✨ TL;DR
This paper extends biologically-informed neural networks (BINNs) from 1D to 2D spatial domains for learning reaction-diffusion equations from data, combining neural network training with symbolic regression to discover closed-form equations. The method is demonstrated on real lung cancer cell microscopy data, successfully recovering interpretable 2D+time reaction-diffusion models.
Physics-informed neural networks (PINNs) have shown promise for learning dynamical systems from data, but existing biologically-informed neural network (BINN) approaches are limited to one-dimensional spatial systems. Real biological systems often exhibit complex spatio-temporal dynamics in two or three spatial dimensions that cannot be adequately captured by 1D models. Furthermore, previous BINN studies have focused primarily on forward prediction tasks rather than explicit equation identification, using governing equations as regularizers rather than as targets for discovery. There is a need for methods that can learn interpretable, closed-form governing equations from real experimental data in higher-dimensional spatial settings.
The authors develop a three-stage framework that extends BINNs to 2D+time reaction-diffusion systems. First, they preprocess experimental data to extract clean spatio-temporal measurements. Second, they employ a BINN architecture that preserves the known reaction-diffusion operator structure while using trainable neural subnetworks to learn the unknown reaction and diffusion terms, enforced through soft residual penalties in the loss function. The neural networks approximate the constitutive relationships (reaction kinetics and diffusion coefficients) while respecting the underlying PDE structure. Third, they apply symbolic regression post-processing to the learned neural network representations to extract closed-form analytical expressions for the governing equations. The framework is demonstrated on time-lapse microscopy data of lung cancer cell populations, where cell density evolves according to unknown reaction-diffusion dynamics.