LFNO: Bridging Laplace and Fourier via Transient-Steady Decomposition

Researchers introduce LFNO (Laplace-Fourier Neural Operator), a unified neural network framework that combines spectral advantages of Laplace and Fourier transforms to model dynamical systems across transient and steady-state phases. The approach significantly outperforms existing methods on ODE benchmarks while remaining competitive on PDE systems, offering improved stability and interpretability for complex systems.

LFNO represents a meaningful advancement in computational modeling of dynamical systems by addressing a fundamental limitation in existing neural operator frameworks. Traditional approaches like Laplace Neural Operators (LNO) excel at capturing transient dynamics while struggling with steady-state behavior, whereas Fourier Neural Operators (FNO) show the opposite characteristics. LFNO's dual-branch architecture explicitly decomposes system dynamics into these two complementary regimes, enabling stronger performance across multiple temporal scales.

The theoretical foundation builds on decades of control theory and signal processing, where Laplace and Fourier transforms serve distinct purposes in analyzing system behavior. By unifying these spectral perspectives within a neural operator framework, the researchers tap into established mathematical principles rather than introducing novel concepts. This approach demonstrates how classical engineering intuition can enhance modern machine learning architectures.

For computational science and engineering communities, LFNO's ability to handle diverse benchmark problems—from nonlinear ODEs like Lorenz and Duffing systems to complex PDEs including Navier-Stokes—suggests practical value across applications ranging from fluid dynamics simulations to structural analysis. The component-wise decomposition improves physical interpretability, allowing practitioners to understand which mechanisms drive system behavior at different time scales, rather than treating the model as a black box.

The competitive results against FNO on PDE benchmarks while dominating ODE systems indicate this framework fills a genuine capability gap. Future development likely involves scaling LFNO to higher-dimensional systems and exploring its potential in real-world engineering applications where transient-steady dynamics decomposition provides engineering insights.

• →LFNO combines Laplace and Fourier transforms in a dual-branch architecture to model both transient and steady-state dynamics.
• →The framework significantly outperforms existing operators on ODE systems and maintains competitive performance on PDE benchmarks.
• →Component-wise decomposition improves physical interpretability compared to black-box neural operator approaches.
• →Tested on nine diverse benchmarks including Navier-Stokes, Burgers, and reaction-diffusion equations.
• →Enhanced stability properties make LFNO suitable for long-horizon predictions across multiple temporal scales.