Physics-Informed Neural Networks for Biological $2\mathrm{D}{+}t$ Reaction-Diffusion Systems

TL;DR

This paper extends physics-informed neural networks to learn and identify 2D reaction-diffusion systems from biological data, enabling discovery of governing equations in complex spatio-temporal systems.

Contribution

It introduces a framework combining BINNs with symbolic regression for explicit equation discovery in 2D+time systems, demonstrated on lung cancer cell dynamics.

Findings

Successfully learned 2D reaction-diffusion models from microscopy data.

Extended BINNs to 2D+time systems with real-world biological applications.

Framework enables fast, interpretable analytic equation discovery.

Abstract

Physics-informed neural networks (PINNs) provide a powerful framework for learning governing equations of dynamical systems from data. Biologically-informed neural networks (BINNs) are a variant of PINNs that preserve the known differential operator structure (e.g., reaction-diffusion) while learning constitutive terms via trainable neural subnetworks, enforced through soft residual penalties. Existing BINN studies are limited to $1\mathrm{D}{+}t$ reaction-diffusion systems and focus on forward prediction, using the governing partial differential equation as a regulariser rather than an explicit identification target. Here, we extend BINNs to $2\mathrm{D}{+}t$ systems within a PINN framework that combines data preprocessing, BINN-based equation learning, and symbolic regression post-processing for closed-form equation discovery. We demonstrate the framework's real-world applicability by learning the governing equations of lung cancer cell population dynamics from time-lapse microscopy data, recovering $2\mathrm{D}{+}t$ reaction-diffusion models from experimental observations. The proposed framework is readily applicable to other spatio-temporal systems, providing a practical and interpretable tool for fast analytic equation discovery from data.