# Ground-state fractional regularizer

- ID: 131
- Canonical URL: https://synthcore.org/idea/131/ground-state-fractional-regularizer
- API JSON: https://synthcore.org/api/idea/131.json
- API Markdown: https://synthcore.org/api/idea/131.md
- Verification status: unverified
- Source: [arXiv:2608.24389](https://arxiv.org/abs/2608.24389)
- Category: regularization
- Solves: accuracy, stability, sample-efficiency
- ML areas: mlp, loss, regularization
- Math tags: functional-analysis, measure-theory, numerical-analysis, approximation-theory
- Ratings: usefulness 5/10; difficulty 5/10; novelty 7/10

## Idea description

For a coordinate network representing a field near a boundary or interface, factor the prediction as u(x)=h(x)v(x), where h is a known fractional-Hardy ground-state profile, and regularize v with a weighted nonlocal difference energy. Add the corresponding critical Hardy penalty to the loss so that the network spends capacity on the nonsingular residual v instead of relearning the boundary singularity.

## Mathematical statement

The paper defines the fractional Laplacian kernel M(x,y)=|x-y|^{-\alpha-1} for 0<\alpha<2, with normalization A_\alpha=2^\alpha\Gamma((1+\alpha)/2)/(\sqrt{\pi}|\Gamma(-\alpha/2)|), and the fractional quadratic energy E[u]=(1/2)\int_{\mathbb R^2}(u(x)-u(y))^2 A_\alpha M(x,y)\,dy\,dx. Its ground-state family is h_{\beta,s}(x)=|x|^\beta for x>0 and s|x|^\beta for x<0; for p=2 the square-root profile is h_{2,\beta,s}(x)=h_{\beta,s}(x)^{1/2}. The ground-state representation decomposes the p-energy into a Hardy-weight term plus a nonnegative Bregman/difference remainder after writing u=h_{2,\beta,s}v. The critical asymmetric optimization has the structure sup_{s>0}\int |u|^pW_{k,p,s}=\int |u|^pW_{k,p,1}+\mathcal G_{k,p}(u), where \mathcal G_{k,p}(u)\ge0; this motivates maximizing or softly approximating the best left/right scale parameter during training. The explicit coefficient function is \gamma(\alpha,\rho)=\int_0^1 ((t^\rho-1)(1-t^{\alpha-1-\rho}))/(1-t)^{\alpha+1}\,dt, equivalently -\gamma(\alpha,\rho)=\sin(\pi(\alpha/2-\rho))B(\rho+1,\alpha-\rho)/\sin(\pi\alpha/2)-1/\alpha, for \rho\in(-1,\alpha), where B is the beta function. In the adaptation, \alpha is the fractional order, \beta is the prescribed singularity exponent, s>0 is a learnable asymmetry, and v_    heta is the neural residual.

## Key formulas

- $$\mathcal{E}[u]=\frac{1}{2}\int_{\mathbb R}\int_{\mathbb R}(u(x)-u(y))^2\,\mathcal A_\alpha |x-y|^{-\alpha-1}\,dy\,dx.$$
- $$h_{\beta,s}(x)=|x|^{\beta}\begin{cases}1,&x>0,\\s,&x<0,\end{cases}\qquad h_{2,\beta,s}(x)=h_{\beta,s}(x)^{1/2}.$$
- $$\mathcal R_{\alpha,\beta,s}[v]=\frac{\mathcal A_\alpha}{2}\int\!\!\int h_{2,\beta,s}(x)h_{2,\beta,s}(y)\bigl(v(x)-v(y)\bigr)^2|x-y|^{-\alpha-1}\,dx\,dy+\lambda\int |h_{2,\beta,s}(x)v(x)|^2|x|^{-\alpha}\,dx.$$
- $$\gamma(\alpha,\rho)=\int_0^1\frac{(t^{\rho}-1)(1-t^{\alpha-1-\rho})}{(1-t)^{\alpha+1}}\,dt,\qquad -\gamma(\alpha,\rho)=\frac{\sin\!\left(\pi(\alpha/2-\rho)\right)}{\sin(\pi\alpha/2)}B(\rho+1,\alpha-\rho)-\frac{1}{\alpha}.$$

## Implementation notes

1. Integration point: use a one-dimensional coordinate MLP or PINN that predicts a scalar field on x\in[-1,1], with a known singular or interface location shifted to x=0. Replace the final output by u_    heta(x)=h_{2,\beta,s}(x)v_    heta(x), where \beta is estimated from boundary asymptotics or selected from a small grid and s=\exp(a) is either fixed or learned. Add the regularizer \mathcal R_{\alpha,\beta,s} to the task loss; the first double integral is evaluated on sampled point pairs. 2. Pseudocode: sample collocation points x_i and pairs (x_i,y_i); compute h_i=(|x_i|^\beta(1_{x_i>0}+s1_{x_i<0}))^{1/2}; compute v_i=v_    heta(x_i), u_i=h_i v_i; set E_pair=mean[h_i h_j (v_i-v_j)^2 |x_i-y_j|^{-\alpha-1}]; set E_hardy=mean[u_i^2 |x_i|^{-\alpha}]; optimize L=L_task+\lambda_f\mathcal A_\alpha E_pair/2+\lambda_h E_hardy. Exclude identical pairs and clip |x-y| below a small \varepsilon for finite-precision stability. 3. Compute \mathcal A_\alpha exactly from the displayed gamma-function formula. Use \gamma(\alpha,\rho) to initialize \lambda_h, with \rho=\beta/2 or a validation-selected value; do not assume the theoretical coefficient remains sharp for arbitrary data. Estimate the optional optimal asymmetry by one or two gradient steps on a=\log s, or by evaluating a small set of s values and choosing the lowest validation loss. 4. First experiment: train a four-layer width-128 tanh MLP on one-dimensional fractional Poisson data whose solution is u(x)=|x|^\beta g(x), comparing ordinary MSE plus an unweighted fractional penalty against the ground-state parameterization. Use 2,048 collocation points and 32,768 random pairs. Measure relative L2 error, boundary-error scaling, gradient variance, and convergence at equal pair-evaluation FLOPs. Success means lower error near x=0 and a smoother loss curve, ideally reaching a fixed relative error with fewer collocation samples; also test sensitivity to the chosen \beta and whether learned s improves asymmetric interface cases.

## Disclaimer

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