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Adaptive Path Integral Diffusion: AdaPID.

Michael Chertkov1, Hamidreza Behjoo1

  • 1Program in Applied Mathematics, Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA.

Entropy (Basel, Switzerland)
|May 26, 2026
PubMed
Summary
This summary is machine-generated.

Harmonic Path Integral Diffusion (H-PID) offers a new way to optimize sampling dynamics by controlling temporal stiffness. This method allows precise control over probability mass evolution for enhanced accuracy and efficiency in complex models.

Keywords:
Gaussian mixturesdiffusion bridgediffusion modelsgenerative modelsoptimal transportpath integralssampling diagnosticsschedule optimizationstochastic optimal control

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Area of Science:

  • Computational Statistics
  • Machine Learning
  • Applied Mathematics

Background:

  • Harmonic Path Integral Diffusion (H-PID) is an analytically tractable framework for sampling from target densities.
  • It functions as a diffusion bridge model, addressing stochastic optimal transport from a delta-density to a target density.
  • The core dynamics involve a controlled stochastic differential equation with a variational objective balancing potential energy and kinetic control costs.

Purpose of the Study:

  • To design and optimize the temporal stiffness protocol (βt) in H-PID for explicit control over intermediate sampling dynamics.
  • To leverage H-PID's integrability for practical protocol optimization methodologies.
  • To develop complementary deterministic and stochastic optimization principles for stiffness schedules.

Main Methods:

  • Exploited H-PID's integrability to derive optimal control in terms of target density and Green functions.
  • Developed a deterministic optimization principle using explicit dynamic marginals and a velocity-gradient-sensitivity objective.
  • Developed a stochastic optimization principle via sampling, using a sharpness-based temporal-memory objective.

Main Results:

  • Demonstrated that piecewise-constant stiffness schedules can be optimized using the developed principles.
  • Showcased control over both terminal sampling accuracy and transient probability mass evolution in Gaussian mixture models.
  • Revealed target-dependent schedule quality and non-universal dependence on stiffness (β), indicating different target geometries favor different stiffness regimes.

Conclusions:

  • The proposed H-PID protocol optimization methodology is computationally light and theoretically transparent.
  • It enables precise control over sampling path organization and transient dynamics, beyond just terminal accuracy.
  • The approach offers a flexible framework for tailoring sampling dynamics to specific target densities and desired path properties.