PINN Framework Pipeline for Microscopy Data

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Paper title: Physics-Informed Neural Networks for Biological $2\mathrm{D}{+}t$ Reaction-Diffusion Systems 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. Passages referencing this figure: ^ ​ ( u ^ ) \hat{G}(\hat{u}) . Here, u ^ \hat{u} , D ^ \hat{D} , and G ^ \hat{G} denote neural network surrogates of the functions u u , D D , and G G , in ( 2 ). All network parameters { θ u , θ D , θ G } \{\theta_{u},\theta_{D},\theta_{G}\} are jointly optimised to fit the data while satisfying the PDE constraints and optional biological restrictions. The general architecture is illustrated in Fig. 1 , and we here follow the BINN design choices and training protocols discussed in our recent study [ 18 ] . These choices and protocols are summarised below. Figure 1: PINN framework developed in this work (top) with the BINN architecture highlighted (bottom). II-B 1 MLP design Each MLP consists of three hidden layers of equal width followed by a single-neuron output layer, providing suffici All network parameters { θ u , θ D , θ G } \{\theta_{u},\theta_{D},\theta_{G}\} are jointly optimised to fit the data while satisfying the PDE constraints and optional biological restrictions. The general architecture is illustrated in Fig. 1 , and we here follow the BINN design choices and training protocols discussed in our recent study [ 18 ] . These choices and protocols are summarised below. Figure 1: PINN framework developed in this work (top) with the BINN architecture highlighted (bottom). II-B 1 MLP design Each MLP consists of three hidden layers of equal width followed by a single-neuron output layer, providing sufficient depth to capture nonlinear relationships while maintaining a compact architecture. All hidden layers use SiLU activations. The density MLP NN u \mathrm{NN}_{u}