# Radial singularity-aware collocation

- ID: 2354
- Canonical URL: https://synthcore.org/idea/2354/radial-singularity-aware-collocation
- API JSON: https://synthcore.org/api/idea/2354.json
- API Markdown: https://synthcore.org/api/idea/2354.md
- Verification status: unverified
- Source: [arXiv:2608.15563](https://arxiv.org/abs/2608.15563)
- Category: sampling
- Solves: stability, sample-efficiency, accuracy
- ML areas: loss, sampling, mlp, training
- Math tags: numerical-analysis, geometry, approximation-theory, differential-geometry
- Ratings: usefulness 6/10; difficulty 5/10; novelty 8/10

## Idea description

Use the scaled-boundary radial coordinate to control collocation density near a known point singularity or sharp source located at the scaling center. Instead of sampling \(\rho\) uniformly, choose a transformed or Gauss-Jacobi radial rule whose weights analytically cancel the geometric factor \(\rho^{d-1}\), reducing gradient variance and resolving singular fields with fewer points.

## Mathematical statement

The scaled-boundary map has radial Jacobian factor \(\rho^{d-1}\), where \(d\) is the ambient dimension. If an integrand behaves near the center as \(f(\bm\varphi(\rho,\bm t))\sim\rho^{-\beta}g(\rho,\bm t)\), then the volume integrand behaves as \(\rho^{d-1-\beta}g(\rho,\bm t)\). For \(\beta<d\), the integral is finite despite the pointwise singularity. Integrate the radial factor with a Gauss-Jacobi rule for the measure \(\rho^\alpha(1-\rho)^\gamma d\rho\), using \(\alpha=d-1-\beta\) and typically \(\gamma=0\), while evaluating the nonsingular remainder \(g\). If the neural output itself is singular, use \(u_\theta(\bm x)=(\rho(\bm x)+\varepsilon)^{-\beta}v_\theta(\bm x)\), where \(v_\theta\) is a regular MLP, \(\varepsilon>0\) prevents numerical blow-up, and \(\rho(\bm x)\) is the scaled-boundary radial coordinate. The exponent \(\beta\) must come from the PDE's known asymptotics or be treated as an experimental hyperparameter.

## Key formulas

- $$\bm\varphi_i(\rho,\bm t)=\bm x_0+\rho\bigl(\bm s_i(\bm t)-\bm x_0\bigr),\qquad J_i(\rho,\bm t)=\rho^{d-1}\det\Bigl[\bm s_i(\bm t)-\bm x_0\quad \partial_{t_1}\bm s_i(\bm t)\quad\cdots\quad\partial_{t_{d-1}}\bm s_i(\bm t)\Bigr].$$
- $$f\bigl(\bm\varphi_i(\rho,\bm t)\bigr)\sim\rho^{-\beta}g(\rho,\bm t)\quad\Longrightarrow\quad f\bigl(\bm\varphi_i(\rho,\bm t)\bigr)J_i(\rho,\bm t)\sim\rho^{d-1-\beta}g(\rho,\bm t)C_i(\bm t).$$
- $$\int_0^1\rho^{d-1-\beta}g(\rho)\,d\rho\;\approx\;\sum_{q=1}^{p}w_q^{(\alpha,0)}g(\rho_q),\qquad \alpha=d-1-\beta,$$
- $$u_\theta(\bm x)=\bigl(\rho(\bm x)+\varepsilon\bigr)^{-\beta}v_\theta(\bm x),\qquad \rho(\bm x)\text{ defined by }\bm x=\bm x_0+\rho(\bm s_i(\bm t)-\bm x_0).$$

## Implementation notes

1. Integration point: replace only the radial sampler or radial quadrature inside an SBC-based PINN loss for a PDE with a point source, corner singularity, crack tip, or prescribed inverse-power behavior. Set \(\bm x_0\) at the singular location and retain the boundary-patch parameter rule. 2. Pseudocode: specify or estimate `beta`; set `alpha = d - 1 - beta`; generate Gauss-Jacobi nodes and weights for the measure \(\rho^\alpha(1-\rho)^0d\rho\); for every patch and patch parameter node compute `x = x0 + rho*(s_i(t)-x0)`, evaluate the PDE residual, and multiply by the patch determinant and Jacobi weight, without multiplying by \(\rho^\alpha\) a second time. If the solution is singular, train `v_theta` and define `u = (rho + eps)**(-beta) * v_theta`; omit the exact apex because derivatives may be undefined. 3. The paper supplies the radial Jacobian cancellation principle and motivates generalized radial scalings and Gauss-Jacobi quadrature. The exponent \(\beta\), quadrature order, \(\varepsilon\), and any clipping are problem-dependent and must be ablated. Verify the implementation using \(\int_0^1\rho^{d-1-\beta}d\rho=1/(d-\beta)\). 4. First experiment: solve a 2D or 3D Poisson problem with a known Green-function singularity in a polygon or polyhedron. Compare uniform radial sampling, ordinary Gauss-Legendre SBC, and the Jacobi rule at equal residual evaluations. Track relative error near the singularity, global \(L^2\) error, gradient-norm variance, and optimizer stability. Success means lower near-source error and fewer exploding-gradient events at fixed compute, while preserving or improving far-field accuracy.

## Disclaimer

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