Best of Both Worlds: Bridging Laplace and Fourier for Generalizable and Efficient Operator Learning

We introduce Laplace–Fourier Neural Operator (LFNO), a novel operator learning model that bridges the strengths of Laplace Neural Operators (LNO) and Fourier Neural Operators (FNO).Building on LNO’s ability to capture transient and steady-state responses, we augment LNO's steady-state component with a Fourier integral operator and a learnable nonlinearity, the key essence of FNO's resolution invariance and efficiency.This integration enables LFNO to efficiently represent both the transient and steady-state responses, thereby enhancing LFNO's ability to model complex dynamics across ODEs and PDEs.We demonstrate LFNO's effectiveness on solving ODE (Duffing equation) and PDE (Euler-Bernoulli beam) benchmarks, in comparison to LNO.These results suggest LFNO as a versatile and generalizable neural operator framework for modeling dynamical systems in both linear and nonlinear regimes, offering a principled step toward ML-powered scalable methods for physical sciences.