# From Uncertain to Safe: Conformal Adaptation of Diffusion Models for Safe PDE Control

arXiv: https://arxiv.org/abs/2502.02205 · HTML: https://mlsanity.com/paper/2502.02205/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion and flow policies for robotics, one of 14 directions in Diffusion models.
- Builds on 33 of its 54 references that are in the map.
- First posted on arXiv 4 Feb 2025.

## Placed in

- Diffusion models / [Diffusion and flow policies for robotics](https://mlsanity.com/diffusion/diffusion-and-flow-policies-for-robotics/)

## Builds on (most cited first, 30 of 33)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Decision Transformer: Reinforcement Learning via Sequence Modeling](https://mlsanity.com/paper/2106.01345/) (2021)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Planning with Diffusion for Flexible Behavior Synthesis](https://mlsanity.com/paper/2205.09991/) (2022)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [LION: Latent Point Diffusion Models for 3D Shape Generation](https://mlsanity.com/paper/2210.06978/) (2022)
- [Conformal Prediction Under Covariate Shift](https://mlsanity.com/paper/1904.06019/) (2019)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [Risk-Constrained Reinforcement Learning with Percentile Risk Criteria](https://mlsanity.com/paper/1512.01629/) (2015)
- [End-to-End Safe Reinforcement Learning through Barrier Functions for Safety-Critical Continuous Control Tasks](https://mlsanity.com/paper/1903.08792/) (2019)
- [Using Trusted Data to Train Deep Networks on Labels Corrupted by Severe Noise](https://mlsanity.com/paper/1802.05300/) (2018)
- [A Review of Safe Reinforcement Learning: Methods, Theory and Applications](https://mlsanity.com/paper/2205.10330/) (2022)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [Distribution-free uncertainty quantification for classification under label shift](https://mlsanity.com/paper/2103.03323/) (2021)
- [Safe Nonlinear Control Using Robust Neural Lyapunov-Barrier Functions](https://mlsanity.com/paper/2109.06697/) (2021)
- [Constraints Penalized Q-learning for Safe Offline Reinforcement Learning](https://mlsanity.com/paper/2107.09003/) (2021)
- [Robust Validation: Confident Predictions Even When Distributions Shift](https://mlsanity.com/paper/2008.04267/) (2020)
- [COptiDICE: Offline Constrained Reinforcement Learning via Stationary Distribution Correction Estimation](https://mlsanity.com/paper/2204.08957/) (2022)
- [Constrained Decision Transformer for Offline Safe Reinforcement Learning](https://mlsanity.com/paper/2302.07351/) (2023)
- [Exact and Robust Conformal Inference Methods for Predictive Machine Learning With Dependent Data](https://mlsanity.com/paper/1802.06300/) (2018)
- [Safe Offline Reinforcement Learning with Feasibility-Guided Diffusion Model](https://mlsanity.com/paper/2401.10700/) (2024)
- [Datasets and Benchmarks for Offline Safe Reinforcement Learning](https://mlsanity.com/paper/2306.09303/) (2023)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [Synthetic Lagrangian Turbulence by Generative Diffusion Models](https://mlsanity.com/paper/2307.08529/) (2023)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)

## Built on by (most cited first, 3 of 3)

- [HardFlow: Hard-Constrained Sampling for Flow-Matching Models via Trajectory Optimization](https://mlsanity.com/paper/2511.08425/) (2025)
- [PerFlow: Physics-Embedded Rectified Flow for Efficient Reconstruction and Uncertainty Quantification of Spatiotemporal Dynamics](https://mlsanity.com/paper/2605.03548/) (2026)
- [Uncertainty-Calibrated Diffusion for Reliable 3D Molecular Graph Generation](https://mlsanity.com/paper/2606.01595/) (2026)

## Abstract (arXiv)

The application of deep learning for partial differential equation (PDE)-constrained control is gaining increasing attention. However, existing methods rarely consider safety requirements crucial in real-world applications. To address this limitation, we propose Safe Diffusion Models for PDE Control (SafeDiffCon), which introduce the uncertainty quantile as model uncertainty quantification to achieve optimal control under safety constraints through both post-training and inference phases. Firstly, our approach post-trains a pre-trained diffusion model to generate control sequences that better satisfy safety constraints while achieving improved control objectives via a reweighted diffusion loss, which incorporates the uncertainty quantile estimated using conformal prediction. Secondly, during inference, the diffusion model dynamically adjusts both its generation process and parameters through iterative guidance and fine-tuning, conditioned on control targets while simultaneously integrating the estimated uncertainty quantile. We evaluate SafeDiffCon on three control tasks: 1D Burgers' equation, 2D incompressible fluid, and controlled nuclear fusion problem. Results demonstrate that SafeDiffCon is the only method that satisfies all safety constraints, whereas other classical and deep learning baselines fail. Furthermore, while adhering to safety constraints, SafeDiffCon achieves the best control performance. The code can be found at https://github.com/AI4Science-WestlakeU/safediffcon.