PRISMA (2025)

Beyond Loss Guidance: Using PDE Residuals as Spectral Attention in Diffusion Neural Operators
Ilyes Batatia et al.

Diffusion Operator learning PDE residual conditioning
Quick facts

Type: diffusion neural operator
Setting: sparse/partial observations
Key idea: residual-guided spectral attention

← Research · Home

TL;DR

PRISMA reframes "physics guidance" for diffusion neural operators: instead of adding a PDE-residual gradient at sampling time, it feeds residual information into the denoiser as a *frequency-aware attention signal*. This aims to improve high-frequency fidelity and stability for inverse problems under sparse/partial observations.

Problem

Diffusion-based PDE priors are powerful for inverse problems, but *loss guidance* (injecting ∇||F(u)|| during sampling) can be unstable and expensive because it requires repeated PDE residual evaluations/gradients. PRISMA targets more stable and efficient conditioning by integrating residual information into the denoiser through spectral attention.

Benefits vs others

Interesting detail

Core method (math)

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

\[u_t = \alpha_t\,u_0 + \sigma_t\,\varepsilon,\quad \varepsilon\sim\mathcal{N}(0,I)\] \[r(u) = F(u)\quad\text{(PDE residual / constraint operator)}\] \[\widehat r(k) = \mathcal{F}[r(u)](k),\quad w_k = \mathrm{softmax}\big(\phi(|\widehat r(k)|)\big)\] \[\widehat u'(k) = w_k \odot \widehat u(k)\quad\text{(residual-driven spectral reweighting)}\] \[\mathcal{L}_{\text{DM}} = \mathbb{E}\big[\|\varepsilon - \varepsilon_\theta(u_t,t,c,\{w_k\})\|_2^2\big]\]

Main theoretical contribution

Main contribution

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)

  • Partial observation (masked measurements) with diffusion sampling.
  • Reports results across different step counts (1–200).
  • Emphasis on fast sampling with minimal quality degradation.

Comparable baselines

Main results

Reported relative error (examples)

Numbers transcribed from the paper page previously added to this repo; please consult the paper for the complete experimental protocol.

PDEDir.PRISMA (200)PRISMA (8)PRISMA (1)FunDPS (8)DiffusionPDE (8)DiffusionPDE (1)
DarcyFwd4.2%4.1%4.0%4.0%4.1%7.8%
DarcyInv17.5%13.5%13.5%21.6%29.0%79.6%
PoissonInv14.5%11.9%10.8%18.8%28.4%31.9%
HelmholtzInv10.4%5.6%5.0%8.4%8.5%8.9%
Navier–StokesInv13.1%6.3%5.6%11.2%11.7%11.7%
If you replicate, make sure to match the observation masks and the normalization used in the paper.

Citation (BibTeX)

@article{prisma2025,
  title={Beyond Loss Guidance: Using PDE Residuals as Spectral Attention in Diffusion Neural Operators},
  author={Batatia, Ilyes and others},
  journal={arXiv preprint arXiv:2512.01370},
  year={2025}
}