△ Mechanism confirmed, baseline not beaten 2026

Work-trained neural Hamiltonian bridge

Implementation & benchmark of arXiv:2607.15682 — Neural Non-Equilibrium Hamiltonian Monte Carlo for Corrected Boltzmann Sampling

Usefulness7/10

Difficulty6/10

Novelty7/10

Sampling Accuracy Sample efficiency Stability Diffusion sampling Optimizer Training Hamiltonian Stochastic processes Probability Information theory

Source paper: Neural Non-Equilibrium Hamiltonian Monte Carlo for Corrected Boltzmann Sampling arXiv:2607.15682 ⓘ · analyzed Aug 30, 2026

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 Not peer-reviewed.

Idea description

Train a neural finite-time Hamiltonian-style path from an easy base density to a Boltzmann target by minimizing its generalized nonequilibrium work. The work is a path-space log-density ratio, so its mean is a forward KL divergence up to a constant and the endpoint marginal mismatch is bounded by the same quantity. Unlike an uncorrected neural sampler, this produces a global proposal whose bias and overlap can be measured quantitatively.

Formulas

$$\pi(x)=\frac{1}{Z}\exp[-E(x)]$$

$$W_\theta(\tau)=\log Q_\theta(\tau)-\log R(\tilde\tau)+E(x_T)-E(x_0)$$

$$\mathbb{E}_{Q_\theta}[W_\theta]=\mathrm{KL}(Q_\theta\|P^*)+C,\qquad \mathrm{KL}(q_{\theta,T}\|\pi)\leq\mathrm{KL}(Q_\theta\|P^*)$$

$$\theta\leftarrow\theta-\eta\nabla_\theta\frac{1}{B}\sum_{b=1}^{B}W_\theta(\tau^{(b)})$$

Mathematical statement

The target configuration density is $\pi(x)=Z^{-1}\exp[-E(x)]$, where $x$ is the configuration, $E(x)$ is its energy, and $Z=\int\exp[-E(x)]dx$ is the unknown normalizer. Let $\rho_0(z)$ be a tractable base density over an augmented state $z=(x,p)$, let $Q_\theta(\tau)$ be the normalized density of a complete neural trajectory $\tau=(z_0,z_1,\ldots,z_T)$, and let $R(\tilde\tau)$ be the density of a time-reversed reference trajectory $\tilde\tau=(\mathcal Rz_T,\ldots,\mathcal Rz_0)$, where $\mathcal R(x,p)=(x,-p)$. The generalized work is the computable path ratio $W_\theta(\tau)=\log Q_\theta(\tau)-\log R(\tilde\tau)+E(x_T)-E(x_0)$, up to a parameter-independent free-energy constant. If $P^*(\tau)\propto R(\tilde\tau)\exp[-E(x_T)+E(x_0)]$ is the reverse-compatible target path law, then $\mathbb E_{Q_\theta}[W_\theta]=\mathrm{KL}(Q_\theta\|P^*)+C$, with $C$ independent of $\theta$. By data processing, $\mathrm{KL}(q_{\theta,T}\|\pi)\leq\mathrm{KL}(Q_\theta\|P^*)$, where $q_{\theta,T}$ is the endpoint density.

Implementation notes

Integrate this at the sampler level. Use an augmented state $z_t=(x_t,p_t)$, sample $z_0\sim\rho_0$, and run $T$ learned transitions. A simple starting transition is $p_{t+1}=p_t-\epsilon_t\nabla_xE(x_t)-\epsilon_t f_\theta(x_t,p_t,t)+\sigma_t\xi_t$ followed by $x_{t+1}=x_t+\epsilon_t p_{t+1}$, with $\xi_t\sim\mathcal N(0,I)$. Accumulate $\log Q_\theta(\tau)=\log\rho_0(z_0)+\sum_t\log q_\theta(z_{t+1}|z_t)$. Define a fixed reverse transition $r(z_t|z_{t+1})$, accumulate $\log R(\tilde\tau)=\sum_t\log r(\mathcal Rz_t|\mathcal Rz_{t+1})+\log\rho_T(\mathcal Rz_T)$, and compute the stated work from the two path log densities and endpoint energies. Backpropagate the minibatch mean work through reparameterized Gaussian noise. The exact mathematical quantities are path log probabilities, endpoint energy differences, and work; endpoint KL and mixing must be estimated empirically. First test on a 2-D double-well and a 2-D Gaussian mixture, comparing against HMC and an uncorrected neural proposal at equal energy-gradient calls. Success means lower mean and variance of work, more frequent mode switching, and lower endpoint MMD. Failure signals are exploding work variance, momentum collapse, or endpoint mode imbalance.

