We investigate image segmentation of cells under the lens of scalar fields. Our goal is to learn a continuous scalar field on image domains such that its segmentation produces robust instances for cells present in images. This field is a function parameterized by the trained network, and its segmentation is realized by the watershed method. The fields we experiment with are solutions to the Poisson partial differential equation and a diffusion mimicking the steady-state solution of the heat equation. These solutions are obtained by minimizing just the field residuals, no regularization is needed, providing a robust regression capable of diminishing the adverse impacts of outliers in the training data and allowing for sharp cell boundaries. A single tensor is all that is needed to train a \unet\ thus simplifying implementation, lowering training and inference times, hence reducing energy consumption, and requiring a small memory footprint, all attractive features in edge computing. We present competitive results on public datasets from the literature and show that our novel, simple yet geometrically insightful approach can achieve excellent cell segmentation results.