# Spectral-cusp-factored neural wavefunction

- ID: 368
- Canonical URL: https://synthcore.org/idea/368/spectral-cusp-factored-neural-wavefunction
- API JSON: https://synthcore.org/api/idea/368.json
- API Markdown: https://synthcore.org/api/idea/368.md
- Verification status: failed_benchmark
- Source: [arXiv:2607.02492](https://arxiv.org/abs/2607.02492)
- Category: architecture
- Solves: accuracy, stability, sample-efficiency
- ML areas: mlp, loss, regularization, initialization
- Math tags: pde, harmonic-analysis, approximation-theory, functional-analysis
- Ratings: usefulness 7/10; difficulty 4/10; novelty 6/10

## 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.

## 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.

## Key formulas

- $$H=-\sum_{i=1}^{N}\Delta_{x^{i}}+V(x),\qquad V(x)=-\sum_{i=1}^{N}\sum_{\ell=1}^{L}\frac{Z^{\ell}}{|x^{i}-R^{\ell}|}+\sum_{1\leq i<j\leq N}\frac{1}{|x^{i}-x^{j}|}.$$
- $$\phi=e^{-F_{\rm cut}}\psi,\qquad \psi\in\mathcal{B}_{\mathrm{sp}}^{s}\ (0\leq s<1),\qquad \phi\in\mathcal{B}_{\mathrm{sp}}^{s}\ (0\leq s<2).$$
- $$F_{\rm cut}(x)= -\sum_{i=1}^{N}\sum_{\ell=1}^{L}Z^{\ell}\,\chi_{\rm en}(|x^{i}-R^{\ell}|)|x^{i}-R^{\ell}|+\frac12\sum_{1\leq i<j\leq N}\chi_{\rm ee}(|x^{i}-x^{j}|)|x^{i}-x^{j}|,$$
- $$\psi_{\theta}(x)=\exp(F_{\rm cut}(x))u_{\theta}(x),\qquad u_{\theta}(x)=\exp(g_{\theta}(x))\,\Phi_{\theta}(x)$$

## 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

- Status: failed_benchmark
- Mechanism evidence: yes
- Mechanism confirmed: no
- Practical verdict: harms
- Verdict: 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.

### Mechanism check

- Verdict: 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.
- Confidence: 8/10
- 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.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 0/0
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_report.json](https://synthcore.org/code/199/bench_report.json)
- [coulomb_track.py](https://synthcore.org/code/199/coulomb_track.py)
- [cusp_experiment.py](https://synthcore.org/code/199/cusp_experiment.py)
- [report.md](https://synthcore.org/code/199/report.md)
- [report_bench_2026-08-30T195653.md](https://synthcore.org/code/199/report_bench_2026-08-30T195653.md)
- [results.json](https://synthcore.org/code/199/results.json)
- [run_bench.py](https://synthcore.org/code/199/run_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/199)

## Disclaimer

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