FastLSQ: Solving PDEs in One Shot via Fourier Features with Exact Analytical Derivatives
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908904523300864 |
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| author | Sulc, Antonin |
| author_facet | Sulc, Antonin |
| contents | 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}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_10541 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | FastLSQ: Solving PDEs in One Shot via Fourier Features with Exact Analytical Derivatives Sulc, Antonin Numerical Analysis Machine Learning 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}. |
| title | FastLSQ: Solving PDEs in One Shot via Fourier Features with Exact Analytical Derivatives |
| topic | Numerical Analysis Machine Learning |
| url | https://arxiv.org/abs/2602.10541 |