Dilation-Matched Metropolized Dynamics

Integrate the method at the sampler force-evaluation point, rather than inside the neural-network forward pass. Begin with a scalar or low-dimensional latent target so the dilation operation $x\mapsto bx$ is well defined. Let the target energy be $U_N(x)=E_\theta(x)+\kappa W_N(x)$, where $E_\theta$ is a differentiable neural energy and $W_N$ is the explicit rough feature. At each leapfrog step, use autodiff to compute $g_\theta=\nabla E_\theta(x)$. Compute the rough field outside autodiff as $d_N=[W_N(x)-aW_N(bx)]/x$, or use the finite-closure expression $d_N=[\phi(x)-a^{N+1}\phi(b^{N+1}x)]/x$. Set $G_N=g_\theta+\kappa d_N$, then apply the kick-drift-kick equations above. At the end, optionally negate momentum to make reversibility explicit, and accept using the full target energy $U_N$, not the surrogate field. Restrict the initial test to $x\geq x_{\min}>0$ to avoid the quotient singularity; separately test an additive quotient near zero. The paper determines the scale recursion and closure, while acceptance rate, integrated autocorrelation time, effective sample size, and stable step size are measured empirically. Use a 2D target with a small MLP energy and resolutions $N\in\{4,8,12,16\}$. Compare classical autodiff HMC, the matched-quotient sampler, and random-walk Metropolis at equal energy evaluations. Success means that quotient-sampler acceptance, posterior moments, and ESS per evaluation vary substantially less with $N$ than classical HMC, whose step size should need resolution-dependent shrinking.