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.

翻译：物理信息神经网络（PINNs）为从数据中学习动态系统的控制方程提供了强大框架。生物信息神经网络（BINNs）作为PINNs的变体，在保留已知微分算子结构（如反应-扩散）的同时，通过可训练神经网络子网络学习本构项，并采用软残差惩罚进行约束。现有BINN研究局限于一维加时间反应-扩散系统，且聚焦于正向预测，将控制偏微分方程作为正则化项而非显式辨识目标。本文在PINN框架内将BINN扩展至二维加时间系统，该框架整合了数据预处理、基于BINN的方程学习以及用于闭式方程发现的符号回归后处理。我们通过从延时显微数据中学习肺癌细胞种群动态的控制方程，展示了该框架的实际应用价值，从实验观测中恢复了二维加时间反应-扩散模型。所提框架可直接应用于其他时空系统，为从数据中快速发现解析方程提供了一种实用且可解释的工具。