FastLSQ: Solving PDEs in One Shot via Fourier Features with Exact Analytical Derivatives
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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.
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References · 6
- 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
- 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
- 2023PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEscited by 87
- 2019Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equationscited by 67
- 2017Bayesian Probabilistic Numerical Methodscited by 51
- 2024Solving Partial Differential Equations with Random Feature Modelscited by 6
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