Verification

Mechanism confirmed, baseline not beaten

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: No measurable effect

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

Implemented a reparameterized neural Hamiltonian-style bridge and an exact Gaussian path-law sanity check. The mechanism manifested: mean work scaled quadratically with displacement (observed slope 0.5366 versus predicted 0.5346, 0.36% error), the endpoint/path KL ratio matched 0.9352, and the endpoint KL bound held with zero violation. In the double-well mini-experiment, training reduced mean work and energy versus the uncorrected proposal, but work variance remained very large and the training loss was not consistently decreasing, so this is mechanism validation rather than a robust sampler win.

Agent confidence: 8/10

Baseline: Uncorrected double-well proposal: work mean 76.19, work std 1562.43, mode-positive fraction 0.5016, mean energy 2.5473.
Idea: Work-trained bridge: work mean 12.33, work std 201.38, mode-positive fraction 0.4929, mean energy 1.2871. Exact checks: quadratic slope 0.5366 observed versus 0.5346 predicted; endpoint/path KL ratio 0.9352 observed versus 0.9352 predicted; maximum endpoint-bound violation 0.

Limitations: The path/reference construction is a small toy approximation rather than an exactly reversible Hamiltonian target path law. Only a 2-D double-well was used for the sampler comparison; HMC, Gaussian-mixture experiments, MMD, effective sample size, and longer stability or mode-mixing studies were not tested. Work variance remained high and optimization was not monotonic.

How to run: python3 experiment.py > results.json && cat results.json

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.15682) Sep 2, 2026

No significant difference vs. the tuned baseline (−-0.0%, p=1, wins 0 / 8 paired seeds).

Benchmark: Dynamics forecasting
Model: rnn_small
Paired seeds: 8
Baseline mean: 0.0014
Idea mean: 0.0014
Effect (Δ): 0 (−-0.0%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 1 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±0.0%
Mechanism: Confirmed ✓
Practical effect: No effect
Baseline tuning: swept over 6 configs

Limitations:

Only the built-in dynamics track was tested; no vision, sequence, tabular, HMC, exact path-density implementation, mode-mixing, MMD, or work-variance evaluation was included. The MVP uses an endpoint energy-change proxy rather than the full normalized trajectory log-density ratio.

How to run: python3 bench_experiment.py

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

Stage 2 — Benchmark Aug 31, 2026 latest Failed ✗

Agent confidence: 9/10

Built a work-inspired Hamiltonian endpoint-energy training intervention for the structurally matched dynamics/control track using the shared rnn_small architecture. The best idea configuration selected lambda=0, exactly matching the baseline; positive work penalties degraded MSE. The endpoint work proxy showed the predicted reduction, but this did not translate into a task-metric improvement, so the idea did not work on this benchmark.

Baseline: 8-seed tuned baseline MSE mean 0.00141070, std 0.00038184; best cfg lr=0.01, weight_decay=0.0001.
Idea: Best 8-seed idea MSE mean 0.00141070, std 0.00038184 at lambda=0; lambda=0.01 mean 0.00221551 and lambda=0.05 mean 0.02021107. Paired delta 0.0, permutation p=1.0.
Mechanism signature: Endpoint work proxy mean -0.09426, std 0.24113, correlation with endpoint squared error -0.50975, observed direction reduced, confirmed=true; no task-metric win.

Benchmark result