✨ TL;DR
This paper extends biologically-informed neural networks (BINNs) from 1D to 2D spatial domains for learning reaction-diffusion equations from data, combining neural network training with symbolic regression to discover closed-form equations. The method is demonstrated on real lung cancer cell microscopy data, successfully recovering interpretable 2D+time reaction-diffusion models.
Physics-informed neural networks (PINNs) have shown promise for learning dynamical systems from data, but existing biologically-informed neural network (BINN) approaches are limited to one-dimensional spatial systems. Real biological systems often exhibit complex spatio-temporal dynamics in two or three spatial dimensions that cannot be adequately captured by 1D models. Furthermore, previous BINN studies have focused primarily on forward prediction tasks rather than explicit equation identification, using governing equations as regularizers rather than as targets for discovery. There is a need for methods that can learn interpretable, closed-form governing equations from real experimental data in higher-dimensional spatial settings.
The authors develop a three-stage framework that extends BINNs to 2D+time reaction-diffusion systems. First, they preprocess experimental data to extract clean spatio-temporal measurements. Second, they employ a BINN architecture that preserves the known reaction-diffusion operator structure while using trainable neural subnetworks to learn the unknown reaction and diffusion terms, enforced through soft residual penalties in the loss function. The neural networks approximate the constitutive relationships (reaction kinetics and diffusion coefficients) while respecting the underlying PDE structure. Third, they apply symbolic regression post-processing to the learned neural network representations to extract closed-form analytical expressions for the governing equations. The framework is demonstrated on time-lapse microscopy data of lung cancer cell populations, where cell density evolves according to unknown reaction-diffusion dynamics.
What the paper shows.
The framework successfully learned governing equations for lung cancer cell population dynamics from time-lapse microscopy data. The method recovered 2D+time reaction-diffusion models that captured the spatio-temporal evolution of cell populations in the experimental observations. By applying symbolic regression to the trained neural networks, the authors obtained closed-form analytical expressions for both the reaction terms (describing cell proliferation and death) and diffusion coefficients (describing cell migration). The discovered equations were interpretable and consistent with the known physics of reaction-diffusion systems, demonstrating that the approach can extract meaningful mathematical models from real-world biological data.
The paper does not provide detailed quantitative metrics comparing the accuracy of discovered equations against ground truth or alternative methods. The framework requires three separate stages (preprocessing, BINN training, symbolic regression), which may introduce errors at each step and require careful tuning. The method's performance on systems with more complex dynamics, multiple interacting species, or highly nonlinear terms is not extensively explored. The reliance on symbolic regression in the final stage may limit the complexity of equations that can be discovered, and the approach may struggle with systems where the reaction-diffusion assumption does not hold. Computational costs and scalability to 3D systems or larger datasets are not discussed.
✨ Generated by Claude · Apr 21, 2026 · Read the PDF for authoritative content.