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

arXiv: https://arxiv.org/abs/2602.10541 · HTML: https://mlsanity.com/paper/2602.10541/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 13 references that are in the map.
- First posted on arXiv 11 Feb 2026.

## Builds on (most cited first, 6 of 6)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [Solving Partial Differential Equations with Random Feature Models](https://mlsanity.com/paper/2501.00288/) (2024)

## Abstract (arXiv)

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 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 textttpip install fastlsq.