PDE-Constrained Optimization for Neural Image Segmentation with Physics Priors

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.

💡 Research Summary

This paper tackles the challenging problem of microscopy image segmentation by reformulating it as a PDE‑constrained inverse problem and embedding physically motivated regularizers directly into a deep neural network. The authors start by defining a continuous segmentation field u(x) ∈ (0, 1) that represents the probability of each pixel belonging to the foreground. A UNet‑style encoder‑decoder network Nθ produces this field from the raw image I(x). Rather than relying solely on empirical risk minimization (e.g., Dice and binary cross‑entropy), the authors augment the loss with two differentiable residual terms that encode prior knowledge derived from partial differential equations.

The first prior is a reaction‑diffusion (RD) model. Assuming steady‑state, the PDE D∇²u + f(u) = 0 must hold, where f(u)=u(1−u)(u−a) is a cubic bistable reaction term that pushes the field toward the binary states 0 and 1. The residual r_RD(u)=D∇²u + f(u) is squared and integrated over the image domain to form L_RD. This term penalizes spurious oscillations and enforces smooth, membrane‑like transitions.

The second prior is a phase‑field (PF) interface energy inspired by the Van der Waals‑Cahn‑Hilliard functional: E_PF(u)=∫