✗ Failed on benchmark 2026

Spectral-cusp-factored neural wavefunction

Implementation & benchmark of arXiv:2607.02492 — Cut-off Jastrow Factors and Spectral Barron Regularity of Coulombic Electronic Wave Functions

Usefulness7/10

Difficulty4/10

Novelty6/10

Architecture Accuracy Stability Sample efficiency Mlp Loss Regularization Initialization Pde Harmonic analysis Approximation theory Functional analysis

Source paper: Cut-off Jastrow Factors and Spectral Barron Regularity of Coulombic Electronic Wave Functions arXiv:2607.02492 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Represent the physical wavefunction as a fixed cusp factor multiplied by a neural residual, rather than forcing the network to learn Coulomb singularities from data. Use cutoff distance features so the factor is nontrivial only near coalescences and remains numerically bounded at long range. The residual should have substantially lighter Fourier tails and therefore require less network capacity to attain a given energy or local-energy accuracy.

Formulas

$$H=-\sum_{i=1}^{N}\Delta_{x^{i}}+V(x),\qquad V(x)=-\sum_{i=1}^{N}\sum_{\ell=1}^{L}\frac{Z^{\ell}}{|x^{i}-R^{\ell}|}+\sum_{1\leq i<j\leq N}\frac{1}{|x^{i}-x^{j}|}.$$

$$\phi=e^{-F_{\rm cut}}\psi,\qquad \psi\in\mathcal{B}_{\mathrm{sp}}^{s}\ (0\leq s<1),\qquad \phi\in\mathcal{B}_{\mathrm{sp}}^{s}\ (0\leq s<2).$$

$$F_{\rm cut}(x)= -\sum_{i=1}^{N}\sum_{\ell=1}^{L}Z^{\ell}\,\chi_{\rm en}(|x^{i}-R^{\ell}|)|x^{i}-R^{\ell}|+\frac12\sum_{1\leq i<j\leq N}\chi_{\rm ee}(|x^{i}-x^{j}|)|x^{i}-x^{j}|,$$

$$\psi_{\theta}(x)=\exp(F_{\rm cut}(x))u_{\theta}(x),\qquad u_{\theta}(x)=\exp(g_{\theta}(x))\,\Phi_{\theta}(x)$$

Mathematical statement

The paper studies the Coulomb Hamiltonian $H=-\sum_{i=1}^{N}\Delta_{x^{i}}+V(x)$ on $\mathbb{R}^{3N}$, with $V(x)=-\sum_{i=1}^{N}\sum_{\ell=1}^{L}Z^{\ell}/|x^{i}-R^{\ell}|+\sum_{1\leq i<j\leq N}1/|x^{i}-x^{j}|$, where $x^{i}\in\mathbb{R}^{3}$ are electron coordinates, $R^{\ell}$ are nuclear positions, and $Z^{\ell}>0$ are nuclear charges. For an eigenfunction $\psi$, it defines the Jastrow quotient $\phi=e^{-F_{\rm cut}}\psi$ and proves $\psi\in\mathcal{B}_{\mathrm{sp}}^{s}$ for every $0\leq s<1$, while $\phi\in\mathcal{B}_{\mathrm{sp}}^{s}$ for every $0\leq s<2$; $\mathcal{B}_{\mathrm{sp}}^{s}$ denotes the paper's weighted Fourier, or spectral Barron, regularity class of order $s$. Implement the same structural cancellation with $\psi_{\theta}(x)=e^{F_{\rm cut}(x)}u_{\theta}(x)$, where $u_{\theta}$ is a neural residual. Since the extracted text does not provide the paper's exact cutoff formula, use a cusp-matched cutoff $F_{\rm cut}(x)=\sum_{i,\ell}(-Z^{\ell})\chi_{\rm en}(|x^{i}-R^{\ell}|)|x^{i}-R^{\ell}|+\sum_{i<j}(1/2)\chi_{\rm ee}(|x^{i}-x^{j}|)|x^{i}-x^{j}|$. Here $\chi_{\rm en},\chi_{\rm ee}$ are smooth functions equal to $1$ below a radius $r_{0}$ and equal to $0$ above $2r_{0}$; the coefficients $-Z^{\ell}$ and $1/2$ encode the electron–nucleus and electron–electron cusp slopes. The target property is that $u_{\theta}\approx e^{-F_{\rm cut}}\psi$ has no leading Coulomb cusp and should empirically exhibit a faster-decaying high-frequency spectrum.

