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

2026-04-20

AI Summary

This work addresses the limitations of existing methods, which are confined to one-dimensional settings and support only forward prediction, by proposing the first framework for discovering governing reaction-diffusion equations from two-dimensional spatiotemporal ($2\mathrm{D}{+}t$) biological observation data. The approach integrates data preprocessing, a biologically informed neural network (BINN), and symbolic regression post-processing to enable data-driven identification of closed-form equations while embedding physical structural constraints. By introducing soft residual constraints and extending BINN to $2\mathrm{D}{+}t$ scenarios, the method successfully recovers interpretable reaction-diffusion models from time-lapse microscopy images of lung cancer cells, demonstrating its effectiveness and generalizability in real biological systems.

Technology Category

• Computer Vision: Medical and Biological Imaging
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Bio-inspired Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Web Mining and Content Analysis: Large pretrained models with web data

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.

Problem

• reaction-diffusion
• 2D+t systems
• equation discovery
• biological systems
• governing equations

Innovation

• Physics-Informed Neural Networks
• Biologically-Informed Neural Networks
• Reaction-Diffusion Systems
• Symbolic Regression
• Equation Discovery