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

2026-02-11

AI Summary

This work proposes a novel approach to solving partial differential equations (PDEs) that overcomes the high computational cost and reliance on iterative optimization or automatic differentiation inherent in traditional physics-informed neural networks (PINNs). By leveraging sine-based random Fourier features with analytically tractable, recursive higher-order derivatives, the method enables direct solution of linear PDEs via a single least-squares solve under frozen features, while efficiently handling nonlinear PDEs through Newton–Raphson iteration. Notably, it achieves operator construction with O(1) complexity and completely eliminates the need for automatic differentiation, establishing a true “solve-once” framework. Evaluated on 17 benchmark PDEs across dimensions one to six, the approach attains relative L² errors of 10⁻⁷ for linear problems in 0.07 seconds and 10⁻⁸–10⁻⁹ for nonlinear cases in under 9 seconds, significantly outperforming existing PINN methods in both accuracy and speed.

Technology Category

• Machine Learning: Feature Construction/Reformulation
• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• partial differential equations
• random Fourier features
• derivative computation
• iterative optimization
• high-order derivatives

Innovation

• Fourier features
• analytical derivatives
• FastLSQ
• one-shot PDE solver
• least-squares method