# PDE-Constrained Optimization for Neural Image Segmentation with Physics Priors

arXiv: https://arxiv.org/abs/2602.01069 · HTML: https://mlsanity.com/paper/2602.01069/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 36 references that are in the map.
- First posted on arXiv 1 Feb 2026.

## Builds on (most cited first, 17 of 17)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Microsoft COCO: Common Objects in Context](https://mlsanity.com/paper/1405.0312/) (2014)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [TransUNet: Transformers Make Strong Encoders for Medical Image Segmentation](https://mlsanity.com/paper/2102.04306/) (2021)
- [UNet++: A Nested U-Net Architecture for Medical Image Segmentation](https://mlsanity.com/paper/1807.10165/) (2018)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [Attention U-Net: Learning Where to Look for the Pancreas](https://mlsanity.com/paper/1804.03999/) (2018)
- [Image Segmentation Using Deep Learning: A Survey](https://mlsanity.com/paper/2001.05566/) (2020)
- [Domain Adaptation for Medical Image Analysis: A Survey](https://mlsanity.com/paper/2102.09508/) (2021)
- [A Probabilistic U-Net for Segmentation of Ambiguous Images](https://mlsanity.com/paper/1806.05034/) (2018)
- [A Review on Deep Learning Techniques Applied to Semantic Segmentation](https://mlsanity.com/paper/1704.06857/) (2017)
- [RTSeg: Real-time Semantic Segmentation Comparative Study](https://mlsanity.com/paper/1803.02758/) (2018)
- [PINNs for Medical Image Analysis: A Survey](https://mlsanity.com/paper/2408.01026/) (2024)
- [Variational reaction-diffusion systems for semantic segmentation](https://mlsanity.com/paper/1604.00092/) (2016)
- [Learning to Solve Optimization Problems Constrained with Partial Differential Equations](https://mlsanity.com/paper/2509.24573/) (2025)
- [A Physics-Informed Loss Function for Boundary-Consistent and Robust Artery Segmentation in DSA Sequences](https://mlsanity.com/paper/2511.20501/) (2025)

## Abstract (arXiv)

Segmentation of microscopy images constitutes an ill-posed inverse problem due to measurement noise, weak object boundaries, and limited labeled data. Although deep neural networks provide flexible nonparametric estimators, unconstrained empirical risk minimization often leads to unstable solutions and poor generalization. In this work, image segmentation is formulated as a PDE-constrained optimization problem that integrates physically motivated priors into deep learning models through variational regularization. The proposed framework minimizes a composite objective function consisting of a data fidelity term and penalty terms derived from reaction-diffusion equations and phase-field interface energies, all implemented as differentiable residual losses. Experiments are conducted on the LIVECell dataset, a high-quality, manually annotated collection of phase-contrast microscopy images. Training is performed on two cell types, while evaluation is carried out on a distinct, unseen cell type to assess generalization. A UNet architecture is used as the unconstrained baseline model. Experimental results demonstrate consistent improvements in segmentation accuracy and boundary fidelity compared to unconstrained deep learning baselines. Moreover, the PDE-regularized models exhibit enhanced stability and improved generalization in low-sample regimes, highlighting the advantages of incorporating structured priors. The proposed approach illustrates how PDE-constrained optimization can strengthen data-driven learning frameworks, providing a principled bridge between variational methods, statistical learning, and scientific machine learning.