Adaptive Path Integral Diffusion: AdaPID

Harmonic Path Integral Diffusion (H-PID) provides an analytically tractable framework for sampling from a target density p(tar)(x)∝exp(-E(x)). H-PID can be viewed as a diffusion bridge model solving a stochastic optimal transport problem from a δ-density at t=0 to the target density at t=1. The dynamics are governed by a controlled stochastic differential equation, and the corresponding variational stochastic optimal transport objective combines a time-dependent quadratic potential, βt∥xt∥2/2, with a kinetic control cost, ∥u(t;xt)∥2/2. The focus of this paper is the design of the temporal stiffness protocol βt, which enables explicit control of intermediate sampling dynamics when the terminal density is fixed. We exploit the central advantage of H-PID-its integrability-which yields an explicit representation of the optimal control in terms of the target density and Green functions of the associated linear forward and backward diffusion-in-a-potential problems. Our main contribution is to convert this integrable structure into a practical methodology for protocol optimization. Specializing to piecewise-constant stiffness schedules and Gaussian-mixture targets, we develop two complementary optimization principles: The first is a deterministic one, relying on explicit evaluation of the dynamic marginals, and exemplified on a velocity-gradient-sensitivity objective, which provides a computationally controlled framework for optimizing transport regularity and stiffness. The second is a stochastic one, implemented via sampling, and exemplified on sharpness-based temporal-memory objective regularized to favor transitions within a prescribed time window that targets the temporal organization of the sampling path. These two objectives illuminate different aspects of the same protocol-design problem. The velocity-gradient-sensitivity objective serves as a clean methodological backbone and supports interpretable optimization and scaling studies. The sharpness-based objective reveals that schedule quality is target-dependent, and that the dependence on β is not universal: different target geometries may favor different stiffness regimes and qualitatively different transient organizations. Examples with low- and moderate-dimensional Gaussian mixtures demonstrate that the proposed approach can control not only the terminal sampling accuracy but also the transient evolution of probability mass, while remaining computationally light and theoretically transparent.