Unverified 2026

Rational-Pole Neural Field Pooling

Implementation & benchmark of arXiv:2607.17851 — Cubature from rational approximation

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Architecture Speedup Accuracy Scalability Mlp Graph nn World model Inference speedup Approximation theory Numerical analysis Linear algebra Measure theory

Source paper: Cubature from rational approximation arXiv:2607.17851 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Replace dense spatial pooling or integral evaluation over a planar domain by a sparse cubature layer whose nodes are poles of a rational approximation fitted only on the domain boundary. For analytic or nearly analytic neural-field channels, the same learned field can then be integrated using substantially fewer evaluations than a uniform grid, while the boundary approximation residual supplies a cheap reliability signal.

Formulas

$$\iint_{\Omega}f(z)\,w(z,\bar{z})\,dA=\frac{1}{2i}\oint_{\gamma}f(z)\,W(z,\bar{z})\,dz,\qquad \frac{\partial W}{\partial\bar z}=w.$$

$$r(z)=q(z)+\sum_{k=1}^{m}\frac{a_k}{z-p_k},\qquad \iint_{\Omega}f(z)w(z,\bar z)\,dA\approx\sum_{k=1}^{m}a_k f(p_k).$$

$$\left|I-I_r\right|\le \frac12\oint_{\gamma}|f(z)|\,|W(z,\bar z)-r(z)|\,|dz|.$$

$$r_{\mathrm{AAA}}(z)=\frac{\displaystyle\sum_{j=1}^{n}\frac{\alpha_j W(z_j)}{z-z_j}}{\displaystyle\sum_{j=1}^{n}\frac{\alpha_j}{z-z_j}},\qquad \alpha=\arg\min_{\alpha}\left\|W(z_i)-r_{\mathrm{AAA}}(z_i)\right\|_{i\in\mathcal R}.$$

Mathematical statement

The construction starts from the Cauchy–Green identity: for a domain $\Omega\subset\mathbb C$ with rectifiable Jordan boundary $\gamma=\partial\Omega$, analytic integrand $f$ on a neighborhood of $\overline\Omega$, weight $w$, and any $\bar\partial$-antiderivative $W$ satisfying $\partial W/\partial\bar z=w$, $\iint_\Omega f(z)w(z,\bar z)dA=(2i)^{-1}\oint_\gamma f(z)W(z,\bar z)dz$. Approximate the boundary function $W|_\gamma$ by a rational function $r$ with free poles. If $r(z)=q(z)+\sum_{k=1}^m a_k/(z-p_k)$, where $q$ is analytic in $\Omega$, $p_k\in\Omega$ are poles, and $a_k$ are residues, the residue theorem gives the neural cubature rule $\iint_\Omega f w\,dA\approx\sum_{k=1}^m a_k f(p_k)$. The approximation error is controlled by the boundary residual through $|I-I_r|\le(1/2)\oint_\gamma |f(z)|\,|W(z,\bar z)-r(z)|\,|dz|$. In practice, sample boundary points $z_j$, fit $r$ using AAA, and estimate the residual by a weighted maximum or RMS of $|W(z_j)-r(z_j)|$.

Implementation notes

Integrate this at the spatial aggregation point of a neural field or neural operator. Let the network receive a complex coordinate $z=x+iy$ and output $f_\theta(z)\in\mathbb R^c$ or $\mathbb C^c$; the desired pooled feature is $u_\theta=|\Omega|^{-1}\iint_\Omega f_\theta(z)w(z,\bar z)dA$. First choose a fixed domain, such as the unit disk or square, and precompute boundary samples $z_j$, arc-length weights $\ell_j$, a boundary antiderivative signal $W_j$, AAA barycentric coefficients $\alpha_j$, and the interior poles $p_k$ and residues $a_k$ of the resulting rational function. During each forward pass, evaluate the network only at the $m$ poles and return $u_\theta=|\Omega|^{-1}\sum_k a_k f_\theta(p_k)$, replacing an $H\times W$ grid reduction. If the geometry or weight changes, recompute AAA periodically rather than differentiating through the pole finder. Pseudocode: boundary = sample_gamma(); W = antiderivative_boundary(weight); r = AAA(boundary,W); poles,residues = interior_poles_residues(r); for batch: vals=f_theta(poles); pooled=sum(residues[:,None]*vals,axis=0)/area. Compute the empirical residual $\hat E=\frac12\sum_j\ell_j|W_j-r(z_j)|$, and optionally estimate the full bound by multiplying $\hat E$ by the largest boundary value of $|f_\theta|$. The mathematical quantities are the Cauchy–Green reduction, poles, residues, and residual-based error certificate; network behavior and held-out integration error are measured empirically. For a first cheap experiment, use a coordinate MLP representing analytic functions on the unit disk, with exact polynomial integrals as labels. Compare 8–32 rational-pole evaluations against 16$\times$16 and 32$\times$32 grid pooling and Monte Carlo at equal evaluation budgets. Success means lower integral error at equal network-call count, or at least 4x fewer calls at equal error; also test whether $\hat E$ predicts held-out integration error and whether pole locations remain stable across training checkpoints.

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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