# Transform Once: Efficient Operator Learning in Frequency Domain

arXiv: https://arxiv.org/abs/2211.14453 · HTML: https://mlsanity.com/paper/2211.14453/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 54 references that are in the map.
- First posted on arXiv 26 Nov 2022.

## Builds on (most cited first, 24 of 24)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Perceiver IO: A General Architecture for Structured Inputs & Outputs](https://mlsanity.com/paper/2107.14795/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [COIN: COmpression with Implicit Neural representations](https://mlsanity.com/paper/2103.03123/) (2021)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Fast Training of Convolutional Networks through FFTs](https://mlsanity.com/paper/1312.5851/) (2013)
- [Spectral Representations for Convolutional Neural Networks](https://mlsanity.com/paper/1506.03767/) (2015)
- [Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networks](https://mlsanity.com/paper/2203.08852/) (2022)
- [Hypersolvers: Toward Fast Continuous-Depth Models](https://mlsanity.com/paper/2007.09601/) (2020)
- [Compressing Convolutional Neural Networks](https://mlsanity.com/paper/1506.04449/) (2015)
- [Self-Similarity Priors: Neural Collages as Differentiable Fractal Representations](https://mlsanity.com/paper/2204.07673/) (2022)
- [Neural Solvers for Fast and Accurate Numerical Optimal Control](https://mlsanity.com/paper/2203.08072/) (2022)

## Built on by (most cited first, 20 of 20)

- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning](https://mlsanity.com/paper/2402.15734/) (2024)
- [Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspective](https://mlsanity.com/paper/2404.07200/) (2024)
- [Sampling weights of deep neural networks](https://mlsanity.com/paper/2306.16830/) (2023)
- [Neural Spectral Methods: Self-supervised learning in the spectral domain](https://mlsanity.com/paper/2312.05225/) (2023)
- [SineNet: Learning Temporal Dynamics in Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2403.19507/) (2024)
- [Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective](https://mlsanity.com/paper/2310.06379/) (2023)
- [Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space](https://mlsanity.com/paper/2410.11382/) (2024)
- [Towards Expressive Spectral-Temporal Graph Neural Networks for Time Series Forecasting](https://mlsanity.com/paper/2305.06587/) (2023)
- [SAOT: An Enhanced Locality-Aware Spectral Transformer for Solving PDEs](https://mlsanity.com/paper/2511.18777/) (2025)
- [Learning Space-Time Continuous Neural PDEs from Partially Observed States](https://mlsanity.com/paper/2307.04110/) (2023)
- [A Two-Phase Deep Learning Framework for Adaptive Time-Stepping in High-Speed Flow Modeling](https://mlsanity.com/paper/2506.07969/) (2025)
- [Learning Efficient Surrogate Dynamic Models with Graph Spline Networks](https://mlsanity.com/paper/2310.16397/) (2023)
- [Hierarchical-embedding autoencoder with a predictor (HEAP) as efficient architecture for learning long-term evolution of complex multi-scale physical systems](https://mlsanity.com/paper/2505.18857/) (2025)
- [Can Data-Driven Dynamics Reveal Hidden Physics? There Is A Need for Interpretable Neural Operators](https://mlsanity.com/paper/2510.02683/) (2025)
- [LASER: Learning Active Sensing for Continuum Field Reconstruction](https://mlsanity.com/paper/2604.19355/) (2026)
- [Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs](https://mlsanity.com/paper/2402.09018/) (2024)
- [Generalizing to New Dynamical Systems via Frequency Domain Adaptation](https://mlsanity.com/paper/2507.00025/) (2025)
- [Deep Wave Network for Modeling Multi-Scale Physical Dynamics](https://mlsanity.com/paper/2605.04198/) (2026)
- [Scale-Split Neural Operator for Memory- and Data-Efficient 3D Turbulence Prediction](https://mlsanity.com/paper/2609.38977/) (2026)

## Abstract (arXiv)

Spectral analysis provides one of the most effective paradigms for information-preserving dimensionality reduction, as simple descriptions of naturally occurring signals are often obtained via few terms of periodic basis functions. In this work, we study deep neural networks designed to harness the structure in frequency domain for efficient learning of long-range correlations in space or time: frequency-domain models (FDMs). Existing FDMs are based on complex-valued transforms i.e. Fourier Transforms (FT), and layers that perform computation on the spectrum and input data separately. This design introduces considerable computational overhead: for each layer, a forward and inverse FT. Instead, this work introduces a blueprint for frequency domain learning through a single transform: transform once (T1). To enable efficient, direct learning in the frequency domain we derive a variance-preserving weight initialization scheme and investigate methods for frequency selection in reduced-order FDMs. Our results noticeably streamline the design process of FDMs, pruning redundant transforms, and leading to speedups of 3x to 10x that increase with data resolution and model size. We perform extensive experiments on learning the solution operator of spatio-temporal dynamics, including incompressible Navier-Stokes, turbulent flows around airfoils and high-resolution video of smoke. T1 models improve on the test performance of FDMs while requiring significantly less computation (5 hours instead of 32 for our large-scale experiment), with over 20% reduction in average predictive error across tasks.