SPADES: A Stochastic Physics-Guided Adaptive Denoising Equilibration Solver for Discretized PDEs

SPADES (Stochastic Physics-Guided Adaptive Denoising Equilibration Solver) is a research framework that formulates finite-difference (FDM) and finite-element (FEM) numerical solutions as Gibbs probability measures defined directly from discrete residuals and variational energies. The framework combines residual-based diffusion models with classical numerical solvers through a physics-initialized correction pipeline rather than replacing existing discretizations. This repository accompanies the SPADES manuscript and contains the reproducible implementation, numerical experiments, and evaluation scripts. The current release includes: • A discrete Gibbs formulation for FDM/FEM residuals and linear SPD problems.• Mean–fluctuation factorization with residual whitening and diffusion-based stochastic correction.• A variable-coefficient Poisson warm-start residual diffusion implementation.• A seed-disjoint random-geometry benchmark for reusable solver-error-law learning.• Reproducible training, evaluation, and benchmark scripts with fixed random seeds. The release is intended as a reproducible research artifact accompanying the manuscript. It is not presented as a production PDE solver, nor as evidence of universal superiority over classical numerical methods, PINNs, neural operators, or other diffusion-based PDE approaches.