# Solving Image PDEs with a Shallow Network

arXiv: https://arxiv.org/abs/2110.08327 · HTML: https://mlsanity.com/paper/2110.08327/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 32 references that are in the map.
- First posted on arXiv 15 Oct 2021.

## Builds on (most cited first, 16 of 16)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Perceptual Losses for Real-Time Style Transfer and Super-Resolution](https://mlsanity.com/paper/1603.08155/) (2016)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Handheld Multi-Frame Super-Resolution](https://mlsanity.com/paper/1905.03277/) (2019)
- [Multi-level Residual Networks from Dynamical Systems View](https://mlsanity.com/paper/1710.10348/) (2017)
- [DeepDownscale: a Deep Learning Strategy for High-Resolution Weather Forecast](https://mlsanity.com/paper/1808.05264/) (2018)
- [RAISR: Rapid and Accurate Image Super Resolution](https://mlsanity.com/paper/1606.01299/) (2016)
- [Learned Perceptual Image Enhancement](https://mlsanity.com/paper/1712.02864/) (2017)
- [BLADE: Filter Learning for General Purpose Computational Photography](https://mlsanity.com/paper/1711.10700/) (2017)
- [Approximating the solution to wave propagation using deep neural networks](https://mlsanity.com/paper/1812.01609/) (2018)
- [Fast, Trainable, Multiscale Denoising](https://mlsanity.com/paper/1802.06130/) (2018)

## Built on by (most cited first, 1 of 1)

- [Unsupervised Learning of the Total Variation Flow](https://mlsanity.com/paper/2206.04406/) (2022)

## Abstract (arXiv)

Partial differential equations (PDEs) are typically used as models of physical processes but are also of great interest in PDE-based image processing. However, when it comes to their use in imaging, conventional numerical methods for solving PDEs tend to require very fine grid resolution for stability, and as a result have impractically high computational cost. This work applies BLADE (Best Linear Adaptive Enhancement), a shallow learnable filtering framework, to PDE solving, and shows that the resulting approach is efficient and accurate, operating more reliably at coarse grid resolutions than classical methods. As such, the model can be flexibly used for a wide variety of problems in imaging.