WeakPINN

Weak-form physics-informed learning for noisy scientific image dynamics.

This repository implements and validates a single methodological idea — moving derivatives off noisy observational data and onto smooth, boundary-vanishing test functions via integration by parts — across two scientifically distinct second-order PDE inverse problems:

Domain	Application	Code	Underlying PDE
Heliophysics	Flare-PINN — operational solar flare forecasting	solar/	Resistive MHD induction
Microscopy	FRAP weak-form PINN — diffusion-coefficient recovery	frap/	Linear reaction-diffusion

WeakPINN is a methodology repository, not a single application. Each domain has its own README, training pipeline, and reproducibility instructions.

Headline results

Flare-PINN (solar, chronological test set, corrected integrated weak-moment arm)

Arm	TSS 6 h	TSS 12 h	TSS 24 h	Grand mean
Integrated weak moments	0.7896 ± 0.0387	0.8141 ± 0.0158	0.7816 ± 0.0099	0.7951 ± 0.0173
Data-only, capped at 40 k	0.7123	0.6809	0.7456	0.7130
Difference	+0.0772	+0.1332	+0.0360	+0.0822

A two-stage recipe: 40,000 data-only steps, then an integrated weak-moment continuation scored at step 44,000. A control trained for the same steps without the physics term reaches only 0.7155, so continued training accounts for +0.0025 of the gain and the physics term for +0.0796. Read in distribution instead, on the validation window adjacent to training, the same term is worth −0.0130: it buys out-of-distribution generalization rather than fit. Receipts in recovery/no_physics_control/.

Superseded. Earlier releases headlined 24 h TSS 0.798 from a local-integrand configuration whose development used test-set information, beside a strong-form configuration never re-run under the corrected protocol.

FRAP cross-domain validation (synthetic, pooled noisy)

Method	D-MAE (norm units)	% err vs. D_norm_true
Weak-form PINN	3.88e-4	1.96 %
Strong-form PINN	2.16e-3	10.94 %
Data-only ablation	— (no D gradient)	n/a

Weak-form reduces D-MAE by 82.1 % relative to strong-form across the noisy synthetic sweep. The same matched comparison on 10 experimental DeepFRAP stacks (top-5 quality-ranked per molecular-weight condition, 5 training seeds per stack per method) shows weak-form is more stable than strong-form on 10/10 stacks (median cross-seed std reduction 60.7%, mean 49.0%). The effect is particularly large in the slow-diffusion 56ww condition (59.6–77.0% reduction on all 5 stacks; mean −66.7%).

Repository layout

WeakPINN/
├── figures/                 all publication figures (one canonical location)
│   ├── main/                figures 1-5 (main paper)
│   └── supplement/          SI figures (S-FRAP, solar SI, methodology benchmarks)
├── solar/                   Flare-PINN: solar flare forecasting
│   ├── src/                 model, training, evaluation, DeFN baseline
│   ├── data_scripts/        JSOC fetch, windowing, splits
│   ├── tools/               analysis, validation, viz, methodology_benchmarks
│   ├── final_results/       curated paper-ready metrics + B1/B4/B5 results,
│   │                        plus benchmarks/ (B2 heat-eq sweep, B3 autodiff)
│   └── README.md
├── frap/                    FRAP weak-form PINN cross-domain validation
│   ├── src/                 models + losses
│   ├── scripts/             data, training, analysis, figure generation
│   ├── tests/
│   ├── results/             curated supplement tables + B6 test-fn sensitivity
│   └── README.md
├── docs/                    shared methodology notes + B1-B6 synthesis
├── environment.yml          conda environment (PyTorch + MPS)
└── LICENSE

Installation

conda env create -f environment.yml
conda activate weakpinn

Tested on macOS / Apple-silicon (MPS) and Linux CUDA. Solar pipeline assumes ≥ 32 GB system RAM during data prep; training itself runs on a single GPU.

Reproducing the paper

Each subdirectory README walks through its domain's pipeline end-to-end:

• solar/README.md — full Flare-PINN pipeline: JSOC fetch → windowing → training → evaluation
• frap/README.md — FRAP weak-form: synthetic generation → λ tuning → main matrix → figures

Larger artifacts (windowed magnetogram data, training checkpoints, raw DeepFRAP unzip) are not in this repo; each domain README points to the corresponding figshare / Zenodo deposit.

Citation

If you use WeakPINN in your research, please cite the methodology paper:

@article{weakpinn2026,
  title   = {Weak-Form Physics-Informed Learning for Noise-Limited Scientific Image Dynamics},
  author  = {Author details withheld for double-anonymized review},
  journal = {Under review},
  year    = {2026}
}

The solar dataset draws on SDO/HMI SHARP cutouts; the FRAP experimental stacks are from the public DeepFRAP dataset (Röding et al., 2020).

License

MIT.