# Adaptive Path Integral Diffusion: AdaPID

arXiv: https://arxiv.org/abs/2512.11858 · HTML: https://mlsanity.com/paper/2512.11858/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 51 references that are in the map.
- First posted on arXiv 5 Dec 2025.

## Builds on (most cited first, 26 of 26)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023)
- [Diffusion Normalizing Flow](https://mlsanity.com/paper/2110.07579/) (2021)
- [Non-Denoising Forward-Time Diffusions](https://mlsanity.com/paper/2312.14589/) (2023)
- [Beyond ELBOs: A Large-Scale Evaluation of Variational Methods for Sampling](https://mlsanity.com/paper/2406.07423/) (2024)
- [Improved off-policy training of diffusion samplers](https://mlsanity.com/paper/2402.05098/) (2024)
- [Transferable Boltzmann Generators](https://mlsanity.com/paper/2406.14426/) (2024)
- [No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplers](https://mlsanity.com/paper/2502.06685/) (2025)
- [Towards Predicting Equilibrium Distributions for Molecular Systems with Deep Learning](https://mlsanity.com/paper/2306.05445/) (2023)
- [Adaptive importance sampling for control and inference](https://mlsanity.com/paper/1505.01874/) (2015)
- [Lipschitz-Guided Design of Interpolation Schedules in Generative Models](https://mlsanity.com/paper/2509.01629/) (2025)
- [Optimal Scheduling of Dynamic Transport](https://mlsanity.com/paper/2504.14425/) (2025)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Harmonic Path Integral Diffusion](https://mlsanity.com/paper/2409.15166/) (2024)
- [U-Turn Diffusion](https://mlsanity.com/paper/2308.07421/) (2023)
- [Distribution estimation via Flow Matching with Lipschitz guarantees](https://mlsanity.com/paper/2509.02337/) (2025)
- [Space-Time Diffusion Bridge](https://mlsanity.com/paper/2402.08847/) (2024)

## Built on by (most cited first, 1 of 1)

- [Analytic Bridge Diffusions for Controlled Path Generation](https://mlsanity.com/paper/2605.02961/) (2026)

## Abstract (arXiv)

Diffusion-based samplers -- Score Based Diffusions, Bridge Diffusions and Path Integral Diffusions -- match a target at terminal time, but the real leverage comes from choosing the schedule that governs the intermediate-time dynamics. We develop a path-wise schedule -- selection gramework for Harmonic PID with a time-varying stiffness, exploiting Piece-Wise-Constant(PWC) parametrizations and a simple hierarchical refinement. We introduce schedule-sensitive Quality-of-Sampling (QoS) diagnostics. Assuming a Gaussian-Mixture (GM) target, we retain closed-form Green functions' ration and numerically stable, Neural-Network free oracles for predicted-state maps and score. Experiments in 2D show that QoS driven PWC schedules consistently improve early-exit fidelity, tail accuracy, conditioning of the dynamics, and speciation (label-selection) timing at fixed integration budgets.