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From Uncertain to Safe: Conformal Adaptation of Diffusion Models for Safe PDE Control
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Proposes SafeDiffCon, a conformal-prediction-based diffusion method that post-trains and fine-tunes diffusion models with an uncertainty quantile to satisfy safety constraints in PDE control.
Problem
Deep learning PDE control rarely enforces safety constraints; offline data is suboptimal and unsafe, causing distribution shift between learned and safe distributions.
Approach
Quantify diffusion model safety uncertainty via weighted (covariate-shift) conformal prediction, then use the resulting uncertainty quantile to reweight the diffusion training loss (post-training) and to guide and fine-tune the model at inference time so that the conformal safety interval stays within bounds.
Key results
- On 1D Burgers', SafeDiffCon achieves J=0.0011 with 0% unsafe trajectories, the only method among those with J comparable to optimum that is fully safe (CDT J=0.0012 but 16% unsafe).
- On 2D fluid, SafeDiffCon is the only method with 0% unsafe samples and SVM 0.0000 (BC-Safe 8% unsafe, TREBI 18%), at J=-0.3548.
- On tokamak control, SafeDiffCon achieves J=0.0094 with 0% unsafe samples, versus CDT J=0.0071 with 8% unsafe samples and TREBI J=0.0261 fully safe.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add SafeDiffCon's conformal uncertainty quantile, post-training, and inference-time fine-tuning | PDE control · 1D Burgers' equation | J: 0.0074 -> 0.0011 | Helped |
| add SafeDiffCon's conformal uncertainty quantile, post-training, and inference-time fine-tuning | PDE control · 2D incompressible fluid | R: 18% -> 0% | Helped |
| add SafeDiffCon's conformal uncertainty quantile, post-training, and inference-time fine-tuning | PDE control · Tokamak fusion reactor | J among methods satisfying safety: 0.0261 -> 0.0094 | Helped |
| add post-training (ablation) | SafeDiffCon · 2D incompressible fluid | R: 12% -> 0% | Helped |
| add inference-time fine-tuning (ablation) | SafeDiffCon · 2D incompressible fluid | R: 8% -> 0% | Helped |
| add uncertainty quantile Q (ablation) | SafeDiffCon · 2D incompressible fluid | R: 12% -> 0% | Helped |
| add post-training (ablation) | SafeDiffCon · 1D Burgers' equation | R_sample: 4% -> 0% | Helped |
| add inference-time fine-tuning (ablation) | SafeDiffCon · 1D Burgers' equation | R_sample: 40% -> 0% | Helped |
| add uncertainty quantile Q (ablation) | SafeDiffCon · 1D Burgers' equation | R_sample: 30% -> 0% | Helped |
| add post-training (ablation) | SafeDiffCon · Tokamak fusion reactor | R_sample: 10% -> 0% | Helped |
| add inference-time fine-tuning (ablation) | SafeDiffCon · Tokamak fusion reactor | R_sample: 50% -> 0% | Helped |
| add uncertainty quantile Q (ablation) | SafeDiffCon · Tokamak fusion reactor | R_sample: 28% -> 0% | Helped |
Surprising findings
- Removing the uncertainty quantile Q yields lower control error J in several ablations but violates safety constraints, showing the safety-performance trade-off.
- Standard behavior cloning and CDT achieve better control error than the safe method but violate safety constraints on most or many trajectories.
Limitations
- Theoretical coverage relies on exchangeability/weighted exchangeability and a KL-based approximation of the weights.
- Evaluation limited to three simulated PDE control tasks; authors note future application to more real-world tasks.
- Test set for Burgers' is entirely unsafe under the chosen bound, and the approach trades off control accuracy for safety in some settings.
Experimental setup
Datasets: 1D Burgers' equation simulations (PDE solver generated) 2D incompressible fluid (PhiFlow Navier-Stokes) KSTAR tokamak LSTM-based simulator data
Benchmarks: 1D Burgers' equation safe control 2D incompressible fluid safe control tokamak fusion reactor control
Metrics: control objective J unsafe sample rate unsafe time rate unsafe point rate safety violation magnitude (SVM)
Baselines: BC BC-Safe PID SL-Lag MPC-Lag CDT TREBI
Builds on: DDPM DDIM conformal prediction under covariate shift Decision diffusion control (PDE-constrained diffusion) CDT
Closest papers by meaning
- Safe and Stable Control via Lyapunov-Guided Diffusion Models (NeurIPS 2025 · similarity 0.87)
- SafeDiffuser: Safe Planning with Diffusion Probabilistic Models (ICLR 2025 · similarity 0.86)
- Calibrated Physics-Informed Uncertainty Quantification (ICML 2025 · similarity 0.86)
- Uncertainty Quantification for Forward and Inverse Problems of PDEs via Latent Global Evolution (AAAI 2024 · similarity 0.85)
- A Physics-Informed Machine Learning Framework for Safe and Optimal Control of Autonomous Systems (ICML 2025 · similarity 0.85)
- DiRecT: Safe Diffusion-Based Planning via Receding-Horizon Denoising (NeurIPS 2026 · similarity 0.85)
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