✗ Failed on benchmark 2026

Response-from-Hessian Regularizer

Implementation & benchmark of arXiv:2608.13506 — Equivariant learning of a transferable three-dimensional classical density functional

Usefulness7/10

Difficulty6/10

Novelty7/10

Regularization Stability Accuracy Generalization Regularization Training dynamics World model Optimizer Functional analysis Linear algebra Optimization Statistical mechanics Bifurcations

Source paper: Equivariant learning of a transferable three-dimensional classical density functional arXiv:2608.13506 ⓘ · analyzed Sep 1, 2026

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 Not peer-reviewed.

Idea description

Use the learned variational functional's second functional derivative as a consistency mechanism: equilibrium susceptibility, forces, and phase stability must all be computed from the same Hessian rather than from independently trained predictors. Penalize negative or excessively ill-conditioned Hessian modes during training, while retaining soft negative modes as a detectable phase-transition signal.

Formulas

$$H_\theta=\left.\frac{\delta^2\Omega_\theta[\rho]}{\delta\rho\,\delta\rho}\right|_{\rho=\rho_*},\qquad H_\theta\,\delta\rho+\delta V=0.$$

$$\delta\rho=-H_\theta^{-1}\delta V,\qquad \|\delta\rho\|\lesssim\frac{\|\delta V\|}{\lambda_{\min}(H_\theta)}.$$

$$\mathcal L_{\mathrm{Hess}}=\alpha\,\mathbb E_v\left[\max\left(0,-\frac{v^\top H_\theta v}{v^\top v}+\epsilon\right)^2\right]+\beta\,\mathbb E_v\left[\max\left(0,\frac{v^\top H_\theta v}{v^\top v}-\kappa\right)^2\right],$$

$$\lambda_{\min}(H_\theta)\rightarrow0^+\quad\Longrightarrow\quad \chi=\left\|\frac{\delta\rho}{\delta V}\right\|\sim\lambda_{\min}(H_\theta)^{-1}\rightarrow\infty.$$

Mathematical statement

Let $\rho_*$ minimize $\Omega_\theta$, and let $H_\theta=\delta^2\Omega_\theta/\delta\rho\delta\rho\vert_{\rho_*}$ be the Hessian operator. A small perturbation $\delta V$ to the external potential induces, to linear order, $H_\theta\delta\rho+\delta V=0$, hence the density response is $\delta\rho=-H_\theta^{-1}\delta V$. For a stable minimum, $H_\theta$ is positive semidefinite after removing any conserved-number null direction; approaching a spinodal or bifurcation corresponds to $\lambda_{\min}(H_\theta)\to0^+$, causing susceptibility $\|H_\theta^{-1}\|$ to grow approximately as $1/\lambda_{\min}$. A practical regularizer estimates Rayleigh quotients or extremal eigenvalues using Hessian-vector products without constructing the dense Hessian.

Implementation notes

Attach this regularizer after each training sample has been approximately minimized. Let the network define $\Omega_\theta$, compute the equilibrium field $\rho_*$, and implement Hessian-vector products as $Hv=\nabla_\rho(\nabla_\rho\Omega_\theta\cdot v)$ using automatic differentiation. Draw 4–16 random normalized vectors $v$, estimate Rayleigh quotients $v^\top Hv$, and add the negative-curvature penalty shown above. For a better estimate of the smallest eigenvalue, run 5–10 Lanczos iterations using Hessian-vector products; never materialize $H$. Set $\epsilon$ to a numerical tolerance and choose $\kappa$ from the curvature distribution of a stable baseline instead of imposing an arbitrary scale. Do not train a separate response head: apply a small known perturbation $\delta V$, re-minimize, and compare the observed $\delta\rho$ with the linear prediction $-H^{-1}\delta V$. The paper supplies the variational and Hessian-response mechanism; the curvature penalty and stochastic spectral estimator are the proposed neural implementation. First test on a 16^3 or 32^3 density grid with synthetic external potentials, comparing unconstrained and regularized functionals. The predicted signature is a response norm with log-log slope near $-1$ versus estimated $\lambda_{\min}$, and solver instability when that eigenvalue crosses zero. Verify that regularization removes spurious negative curvature while preserving deliberately generated near-critical configurations.

Verification

Failed on benchmark

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Not reproduced ✗

Stage 2 · Practical benchmark result: Inconclusive

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

Stage 1 — Mechanism check agent confidence 9/10

Built and ran a differentiable Response-from-Hessian Regularizer MVP with autodiff Hessian-vector products, curvature penalties, susceptibility sweeps, and stability checks. The mechanism manifested clearly: HVP error was 1.1e-16, susceptibility followed the predicted 1/lambda law with log-log slope -1.000 and zero relative error, and explicit-Euler instability began at the predicted gamma=2 boundary. Regularization moved a fitted negative curvature from -0.20 to -0.021 while leaving a deliberately soft positive mode at 0.020 unchanged, indicating selective stabilization rather than indiscriminate stiffening.

Agent confidence: 9/10

Baseline: Negative-curvature fit: final curvature -0.200000, MSE 4.0e-16; soft-mode fit: 0.0200001.
Idea: Negative-curvature fit: final curvature -0.021333, MSE 0.0319; soft-mode fit: 0.0200007. Inverse-curvature sweep observed [50,25,12.5,6.25,3.125,1.5625] for lambda [0.02,0.04,0.08,0.16,0.32,0.64], exactly matching 1/lambda.

Limitations: This is a small synthetic diagonal/quartic functional rather than a learned 16^3 or 32^3 cDFT model. It does not test full approximate minimization, conserved-number projection, Lanczos extremal eigenvalue estimation, response re-minimization, wall-clock overhead, or generalization across external potentials.

How to run: python3 response_hessian_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.13506) Sep 2, 2026

Ahead of the baseline (−15.4%) but not statistically significant (p=0.1789, wins 7 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0014
Idea mean: 0.0012
Effect (Δ): -0.0002 (−15.4%; negative = idea better)
Wins: 7 / 8 paired seeds
p-value: 0.1789 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±23.9%
Mechanism: Not confirmed ✗
Practical effect: Inconclusive
Baseline tuning: swept over 3 configs

Limitations:

The benchmark used the built-in 400-sample dynamics task and 12 epochs, with a two-vector stochastic parameter-space Hessian penalty rather than a functional density Hessian or Lanczos estimator. The response signature was a parameter-space diagnostic, not physical external-potential re-minimization; no PDE/cDFT grid, conserved-number projection, wall-clock comparison, or larger-scale generalization study was performed.

How to run: python3 stage2_bench.py

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

Stage 2 — Benchmark Sep 1, 2026 latest Failed ✗

Agent confidence: 9/10

Implemented and evaluated the Response-from-Hessian Regularizer on the registered dynamics track with matched rnn_small systems, a tuned baseline sweep, and 8 paired seeds. The idea reduced mean test MSE and won 7/8 pairs, but the paired permutation p-value was 0.17895, so it failed the required significant-win criterion; its trained-model mechanism signature was also not confirmed.

Baseline: Best lr=0.01; full mean test MSE 0.0014198302378645167.
Idea: Best lr=0.01; full mean test MSE 0.0012014269523206167; delta_mean -0.00021840328554389998; p_value 0.17895; verdict no significant win.

Benchmark result