Implementation notes

Integrate the factor at the neural-wavefunction output, before evaluating amplitudes, gradients, or the local energy. Inputs are electron coordinates $x$, fixed nuclei $R^{\ell}$, charges $Z^{\ell}$, and cutoff radius $r_{0}$. First compute all electron–nucleus distances $r_{i\ell}=|x^{i}-R^{\ell}|$ and electron-pair distances $r_{ij}=|x^{i}-x^{j}|$. Define a differentiable cutoff, for example $\chi(r)=1$ for $r\le r_{0}$, $\chi(r)=1-3t^{2}+2t^{3}$ for $r_{0}<r<2r_{0}$ with $t=(r-r_{0})/r_{0}$, and $\chi(r)=0$ for $r\ge2r_{0}$. Form $F_{\rm cut}$ using the displayed electron–nucleus coefficient $-Z^{\ell}$ and electron–electron coefficient $1/2$. Let the network output a scalar residual $g_{\theta}(x)$ and, optionally, an antisymmetric determinant/backflow component $\Phi_{\theta}(x)$; set $\log|\psi_{\theta}(x)|=F_{\rm cut}(x)+g_{\theta}(x)+\log|\Phi_{\theta}(x)|$. During variational Monte Carlo, compute $\nabla_x\log\psi_{\theta}$ and $\Delta_x\log\psi_{\theta}$ with automatic differentiation, so the analytic factor contributes its exact cusp derivatives rather than being approximated by the network. Train with the usual energy/local-energy objective, keeping the prefactor fixed initially; a later ablation may learn small corrections to its coefficients. For a spectral diagnostic, collect residual values $u_{\theta}(x)$ on a small coordinate grid or random Fourier probe set and estimate the high-frequency tail $T(K)=\sum_{|k|>K}|\widehat{u}_{\theta}(k)|^{2}$, comparing it with the unfactored network. The mathematics supplies the predicted regularity gain from order below $1$ to order below $2$; the exact Barron norm and constants need not be estimated. First test on hydrogen, helium, and a two-electron fixed-nuclei Coulomb problem using a 2–4 layer MLP with matched parameter count, against the same MLP without $F_{\rm cut}$. Measure variational energy, local-energy variance, gradient norm near coalescences, training steps to a fixed energy threshold, and the Fourier-tail slope. Success means lower local-energy variance and faster energy descent at equal parameters, together with steeper decay of $T(K)$ for the residual and no degradation away from coalescences.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Significantly worse ✗

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 8/10

Built a cusp-factor module and reproducible toy experiment with matched 1,217-parameter MLPs. The primary mechanism predictions were confirmed: residual cusp slope followed |a-Z| across the coefficient sweep to numerical precision, matched cancellation was zero inside the cutoff for every tested r0, and the Fourier amplitude tail steepened from -1.99 to -3.11 on the diagnostic grid. The small supervised MLP comparison did not show a win: factored validation MSE was 0.02337 versus 0.00770 baseline, so the mechanism signal is real but this setup does not establish improved training or energy accuracy.

Agent confidence: 8/10

Baseline: Unfactored MLP: validation log-amplitude MSE 0.00770; near-origin residual gradient magnitudes [0.0351, 0.0497, 0.1141]; Fourier amplitude tail slope -1.99.
Idea: Factored MLP: validation residual/physical log-amplitude MSE 0.02337; near-origin residual gradient magnitudes [0.1905, 0.1712, 0.0844]; Fourier amplitude tail slope -3.11. Analytic sweeps: observed residual slopes matched predicted |a-Z| within about 2e-11, and cutoff-radius sweeps gave inside slope 0 and outside slope 1.70 versus predictions 0 and 1.70.

Limitations: This is a hydrogen-inspired supervised log-amplitude toy problem, not a variational Monte Carlo or local-energy calculation. It uses a 1D radial Fourier proxy and does not test helium, antisymmetry, multidimensional spectral Barron norms, energy variance, optimization speed across seeds, or production-scale wavefunction architectures.

How to run: python3 cusp_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.02492) Aug 30, 2026

Significantly WORSE than the tuned baseline (+309.5%, p=0.0081, wins 0 / 0 paired seeds).

Benchmark: Coulomb_cusp_regression
Model: mlp_tiny
Paired seeds: 8
Baseline mean: 0.0076
Idea mean: 0.031
Effect (Δ): 0.0234 (+309.5%; negative = idea better)
Wins: 0 / 0 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Not confirmed ✗
Practical effect: Hurts
Baseline tuning: swept over 0 configs

Limitations:

This was a small supervised Coulombic regression proxy, not variational Monte Carlo or local-energy variance. Trained-model Fourier tails, multi-electron antisymmetry, and full wavefunction energy optimization were not tested.

How to run: python3 run_bench.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Aug 30, 2026 latest Failed ✗

Agent confidence: 10/10

Implemented a structurally matched custom Coulomb/PDE regression track and an end-to-end cusp-factored MLP using the benchmark training path, with equal 4,481-parameter systems and shared learning-rate sweeps over eight paired seeds. The baseline achieved MSE 0.007565 versus 0.030982 for the idea; paired delta was +0.023417 with permutation p=0.0081, so the idea was significantly worse. Analytic cusp cancellation passed, but the trained-model mechanism signature was not quantitatively confirmed.

Baseline: Best lr=0.01; 8-seed test MSE 0.0075653 ± 0.0014248.
Comparison: Paired delta +0.0234171, p=0.0081, idea wins 0/8.
Idea: Best lr=0.01; 8-seed test MSE 0.0309824 ± 0.0094876.
Mechanism signature: Analytic slope predicted 0.0 and observed 0.0; trained residual slopes remained [0.1113, 0.1113, 0.0313], so confirmed=false.

Benchmark result