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

• URL: http://arxiv.org/abs/2602.10541v1
• Date: Wed, 11 Feb 2026 05:28:58 GMT
• Title: Solving PDEs in One Shot via Fourier Features with Exact Analytical Derivatives
• Authors: Antonin Sulc,
• Abstract summary: Recent random feature methods for solving partial differential equations (PDEs) reduce computational cost compared to physics-informed neural networks (PINNs)<n>We propose FastLSQ, which combines frozen random Fourier features with analytical operator assembly to solve linear PDEs via a single least-squares call.<n>On a benchmark of 17 PDEs spanning 1 to 6 dimensions, FastLSQ achieves relative $L2$ errors of $10-7$ in 0.07,s on linear problems, three orders of magnitude more accurate and significantly faster than state-of-the-art iterative PINN solvers.
• Score: 0.15229257192293197
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Recent random feature methods for solving partial differential equations (PDEs) reduce computational cost compared to physics-informed neural networks (PINNs) but still rely on iterative optimization or expensive derivative computation. We observe that sinusoidal random Fourier features possess a cyclic derivative structure: the derivative of any order of $\sin(\mathbf{W}\cdot\mathbf{x}+b)$ is a single sinusoid with a monomial prefactor, computable in $O(1)$ operations. Alternative activations such as $\tanh$, used in prior one-shot methods like PIELM, lack this property: their higher-order derivatives grow as $O(2^n)$ terms, requiring automatic differentiation for operator assembly. We propose FastLSQ, which combines frozen random Fourier features with analytical operator assembly to solve linear PDEs via a single least-squares call, and extend it to nonlinear PDEs via Newton--Raphson iteration where each linearized step is a FastLSQ solve. On a benchmark of 17 PDEs spanning 1 to 6 dimensions, FastLSQ achieves relative $L^2$ errors of $10^{-7}$ in 0.07\,s on linear problems, three orders of magnitude more accurate and significantly faster than state-of-the-art iterative PINN solvers, and $10^{-8}$ to $10^{-9}$ on nonlinear problems via Newton iteration in under 9s.

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