# Wide-band butterfly network: stable and efficient inversion via multi-frequency neural networks

arXiv: https://arxiv.org/abs/2011.12413 · HTML: https://mlsanity.com/paper/2011.12413/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Nov 2020.

## Built on by (most cited first, 7 of 7)

- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Operator learning for hyperbolic partial differential equations](https://mlsanity.com/paper/2312.17489/) (2023)
- [Accurate and Robust Deep Learning Framework for Solving Wave-Based Inverse Problems in the Super-Resolution Regime](https://mlsanity.com/paper/2106.01143/) (2021)
- [Back-Projection Diffusion: Solving the Wideband Inverse Scattering Problem with Diffusion Models](https://mlsanity.com/paper/2408.02866/) (2024)
- [Solving the Wide-band Inverse Scattering Problem via Equivariant Neural Networks](https://mlsanity.com/paper/2212.06068/) (2022)
- [Efficient Tensor Completion Algorithms for Highly Oscillatory Operators](https://mlsanity.com/paper/2510.17734/) (2025)
- [A hybrid CNN-adjoint optimization framework for reconstruction of viscoelastic tissue properties in magnetic resonance elastography](https://mlsanity.com/paper/2610.02634/) (2026)

## Abstract (arXiv)

We introduce an end-to-end deep learning architecture called the wide-band butterfly network (WideBNet) for approximating the inverse scattering map from wide-band scattering data. This architecture incorporates tools from computational harmonic analysis, such as the butterfly factorization, and traditional multi-scale methods, such as the Cooley-Tukey FFT algorithm, to drastically reduce the number of trainable parameters to match the inherent complexity of the problem. As a result WideBNet is efficient: it requires fewer training points than off-the-shelf architectures, and has stable training dynamics, thus it can rely on standard weight initialization strategies. The architecture automatically adapts to the dimensions of the data with only a few hyper-parameters that the user must specify. WideBNet is able to produce images that are competitive with optimization-based approaches, but at a fraction of the cost, and we also demonstrate numerically that it learns to super-resolve scatterers in the full aperture scattering setup.