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Guided Diffusion Sampling on Function Spaces with Applications to PDEs
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Proposes FunDPS, a function-space diffusion posterior sampler that trains an unconditional neural-operator denoiser with GRF noise and guides sampling with a gradient-based measurement likelihood, backed by an infinite-dimensional Tweedie formula.
Problem
Conditional sampling / posterior inference for PDE-based inverse and forward problems from extremely sparse or noisy observations, in a discretization-agnostic way that does not require retraining for each sensor configuration.
Approach
Train one unconditional function-space diffusion model (U-shaped neural operator denoiser with Gaussian random field noise) and at inference approximate the time-dependent likelihood via the denoiser's conditional expectation, extending Tweedie's formula to separable Banach spaces so that measurement guidance can be plugged in without task-specific retraining.
Key results
- FunDPS with 500 steps reaches 2.49% forward and 5.18% inverse L2 error on Darcy Flow versus 6.07% and 7.87% for DiffusionPDE (2000 steps).
- Across five PDE tasks with 3% observations, the abstract reports an average 32% accuracy improvement over fixed-resolution diffusion baselines while reducing sampling steps by 4x.
- FunDPS runs at about 15 s/sample for 500 steps on one RTX 4090, versus 190 s/sample for DiffusionPDE; multi-resolution inference reaches 7.5 s/sample.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use FunDPS instead of DiffusionPDE | PDE inverse and forward sampling · Five PDE tasks with 3% observations | average error: roughly 32% | Helped |
| Use FunDPS instead of DiffusionPDE | PDE sampling · Five PDE tasks with 3% observations | sampling steps: one-fourth steps | Helped |
| Train primarily at low resolution, then fine-tune at high resolution | FunDPS · PDE training | GPU training hours: 25% of the GPU hours | Helped |
| Perform 80% of sampling steps at low resolution and use ReNoise after upsampling | FunDPS · Multi-resolution inference | inference time: 2x speed improvement to 7.5s/sample | Helped |
| Use FunDPS rather than DiffusionPDE | PDE sampling · 128 × 128 discretization | runtime: 15s/sample for 500 steps; 190s/sample for 2000 steps | Helped |
| Increase FunDPS sampling steps from 200 to 500 | FunDPS · Darcy Flow forward | L2 relative error: 2.88% -> 2.49% | Helped |
| Increase FunDPS sampling steps from 200 to 500 | FunDPS · Poisson inverse | L2 relative error: 24.04% -> 20.47% | Helped |
| Add PDE loss to observation-based guidance (ablation) | FunDPS · PDE inverse problems | performance | Helped |
Surprising findings
- Incorporating PDE loss in guidance yields only marginal improvements over pure observation-based guidance, attributed partly to finite-difference numerical errors.
Limitations
- Guidance weights require manual tuning for each problem.
- PDE-loss guidance gives only marginal gains, likely due to finite-difference errors.
- Approximation of likelihood via conditional expectation is less accurate at high noise levels.
Experimental setup
Datasets: Darcy Flow Poisson Helmholtz Navier-Stokes Navier-Stokes (BCs)
Benchmarks: Darcy Flow Poisson Helmholtz Navier-Stokes Navier-Stokes (BCs)
Metrics: L2 relative error
Baselines: DiffusionPDE FNO PINO DeepONet PINN
Builds on: DiffusionPDE EDM Denoising Diffusion Operator Tweedie's formula DPS
Closest papers by meaning
- A Variational Perspective on Solving Inverse Problems with Diffusion Models (ICLR 2024 · similarity 0.90)
- Diffusion State-Guided Projected Gradient for Inverse Problems (ICLR 2025 · similarity 0.90)
- Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data (ICLR 2025 · similarity 0.90)
- Divide-and-Conquer Posterior Sampling for Denoising Diffusion priors (NeurIPS 2024 · similarity 0.89)
- Coupled Data and Measurement Space Dynamics for Enhanced Diffusion Posterior Sampling (NeurIPS 2025 · similarity 0.89)
- Analyzing and Guiding Zero-Shot Posterior Sampling in Diffusion Models (NeurIPS 2026 · similarity 0.89)
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