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DiffusionPDE: Generative PDE-Solving under Partial Observation
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Proposes DiffusionPDE, a diffusion model trained on the joint distribution of PDE coefficients and solutions, which at inference is guided by sparse observations and PDE residual losses to solve forward and inverse PDE problems under partial observation.
Problem
Solving PDEs (forward and inverse) when only sparse observations (about 1-3% of points) of coefficients or solutions are available, where existing learning-based neural PDE solvers assume full observations.
Approach
Train one unconditional diffusion model (EDM-style) on the concatenated joint distribution of coefficient and solution fields, then at sampling time apply diffusion posterior sampling-style guidance with an observation MSE loss and a PDE residual loss, so one network handles any observation side and pattern.
Key results
- With 500 sparse observations, DiffusionPDE achieves 2.5% forward and 3.2% inverse error on Darcy Flow versus 28.2% and 49.3% for the best baseline (FNO).
- On bounded Navier-Stokes inverse with 1% observations, relative error drops from 12.0% (GraphPDE) to 2.7%.
- On Burgers' equation, recovering the full space-time field from 5 sensors yields a 2.68% test relative error.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use DiffusionPDE instead of the strongest listed baseline | Darcy Flow forward solving with sparse observations · Darcy Flow forward, sparse observations | relative error: 28.2% -> 2.5% | Helped |
| use DiffusionPDE instead of the strongest listed baseline | Darcy Flow inverse solving with sparse observations · Darcy Flow inverse, sparse observations | error rate: 41.1% -> 3.2% | Helped |
| use DiffusionPDE instead of the strongest listed baseline | Poisson forward solving with sparse observations · Poisson forward, sparse observations | relative error: 100.9% -> 4.5% | Helped |
| use DiffusionPDE instead of the strongest listed baseline | Poisson inverse solving with sparse observations · Poisson inverse, sparse observations | relative error: 105.8% -> 20.0% | Helped |
| use DiffusionPDE instead of the strongest listed baseline | Helmholtz forward solving with sparse observations · Helmholtz forward, sparse observations | relative error: 98.2% -> 8.8% | Helped |
| use DiffusionPDE instead of the strongest listed baseline | Helmholtz inverse solving with sparse observations · Helmholtz inverse, sparse observations | relative error: 132.8% -> 22.6% | Helped |
| use DiffusionPDE instead of the strongest listed baseline | Non-bounded Navier-Stokes forward solving with sparse observations · Non-bounded Navier-Stokes forward, sparse observations | relative error: 101.4% -> 6.9% | Helped |
| use DiffusionPDE instead of the strongest listed baseline | Non-bounded Navier-Stokes inverse solving with sparse observations · Non-bounded Navier-Stokes inverse, sparse observations | relative error: 96.0% -> 10.4% | Helped |
| use DiffusionPDE instead of the strongest listed baseline | Bounded Navier-Stokes forward solving with sparse observations · Bounded Navier-Stokes forward, sparse observations | relative error: 81.1% -> 3.9% | Helped |
| use DiffusionPDE instead of the strongest listed baseline | Bounded Navier-Stokes inverse solving with sparse observations · Bounded Navier-Stokes inverse, sparse observations | relative error: 69.5% -> 2.7% | Helped |
| add PDE loss guidance to observation loss guidance (ablation) | DiffusionPDE · Helmholtz solution, sparse observations on coefficient and solution | relative error: 9.3% -> 0.6% | Helped |
| add PDE loss guidance to observation loss guidance (ablation) | DiffusionPDE · Helmholtz coefficient, sparse observations on coefficient and solution | relative error: 13.2% -> 9.4% | Helped |
Surprising findings
- Baselines trained on full observations degrade to near-constant predictions when trained on sparse inputs, so the authors trained them only on full data.
- Under full observation, PINO outperforms DiffusionPDE on the Navier-Stokes forward and inverse problems (1.1% vs 6.1%, 6.8% vs 8.6%).
Limitations
- Inverse problems for Poisson and Helmholtz have high errors (20.0% and 22.6%) due to weak constraints in the coefficient space.
- Slow sampling: inference takes about 140 s per sample on an A40 and the model has 54M parameters.
- Currently limited to 2D dynamic PDE slices; full time interval recovery for dynamic PDEs is not fully addressed.
Experimental setup
Datasets: Darcy Flow (FNO scripts) Poisson (finite difference) Helmholtz (finite difference) Non-bounded Navier-Stokes (FNO scripts) Bounded Navier-Stokes (Difftaichi) Burgers' equation (FNO scripts)
Benchmarks: Darcy Flow Poisson Helmholtz Non-bounded Navier-Stokes Bounded Navier-Stokes Burgers' equation
Metrics: relative error error rate (Darcy coefficient) standard deviation runtime
Baselines: PINO DeepONet PINNs FNO GraphPDE Shu et al. (2023) optimized CFG OFormer Diffusion with CFG RBF kernel U-Net
Builds on: EDM diffusion posterior sampling FNO PINO GraphPDE
Closest papers by meaning
- Ultra Fast PDE Solving via Physics Guided Few-step Diffusion (NeurIPS 2026 · similarity 0.90)
- Text2PDE: Latent Diffusion Models for Accessible Physics Simulation (ICLR 2025 · similarity 0.89)
- Learning to Solve Generative ODEs Beyond the Linear Span (NeurIPS 2026 · similarity 0.89)
- Physics-Informed Diffusion Models (ICLR 2025 · similarity 0.89)
- Adaptive Stochastic Coefficients for Accelerating Diffusion Sampling (NeurIPS 2025 · similarity 0.89)
- Guided Diffusion Sampling on Function Spaces with Applications to PDEs (NeurIPS 2025 · similarity 0.89)
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