FastLSQ: Solving PDEs in One Shot via Fourier Features with Exact Analytical Derivatives

TL;DR

FastLSQ introduces a Fourier feature-based PDE solving framework that computes exact derivatives efficiently, enabling rapid, accurate solutions and inverse problem applications without autodiff.

Contribution

It develops a novel PDE solver using trigonometric Fourier features with exact derivatives, significantly improving speed and accuracy over existing methods like PINNs.

Findings

Achieves $10^{-7}$ accuracy in 0.07s for linear PDEs

Attains $10^{-8}$--$10^{-9}$ accuracy in under 9s for nonlinear PDEs

Demonstrates successful inverse problems and PDE discovery applications

Abstract

We present FastLSQ, a framework for PDE solving and inverse problems built on trigonometric random Fourier features with exact analytical derivatives. Trigonometric features admit closed-form derivatives of any order in $\mathcal{O}(1)$, enabling graph-free operator assembly without autodiff. Linear PDEs: one least-squares call; nonlinear: Newton--Raphson reusing analytical assembly. On 17 PDEs (1--6D), FastLSQ achieves $10^{-7}$ in 0.07s (linear) and $10^{-8}$--$10^{-9}$ in $<$9s (nonlinear), orders of magnitude faster and more accurate than iterative PINNs. Analytical higher-order derivatives yield a differentiable digital twin; we demonstrate inverse problems (heat-source, coil recovery) and PDE discovery. Code: github.com/sulcantonin/FastLSQ and \texttt{pip install fastlsq}.