[2602.10541] FastLSQ: A Framework for One-Shot PDE Solving

Mathematics > Numerical Analysis arXiv:2602.10541 (math) [Submitted on 11 Feb 2026 (v1), last revised 4 Mar 2026 (this version, v2)] Title:FastLSQ: A Framework for One-Shot PDE Solving Authors:Antonin Sulc View a PDF of the paper titled FastLSQ: A Framework for One-Shot PDE Solving, by Antonin Sulc View PDF HTML (experimental) Abstract:We present FastLSQ, a framework for fast PDE solving and inverse problems built on sinusoidal random Fourier features with exact analytical derivatives. Sinusoids are eigenfunctions of differentiation: derivatives of any order admit closed-form evaluation in $O(1)$ operations, enabling graph-free operator assembly without automatic differentiation. Linear PDEs are solved in a single least-squares call; nonlinear PDEs via Newton--Raphson iteration where each step reuses the analytical assembly. On 17 PDEs spanning 1 to 6 dimensions, FastLSQ achieves $10^{-7}$ in 0.07\,s on linear problems and $10^{-8}$ to $10^{-9}$ on nonlinear problems in under 9\,s -- orders of magnitude faster and more accurate than iterative PINN solvers. The framework extends to inverse problems (heat-source localisation, coil recovery from sparse sensors) and PDE discovery via analytical derivative dictionaries. Code is publicly available at this http URL and via pip install fastlsq. Comments: Subjects: Numerical Analysis (math.NA); Machine Learning (cs.LG) Cite as: arXiv:2602.10541 [math.NA]   (or arXiv:2602.10541v2 [math.NA] for this version)   https://doi.org/10.4855...