# Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts

arXiv: https://arxiv.org/abs/2503.02819 · HTML: https://mlsanity.com/paper/2503.02819/

## Key facts

- Cited by 97 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 52 of its 84 references that are in the map.
- First posted on arXiv 4 Mar 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 52)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [ImageReward: Learning and Evaluating Human Preferences for Text-to-Image Generation](https://mlsanity.com/paper/2304.05977/) (2023)
- [GenEval: An Object-Focused Framework for Evaluating Text-to-Image Alignment](https://mlsanity.com/paper/2310.11513/) (2023)
- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Compositional Visual Generation with Composable Diffusion Models](https://mlsanity.com/paper/2206.01714/) (2022)
- [Analyzing and Improving the Training Dynamics of Diffusion Models](https://mlsanity.com/paper/2312.02696/) (2023)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Guiding a Diffusion Model with a Bad Version of Itself](https://mlsanity.com/paper/2406.02507/) (2024)
- [Therapeutics Data Commons: Machine Learning Datasets and Tasks for Drug Discovery and Development](https://mlsanity.com/paper/2102.09548/) (2021)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Practical and Asymptotically Exact Conditional Sampling in Diffusion Models](https://mlsanity.com/paper/2306.17775/) (2023)
- [A General Framework for Inference-time Scaling and Steering of Diffusion Models](https://mlsanity.com/paper/2501.06848/) (2025)
- [Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMC](https://mlsanity.com/paper/2302.11552/) (2023)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [3D Equivariant Diffusion for Target-Aware Molecule Generation and Affinity Prediction](https://mlsanity.com/paper/2303.03543/) (2023)
- [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023)
- [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015)
- [Derivative-Free Guidance in Continuous and Discrete Diffusion Models with Soft Value-Based Decoding](https://mlsanity.com/paper/2408.08252/) (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)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [Understanding Reinforcement Learning-Based Fine-Tuning of Diffusion Models: A Tutorial and Review](https://mlsanity.com/paper/2407.13734/) (2024)
- [Test-time Alignment of Diffusion Models without Reward Over-optimization](https://mlsanity.com/paper/2501.05803/) (2025)
- [Inference-Time Alignment in Diffusion Models with Reward-Guided Generation: Tutorial and Review](https://mlsanity.com/paper/2501.09685/) (2025)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024)
- [Classifier-Free Guidance is a Predictor-Corrector](https://mlsanity.com/paper/2408.09000/) (2024)

## Built on by (most cited first, 30 of 97)

- [Reasoning with Sampling: Your Base Model is Smarter Than You Think](https://mlsanity.com/paper/2510.14901/) (2025)
- [Q-learning with Adjoint Matching](https://mlsanity.com/paper/2601.14234/) (2026)
- [Diffusion Tree Sampling: Scalable inference-time alignment of diffusion models](https://mlsanity.com/paper/2506.20701/) (2025)
- [Diamond Maps: Efficient Reward Alignment via Stochastic Flow Maps](https://mlsanity.com/paper/2602.05993/) (2026)
- [Progressive Inference-Time Annealing of Diffusion Models for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2506.16471/) (2025)
- [DriftLite: Lightweight Drift Control for Inference-Time Scaling of Diffusion Models](https://mlsanity.com/paper/2509.21655/) (2025)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [RNE: plug-and-play diffusion inference-time control and energy-based training](https://mlsanity.com/paper/2506.05668/) (2025)
- [GLASS Flows: Transition Sampling for Alignment of Flow and Diffusion Models](https://mlsanity.com/paper/2509.25170/) (2025)
- [Test-time scaling of diffusions with flow maps](https://mlsanity.com/paper/2511.22688/) (2025)
- [Inference-Time Scaling of Discrete Diffusion Models via Importance Weighting and Optimal Proposal Design](https://mlsanity.com/paper/2505.22524/) (2025)
- [TR2-D2: Tree Search Guided Trajectory-Aware Fine-Tuning for Discrete Diffusion](https://mlsanity.com/paper/2509.25171/) (2025)
- [Flow Matching with Semidiscrete Couplings](https://mlsanity.com/paper/2509.25519/) (2025)
- [Learn to Guide Your Diffusion Model](https://mlsanity.com/paper/2510.00815/) (2025)
- [Progressive Tempering Sampler with Diffusion](https://mlsanity.com/paper/2506.05231/) (2025)
- [Enhanced Diffusion Sampling: Efficient Rare Event Sampling and Free Energy Calculation with Diffusion Models](https://mlsanity.com/paper/2602.16634/) (2026)
- [Discrete Feynman-Kac Correctors](https://mlsanity.com/paper/2601.10403/) (2026)
- [CREPE: Controlling Diffusion with Replica Exchange](https://mlsanity.com/paper/2509.23265/) (2025)
- [Adaptive Diffusion Guidance via Stochastic Optimal Control](https://mlsanity.com/paper/2505.19367/) (2025)
- [Inference-Time Alignment Control for Diffusion Models with Reinforcement Learning Guidance](https://mlsanity.com/paper/2508.21016/) (2025)
- [Riemannian MeanFlow](https://mlsanity.com/paper/2602.07744/) (2026)
- [Temporal Score Rescaling for Temperature Sampling in Diffusion and Flow Models](https://mlsanity.com/paper/2510.01184/) (2025)
- [Reinforced sequential Monte Carlo for amortised sampling](https://mlsanity.com/paper/2510.11711/) (2025)
- [Conditional Diffusion Models with Classifier-Free Gibbs-like Guidance](https://mlsanity.com/paper/2505.21101/) (2025)
- [Product of Experts for Visual Generation](https://mlsanity.com/paper/2506.08894/) (2025)
- [HollowFlow: Efficient Sample Likelihood Evaluation using Hollow Message Passing](https://mlsanity.com/paper/2510.21542/) (2025)
- [Contrastive Distribution Matching for Amortized Sequential Monte Carlo in Discrete Diffusion](https://mlsanity.com/paper/2605.23346/) (2026)
- [Controllable protein design with particle-based Feynman-Kac steering](https://mlsanity.com/paper/2511.09216/) (2025)
- [Aligning Few-Step Generative Models by Amortizing Sample-based Variational Inference](https://mlsanity.com/paper/2605.26552/) (2026)
- [Wavefunction Flows: Efficient Quantum Simulation of Continuous Flow Models](https://mlsanity.com/paper/2510.08462/) (2025)

## Abstract (arXiv)

While score-based generative models are the model of choice across diverse domains, there are limited tools available for controlling inference-time behavior in a principled manner, e.g. for composing multiple pretrained models. Existing classifier-free guidance methods use a simple heuristic to mix conditional and unconditional scores to approximately sample from conditional distributions. However, such methods do not approximate the intermediate distributions, necessitating additional `corrector' steps. In this work, we provide an efficient and principled method for sampling from a sequence of annealed, geometric-averaged, or product distributions derived from pretrained score-based models. We derive a weighted simulation scheme which we call Feynman-Kac Correctors (FKCs) based on the celebrated Feynman-Kac formula by carefully accounting for terms in the appropriate partial differential equations (PDEs). To simulate these PDEs, we propose Sequential Monte Carlo (SMC) resampling algorithms that leverage inference-time scaling to improve sampling quality. We empirically demonstrate the utility of our methods by proposing amortized sampling via inference-time temperature annealing, improving multi-objective molecule generation using pretrained models, and improving classifier-free guidance for text-to-image generation. Our code is available at https://github.com/martaskrt/fkc-diffusion.