Dynamic frequency-gated fourier neural operators: architecture, principles, and applications

Abstract

Solving parametric partial differential equations (PDEs) has historically been an area of research focused on identifying computationally efficient methods. Because analytic solutions are often impractical, numerical solvers have traditionally been used to approximate these functions. However, with the rise of deep learning, data-driven methods are now widely used and can learn the underlying PDE operators that govern complex fluid mechanics [1, 2]. In this thesis, we propose a dynamic adaptive Fourier neural operator (DA-FNO) that employs frequency gating as a masking mechanism to modulate spectral content directly in the frequency domain. The prevailing “one-size-fits-all” treatment of Fourier modes in the original FNO [3] limits dynamism when certain instances demand finer spectral focus. By learning a frequency gate 𝐺 that controls both the count of modes and their granularity—per quadrant, per sample—within each Fourier layer, the model bypasses costly convolutions on irrelevant modes, yielding a more efficient and adaptive architecture. We train the baseline FNO, our DA-FNO, and several ablated variants on the publicly released MegaFlow2D benchmark dataset.