FunDPS (2025) — Research — PartialObs-PDEBench

TL;DR

FunDPS adapts diffusion posterior sampling to PDE solution *fields*: sample from a learned diffusion prior while enforcing observations and (optionally) physics constraints through guidance terms. The key idea is to treat the unknown as a function (field) and make the guidance compatible with PDE operators and masks.

Problem

Inverse problems for PDEs (inpainting, sparse sensors, data assimilation) require drawing samples of solution fields consistent with partial observations. Standard diffusion priors sample unconditionally; naive conditioning can be weak or expensive. FunDPS targets practical posterior sampling for PDE fields by combining diffusion priors with measurement/physics guidance in a function-space framing.

Benefits vs others

Works naturally with **arbitrary masks / sparse sensors** via a measurement operator M.
Guidance can incorporate **physics residuals** and/or observation likelihoods without retraining a new model for every mask.
Function-space viewpoint encourages discretization-robust formulations (same method across grid resolutions).

Interesting detail

Guidance terms are modular: you can swap in different observation models (noise, mask, forward operators) without retraining the prior.
This makes FunDPS a good *infrastructure* baseline for future partial-observation PDE benchmarks.

Core method (math)

Template for Diffusion. Paper-specific equations are added when manually curated.

\[u_t = \alpha_t\,u_0 + \sigma_t\,\varepsilon,\quad \varepsilon\sim\mathcal{N}(0,I)\] \[s_\theta(u_t,t) \approx \nabla_{u_t}\log p_t(u_t)\quad\text{(score / denoiser)}\] \[y = M u_0 + \eta,\quad \log p(y\mid u) \propto -\|Mu-y\|_2^2/(2\sigma_y^2)\] \[J_{\text{phys}}(u) = \|F(u)\|_2^2\quad\text{(PDE residual penalty)}\] \[u_{t-\Delta} \leftarrow u_t + \underbrace{\text{DDPM/ODE step}}_{\text{prior}} + \lambda\nabla_{u_t}\log p(y\mid u_t) - \mu\nabla_{u_t}J_{\text{phys}}(u_t)\]

Main theoretical contribution

Posterior sampling decomposes into a learned *prior* step (diffusion) plus *guidance* terms encoding measurements/constraints.
Guidance with linear measurement operators M enables efficient masked inpainting and sparse-sensor conditioning.

Main contribution

Extends DPS to **function/field-valued unknowns** with PDE-relevant operators (masking, sensors, residuals).
Demonstrates PDE inverse tasks where guidance significantly improves fidelity under partial observations.
Provides empirical comparisons across PDE datasets and masking patterns (see tables).

Main results (headline)

(Optional) Add main_results for a quick headline summary.

Experiments

PDE problems

Darcy flow
Poisson
Helmholtz
Navier–Stokes

Tasks

Forward operator learning
Inverse / partial-observation reconstruction

Experiment setting (high level)

Guided diffusion sampling; compares different step counts.
Evaluated on standard PDE operator benchmarks; reports relative errors.
Includes both forward and inverse settings depending on PDE.

Comparable baselines

DiffusionPDE
PINO
FNO
DeepONet

Main results

Reported relative error (examples)

Transcribed from the earlier site draft; see the paper for full tables and exact splits.

PDE	Dir.	FunDPS (1)	FunDPS (2)	DiffPDE (10)	DiffPDE (50)	PINO	FNO
Darcy	Fwd	4.5%	4.2%	4.1%	4.1%	4.8%	4.9%
Darcy	Inv	18.8%	18.4%	22.2%	21.7%	—	—
Poisson	Inv	14.9%	14.1%	19.5%	19.3%	—	—
Helmholtz	Inv	8.4%	7.8%	8.5%	8.5%	—	—
Navier–Stokes	Inv	12.5%	11.8%	11.8%	11.7%	—	—

Check the mask generator and the measurement noise level when comparing to other methods.

Citation (BibTeX)

@article{fundps2025,
  title={Guided Diffusion Sampling on Function Spaces with Applications to PDEs},
  author={Tong, Alex and others},
  journal={arXiv preprint arXiv:2505.17004},
  year={2025}
}