": "Integrate the regularizer into the loss of a coordinate MLP f_theta(x), neural SDF, or other implicit field with spatial automatic differentiation. At each training step, sample centers x_0 from valid domain locations and radii r_j=r_min*2^j that remain inside the domain. For every radius, draw M approximately uniform interior points y_m in the disk using y_m=x_0+r_j*sqrt(U_m)*[cos(theta_m),sin(theta_m)], and Q approximately uniform boundary points z_q=x_0+r_j*[cos(phi_q),sin(phi_q)]. Evaluate u=f_theta(x), retain u_0=f_theta(x_0), and obtain Du through automatic differentiation. Estimate the interior energy by E_hat=(pi*r_j^2/M)*sum_m ||Du(y_m)||^p. Estimate the boundary term by H_hat=(2*pi*r_j/Q)*sum_q (||Du(z_q)||+epsilon)^(p-2)*(u(z_q)-u_0)^2. Form N_hat[j]=r_j*E_hat/(H_hat+epsilon), where epsilon is a small positive stabilizer such as 1e-8 times a running energy scale. Compute L_freq as the mean squared difference between adjacent N_hat values, optionally plus lambda_k times the squared deviation from a known degree k. Optimize L_task+lambda_freq*L_freq. The paper's mathematical object is N_*; Monte Carlo quadrature, epsilon, p, and loss weights are engineering choices. Begin with p=2 or p=3 and gradient clipping because small boundary denominators can create large derivatives. The first experiment should fit a 2D coordinate MLP to analytic homogeneous fields such as u(x,y)=x^2-y^2 or u=r^k cos(k theta), comparing vanilla MSE with MSE plus the regularizer at equal model size and FLOPs. Evaluate interpolation and extrapolation error across radii, variance of N_hat across scales, gradient smoothness, and optimization stability. Then test a neural SDF on simple shapes under sparse point supervision; success means lower unseen-point error and fewer small-scale geometric artifacts without worse zero-level-set accuracy.",
 "math_summary": "The paper defines the planar p-harmonic equation div(|Du|^{p-2}Du)=0 for 1\u003cp\u003cinfty, where u is a scalar field, Du is its spatial gradient, and div is divergence. For a center x_0 and radius r, it defines the flux-normalized frequency N_*(x_0,r)=r E_p(x_0,r)/H_p(x_0,r), with E_p(x_0,r)=integral over B_r(x_0) of |Du(x)|^p dx and H_p(x_0,r)=integral over the boundary partial B_r(x_0) of |Du(x)|^{p-2}(u(x)-u(x_0))^2 dS. Here B_r(x_0) is the disk or ball of radius r, partial B_r(x_0) is its boundary, dS is surface measure, and p is a tunable exponent. The paper states that N_* exactly recovers the degree of homogeneity for homogeneous p-harmonic profiles. We use this scale behavior as a regularization signal: when the desired degree k is known, penalize (N_*(x_0,r)-k)^2; when it is unknown, penalize log-scale variation [N_*(x_0,2r)-N_*(x_0,r)]^2. Neural fields are not automatically p-harmonic, so this is an inductive bias rather than a guaranteed monotonicity theorem.",
 "math_tags": [
  "pde",
  "geometry",
  "calculus-of-variations"
 ],
 "ml_areas": [
  "loss",
  "regularization",
  "mlp",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.30847",
  "arxiv_url": "https://arxiv.org/abs/2608.30847",
  "summary_what_math_gives_to_ml": "The paper supplies a nonlinear, gradient-weighted analogue of Almgren frequency that measures the local homogeneity degree of a function without requiring a linear PDE. Its transferable asset is the scale-invariant ratio between interior p-energy and a boundary amplitude normalization weighted by |Du|^{p-2}; unlike an ordinary gradient penalty, it directly tests whether a learned field has a prescribed power-law behavior around a point. The most practical neural-network use is as a local scale-consistency regularizer for neural implicit fields, signed-distance models, or multiscale coordinate networks, with the ratio evaluated on differentiable concentric patches. This should be treated as an empirical regularizer rather than assuming exact monotonicity, because the paper's theorem is specific to planar p-harmonic functions.",
  "title": "An Almgren-type formula for planar $p$-harmonic functions",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 5
 },
 "solves": [
  "accuracy",
  "stability",
  "generalization"
 ],
 "title": "Flux-Frequency Homogeneity Regularizer",
 "url": "https://synthcore.org/idea/2905/flux-frequency-homogeneity-regularizer",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}