Dissipation-Budgeted Nonreversible Sampling

(1) Integration point: modify the reverse-time drift of a small score-based diffusion sampler or the drift of an overdamped Langevin sampler. Keep the learned baseline drift $b_0$ fixed and add $u_\theta(x,t)$, represented by a small MLP. Define a target set as a ball around a desired mode, or as a high-density region under a known toy target. (2) Pseudocode: initialize paired baseline and driven states with the same $x_0$ and Brownian increments; at each step compute $x\leftarrow x+[b_0(x,t)+u_\theta(x,t)]\Delta t+\sqrt{2D\Delta t}\epsilon$; accumulate $q\leftarrow q+u_\theta^\top(2D)^{-1}u_\theta\Delta t$; stop a trajectory at its first entry into the target set; estimate $S(t)$ and $r(t)=-\Delta\log S(t)/\Delta t$. Train or tune $u_\theta$ using mean hitting time minus $\lambda q$, or enforce $q\leq Q_{\max}$ with a Lagrange multiplier or per-step projection. (3) Compute survival curves, first-passage times, terminal sample quality, and quadratic dissipation directly. Fit the unknown conversion coefficient $C$ only on calibration trajectories, then evaluate the bound on held-out trajectories. (4) First cheap experiment: use a two-dimensional double-well potential and compare baseline Langevin, an unconstrained driven sampler, and the dissipation-budgeted sampler over a sweep of drift scales. The falsifiable prediction is that $\log(r_1/r_0)$ versus $Q_{\rm ex}$ is approximately linear for rare events and does not systematically exceed the fitted exponential ceiling; for nonconstant hazards, integrated hazard gain should initially grow linearly with perturbation strength and then saturate. Also verify that faster hitting does not cause an unacceptable increase in terminal KL divergence.