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On conditional diffusion models for PDE simulations
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Comparative study of score-based diffusion models for PDE forecasting and data assimilation, proposing autoregressive sampling for joint models, a history-length-stable universal amortised conditional training scheme, and a hybrid amortised-plus-reconstruction-guidance model.
Problem
Diffusion-based PDE surrogates typically handle either forecasting or data assimilation (DA), not both; existing joint and amortised conditional approaches fail on forecasting or are unstable across history lengths.
Approach
Use autoregressive sampling with a locally-factorised (Markov) joint score instead of all-at-once sampling, and train a universal amortised conditional score that samples the conditioning history length C at random so one model handles any forecast/history split, combined with reconstruction guidance for sparse observations.
Key results
- On online DA, universal amortised model achieves RMSD 7.7 on KS vs 11.6 for joint AR and 25.3 for joint AAO (0 corrections).
- On Kolmogorov online DA, universal amortised reaches RMSD 4.9 vs 7.8 for joint AR and 27.4 for joint AAO (0 corrections).
- Universal amortised model gives stable forecasting performance across P|C history-length conditioning scenarios on KS, while joint AAO diverges from ground truth in forecasting.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace all-at-once sampling with autoregressive sampling (ablation) | Joint model · Forecasting, KS and Kolmogorov | High correlation time | Helped |
| Replace all-at-once sampling with autoregressive sampling (ablation) | Joint model · Offline DA, sparse observations | RMSD | Helped |
| Use autoregressive sampling rather than all-at-once sampling (ablation) | Joint model · Offline DA, dense observations | RMSD | No clear effect |
| Add 1 corrector step to all-at-once sampling (ablation) | Joint model · Online DA, KS | RMSD: 25.3 ± 0.1 -> 26.3 ± 0.1 | Hurt |
| Add 1 corrector step to all-at-once sampling (ablation) | Joint model · Online DA, Kolmogorov | RMSD: 27.4 ± 0.2 -> 30.1 ± 0.2 | Hurt |
| Replace all-at-once sampling with autoregressive sampling (ablation) | Joint model · Online DA, KS | RMSD: 25.3 ± 0.1 -> 11.6 ± 0.6 | Helped |
| Replace all-at-once sampling with autoregressive sampling (ablation) | Joint model · Online DA, Kolmogorov | RMSD: 27.4 ± 0.2 -> 7.8 ± 0.5 | Helped |
| Amortise the conditional model over a variety of history lengths (ablation) | Amortised model · Forecasting, KS | High correlation time | Helped |
| Amortise the conditional model over a variety of history lengths (ablation) | Amortised model · Forecasting, Kolmogorov | High correlation time | No clear effect |
| Add reconstruction guidance to architecture conditioning (ablation) | Universal amortised model · Offline DA, KS | RMSD | Mixed |
| Add reconstruction guidance to architecture conditioning (ablation) | Universal amortised model · Offline DA, Kolmogorov | RMSD | No clear effect |
| Use universal amortised conditioning instead of joint autoregressive sampling | Diffusion PDE surrogate · Online DA, KS | RMSD: 11.6 ± 0.6 -> 7.7 ± 0.6 | Helped |
Surprising findings
- AAO sampling for the joint model rapidly diverges in forecasting, so AR querying is essential.
- Longer history degrades plain amortised models on KS but not on Kolmogorov, i.e. the effect is dataset-dependent.
Limitations
- Forecasting still lacks state-of-the-art performance versus MSE-trained baselines.
- Autoregressive sampling is computationally intensive, requiring a full denoising process per step.
- Guidance strength must be tuned by grid search.
Experimental setup
Datasets: Kuramoto-Sivashinsky (generated) Kolmogorov flow (generated following prior work) Burgers' equation (Li et al. PDE-operator dataset)
Benchmarks: Kuramoto-Sivashinsky Kolmogorov flow
Metrics: high correlation time RMSD MSE Pearson correlation
Baselines: Joint AAO PDE-Refiner MSE-trained U-Net Fourier Neural Operator interpolation baseline climatology persistence
Builds on: score-based SDE diffusion models reconstruction guidance all-at-once sampling of local scores (Rozet & Louppe-style) PDE-Refiner DPM-Solver
Closest papers by meaning
- DiffusionPDE: Generative PDE-Solving under Partial Observation (NeurIPS 2024 · similarity 0.88)
- DiffDA: a Diffusion model for weather-scale Data Assimilation (ICML 2024 · similarity 0.88)
- Zero-Shot Conditioning of Score-Based Diffusion Models by Neuro-Symbolic Constraints (AAAI 2025 · similarity 0.87)
- Guided Diffusion Sampling on Function Spaces with Applications to PDEs (NeurIPS 2025 · similarity 0.87)
- ABC: Any-Subset Autoregression via Non-Markovian Diffusion Bridges in Continuous Time and Space (NeurIPS 2026 · similarity 0.87)
- No Training, No Problem: Rethinking Classifier-Free Guidance for Diffusion Models (ICLR 2025 · similarity 0.87)
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