# Double-well Net for Image Segmentation

arXiv: https://arxiv.org/abs/2401.00456 · HTML: https://mlsanity.com/paper/2401.00456/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 95 references that are in the map.
- First posted on arXiv 31 Dec 2023.

## Builds on (most cited first, 30 of 38)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Encoder-Decoder with Atrous Separable Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1802.02611/) (2018)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [UNet++: A Nested U-Net Architecture for Medical Image Segmentation](https://mlsanity.com/paper/1807.10165/) (2018)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Swin-Unet: Unet-like Pure Transformer for Medical Image Segmentation](https://mlsanity.com/paper/2105.05537/) (2021)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [UNet 3+: A Full-Scale Connected UNet for Medical Image Segmentation](https://mlsanity.com/paper/2004.08790/) (2020)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Deep Network Approximation Characterized by Number of Neurons](https://mlsanity.com/paper/1906.05497/) (2019)
- [Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recovery](https://mlsanity.com/paper/1908.01842/) (2019)
- [RGCNN: Regularized Graph CNN for Point Cloud Segmentation](https://mlsanity.com/paper/1806.02952/) (2018)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [MgNet: A Unified Framework of Multigrid and Convolutional Neural Network](https://mlsanity.com/paper/1901.10415/) (2019)
- [Approximation and Non-parametric Estimation of ResNet-type Convolutional Neural Networks](https://mlsanity.com/paper/1903.10047/) (2019)
- [Deep learning as optimal control problems: models and numerical methods](https://mlsanity.com/paper/1904.05657/) (2019)
- [Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks](https://mlsanity.com/paper/2109.02832/) (2021)
- [Deep Convolutional Neural Networks with Spatial Regularization, Volume and Star-shape Priori for Image Segmentation](https://mlsanity.com/paper/2002.03989/) (2020)
- [Deep nets for local manifold learning](https://mlsanity.com/paper/1607.07110/) (2016)
- [A Neural Network Approach for High-Dimensional Optimal Control Applied to Multi-Agent Path Finding](https://mlsanity.com/paper/2104.03270/) (2021)
- [PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks](https://mlsanity.com/paper/2307.09039/) (2023)
- [Multiclass Data Segmentation using Diffuse Interface Methods on Graphs](https://mlsanity.com/paper/1302.3913/) (2013)
- [Curvature Regularized Surface Reconstruction from Point Cloud](https://mlsanity.com/paper/2001.07884/) (2020)
- [Topology-Preserving 3D Image Segmentation Based On Hyperelastic Regularization](https://mlsanity.com/paper/2103.16768/) (2021)
- [A Regularized Convolutional Neural Network for Semantic Image Segmentation](https://mlsanity.com/paper/1907.05287/) (2019)
- [A Color Elastica Model for Vector-Valued Image Regularization](https://mlsanity.com/paper/2008.08255/) (2020)
- [DOSnet as a Non-Black-Box PDE Solver: When Deep Learning Meets Operator Splitting](https://mlsanity.com/paper/2212.05571/) (2022)

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

- [Deep Convolutional Neural Networks Meet Variational Shape Compactness Priors for Image Segmentation](https://mlsanity.com/paper/2406.19400/) (2024)
- [Euler's Elastica Based Cartoon-Smooth-Texture Image Decomposition](https://mlsanity.com/paper/2407.02794/) (2024)
- [A Mathematical Explanation of UNet](https://mlsanity.com/paper/2410.04434/) (2024)
- [A Mathematical Explanation of Transformers](https://mlsanity.com/paper/2510.03989/) (2025)
- [Image Segmentation via Variational Model Based Tailored UNet: A Deep Variational Framework](https://mlsanity.com/paper/2505.05806/) (2025)
- [Robust Variational Model Based Tailored UNet: Leveraging Edge Detector and Mean Curvature for Improved Image Segmentation](https://mlsanity.com/paper/2512.07590/) (2025)
- [Propagate, Then Sharpen: Post-Hoc Refinement of Frozen Node Classifiers](https://mlsanity.com/paper/2609.35080/) (2026)

## Abstract (arXiv)

In this study, our goal is to integrate classical mathematical models with deep neural networks by introducing two novel deep neural network models for image segmentation known as Double-well Nets. Drawing inspirations from the Potts model, our models leverage neural networks to represent a region force functional. We extend the well-know MBO (Merriman-Bence-Osher) scheme to solve the Potts model. The widely recognized Potts model is approximated using a double-well potential and then solved by an operator-splitting method, which turns out to be an extension of the well-known MBO scheme. Subsequently, we replace the region force functional in the Potts model with a UNet-type network, which is data-driven and is designed to capture multiscale features of images, and also introduce control variables to enhance effectiveness. The resulting algorithm is a neural network activated by a function that minimizes the double-well potential. What sets our proposed Double-well Nets apart from many existing deep learning methods for image segmentation is their strong mathematical foundation. They are derived from the network approximation theory and employ the MBO scheme to approximately solve the Potts model. By incorporating mathematical principles, Double-well Nets bridge the MBO scheme and neural networks, and offer an alternative perspective for designing networks with mathematical backgrounds. Through comprehensive experiments, we demonstrate the performance of Double-well Nets, showcasing their superior accuracy and robustness compared to state-of-the-art neural networks. Overall, our work represents a valuable contribution to the field of image segmentation by combining the strengths of classical variational models and deep neural networks. The Double-well Nets introduce an innovative approach that leverages mathematical foundations to enhance segmentation performance.