DiffusionPDE (2024) — Research — PartialObs-PDEBench

TL;DR

Train a diffusion model on joint fields (coefficients + solutions) and perform PDE reconstruction under sparse observations via guided sampling that combines observation consistency and PDE-residual guidance.

Problem

Given partial observations of a PDE solution (e.g., sparse sensors / masked pixels) and optionally unknown PDE coefficients, reconstruct the full field (and the hidden coefficient field) in a way that is consistent with both the observations and the governing PDE.

Benefits vs others

Handles severe sparsity (e.g., 1% observations) by leveraging a learned generative prior rather than purely supervised regression.
Physics guidance improves physical consistency (PDE residual) and can help stabilize reconstructions compared to observation-only guidance.
Unified framework supports both forward (known coefficient) and inverse (unknown coefficient) reconstruction under partial observation.

Interesting detail

Because conditioning happens at sampling time, the same trained prior can be reused across different mask patterns (random vs structured) and potentially different noise levels.
The ablation suggests observation guidance is necessary for matching measurements, while PDE guidance improves physical plausibility and reduces error when combined.

Core method (math)

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

\[q(x_t\mid x_{t-1}) = \mathcal{N}(\sqrt{1-\beta_t}\,x_{t-1},\ \beta_t I),\quad x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon\] \[p_\theta(x_{t-1}\mid x_t) = \mathcal{N}(\mu_\theta(x_t,t),\ \sigma_t^2 I)\] \[\mu_\theta(x_t,t) = \frac{1}{\sqrt{1-\beta_t}}\left(x_t - \frac{\beta_t}{\sqrt{1-\bar\alpha_t}}\,\epsilon_\theta(x_t,t)\right)\] \[\mathcal{L}_{\text{obs}} = \lVert M\odot(u - y)\rVert^2,\quad \mathcal{L}_{\text{PDE}} = \lVert \mathcal{N}(u,a) \rVert^2\] \[x_{t-1} \leftarrow x_{t-1} - \eta\,\nabla_{x_t}\big( \lambda_{\text{obs}}\mathcal{L}_{\text{obs}} + \lambda_{\text{PDE}}\mathcal{L}_{\text{PDE}} \big)\ \text{(guided sampling)}\]

Main theoretical contribution

Not curated yet. Add bullet points under <code>theory</code> in JSON.

Main contribution

Formulate partial-observation PDE solving as conditional generation: learn a prior over joint fields and condition on sparse observations via guidance at sampling time.
Introduce physics guidance via PDE residual loss and data guidance via observation loss, combined to steer diffusion sampling toward physically valid reconstructions.
Demonstrate strong performance across multiple PDEs and settings (forward/inverse; 1%–3% observations; random masks / row-column masks) compared to operator-learning and inpainting-style baselines.

Main results (headline)

(Optional) Add main_results for a quick headline summary.

Experiments

PDE problems

Darcy flow
Poisson equation
Helmholtz equation
Burgers equation
Navier–Stokes

Tasks

Partial observation reconstruction
Inverse problems
Generative PDE inference

Experiment setting (high level)

Mask-conditioned diffusion; partial measurements of inputs/outputs.
Evaluates forward and inverse settings with relative error (%).
Uses iterative sampling (multiple diffusion steps).

Comparable baselines

FNO
PINO
U-NO
U-WNO
Diffusion inpainting baseline (as described in paper)

Main results

Partial observation (random masks): 1% and 3% observation ratio

Metric: relative error (lower is better). Values are copied from the paper’s Table 1 (forward and inverse settings).

Setting	Method	Darcy (Fwd)	Poisson (Fwd)	Helmholtz (Fwd)	Burgers (Fwd)	Navier–Stokes bounded (Fwd)	Navier–Stokes unbounded (Fwd)	Darcy (Inv)	Poisson (Inv)	Helmholtz (Inv)	Navier–Stokes bounded (Inv)	Navier–Stokes unbounded (Inv)
1% obs	DiffusionPDE	0.043	0.136	0.015	0.012	0.071	0.081	0.048	0.196	0.016	0.071	0.086
1% obs	FNO	0.395	0.637	0.055	0.264	0.187	0.244	0.400	0.640	0.099	0.214	0.283
1% obs	PINO	0.552	0.572	0.069	0.436	0.219	0.261	0.553	0.574	0.107	0.253	0.300
1% obs	U-NO	0.204	0.816	0.022	0.072	0.290	0.183	0.206	0.818	0.026	0.329	0.207
1% obs	U-WNO	0.287	0.367	0.044	0.059	0.111	0.154	0.288	0.370	0.076	0.128	0.182
3% obs	DiffusionPDE	0.031	0.072	0.014	0.005	0.018	0.025	0.032	0.105	0.014	0.018	0.026
3% obs	FNO	0.227	0.511	0.044	0.183	0.130	0.184	0.228	0.514	0.081	0.150	0.215
3% obs	PINO	0.320	0.510	0.066	0.337	0.201	0.226	0.322	0.513	0.103	0.232	0.260
3% obs	U-NO	0.072	0.411	0.020	0.026	0.028	0.030	0.076	0.413	0.023	0.031	0.033
3% obs	U-WNO	0.060	0.238	0.033	0.033	0.027	0.030	0.063	0.241	0.057	0.032	0.036

Full observation (no masks)

Metric: relative error (lower is better). Values are copied from the paper’s Table 4 (forward and inverse settings).

Setting	Method	Darcy (Fwd)	Poisson (Fwd)	Helmholtz (Fwd)	Burgers (Fwd)	Navier–Stokes bounded (Fwd)	Navier–Stokes unbounded (Fwd)	Darcy (Inv)	Poisson (Inv)	Helmholtz (Inv)	Navier–Stokes bounded (Inv)	Navier–Stokes unbounded (Inv)
Full obs	DiffusionPDE	0.009	0.013	0.014	0.002	0.002	0.003	0.010	0.015	0.014	0.002	0.004
Full obs	FNO	0.014	0.042	0.044	0.023	0.028	0.064	0.015	0.049	0.081	0.030	0.081
Full obs	PINO	0.011	0.016	0.052	0.010	0.006	0.010	0.012	0.019	0.086	0.009	0.014
Full obs	U-NO	0.009	0.016	0.014	0.002	0.004	0.009	0.010	0.020	0.014	0.005	0.011
Full obs	U-WNO	0.010	0.015	0.030	0.002	0.002	0.004	0.011	0.020	0.064	0.002	0.007

Guidance ablation (1% observation, random mask, Darcy)

Metric: relative error (lower is better). Values are copied from the paper’s Table 3.

Guidance	Darcy (Fwd)	Darcy (Inv)
Observation loss only	0.065	0.073
PDE loss only	0.124	0.141
Observation + PDE loss	0.043	0.048

To reproduce, align observation masks, diffusion step schedule, and normalization.

Citation (BibTeX)

@article{tang2024diffusionpde,
  title={DiffusionPDE: Generative PDE-Solving Under Partial Observation},
  author={Tang, Wenlong and Fan, Lingling and Sun, Haoyu and Zhang, Yichi and Chen, Yixin},
  journal={arXiv preprint arXiv:2406.17763},
  year={2024}
}