LFNO通过分解瞬态与稳态动态，统一建模复杂系统演化过程。
LFNO: Bridging Laplace and Fourier via Transient-Steady Decomposition
- 双分支结构分离瞬态与稳态成分，提升建模精度。
- 在9个基准上优于现有方法，尤其在瞬态主导的常微分方程中表现突出。
- 兼具物理可解释性与稳定性，适合多时标动力系统研究者。
我们提出拉普拉斯-傅里叶神经算子（LFNO），一种统一框架，用于建模跨瞬态与稳态阶段的动力系统，融合拉普拉斯与傅里叶神经算子的谱优势。LFNO采用双分支架构，显式分解系统动态为瞬态与稳态分量。我们在九个基准上评估：三个常微分方程系统（Duffing、Lorenz、Pendulum）和六个偏微分方程系统（Euler-Bernoulli梁、热方程、反应-扩散、Brusselator、Burgers、Navier-Stokes）。LFNO在常微分方程系统中显著优于现有算子，且在瞬态动态占主导时表现更优；同时持续超越LNO，并在偏微分方程基准上达到与FNO相当的性能。此外，其分量分解提升了稳定性和物理可解释性。结果表明，LFNO为学习多时标复杂动力系统提供了一种鲁棒且统一的方法。
原文摘要 · Abstract (English)
We introduce the Laplace-Fourier Neural Operator (LFNO), a unified framework for modeling dynamical systems across transient and steady-state regimes by integrating the spectral advantages of Laplace and Fourier Neural Operators. LFNO employs a dual-branch architecture that explicitly decomposes system dynamics into transient and steady-state components. We evaluate LFNO on nine benchmarks, including three ODE systems (Duffing, Lorenz, and Pendulum) and six PDE systems (Euler-Bernoulli beam, Heat, Reaction-diffusion, Brusselator, Burgers, and Navier-Stokes). LFNO significantly outperforms existing operators on ODE systems, where transient dynamics dominate, and consistently surpasses LNO while achieving performance competitive with FNO on PDE benchmarks. Furthermore, LFNO offers improved stability and physical interpretability through its component-wise decomposition. These results demonstrate that LFNO provides a robust and unified approach for learning complex dynamical systems across multiple temporal scales.
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