Physics-Informed Neural Networks for Biological $2\mathrm{D}{+}t$ Reaction-Diffusion Systems
2026-04-20 • Machine Learning
Machine Learning
AI summaryⓘ
The authors extended a special type of neural network called biologically-informed neural networks (BINNs) to work with more complex systems that change over space and time in two dimensions. Unlike before, their method not only predicts outcomes but also helps discover the actual equations governing these systems by combining data processing, learning, and a step that finds simple mathematical formulas. They tested their approach on data from lung cancer cell growth, successfully uncovering the underlying reaction-diffusion equations. This framework can be used for other systems that evolve over space and time and helps scientists find understandable mathematical models from data.
Physics-Informed Neural Networks (PINNs)Biologically-Informed Neural Networks (BINNs)Reaction-Diffusion SystemsPartial Differential EquationsSpatio-Temporal ModelingNeural Network Equation LearningSymbolic RegressionTime-Lapse MicroscopyMathematical Biology
Authors
William Lavery, Jodie A. Cochrane, Christian Olesen, Dagim S. Tadele, John T. Nardini, Sara Hamis
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.