Solving PDEs in One Shot via Fourier Features with Exact Analytical Derivatives
This provides a faster and more accurate solver for PDEs in scientific computing, though it is incremental as it builds on prior random feature methods.
The paper tackled the computational inefficiency of solving partial differential equations (PDEs) by introducing FastLSQ, a method that uses frozen random Fourier features with exact analytical derivatives to enable one-shot least-squares solves for linear PDEs and Newton-Raphson iteration for nonlinear ones, achieving relative L^2 errors as low as 10^-9 in under 9 seconds on a benchmark of 17 PDEs.
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.