John T. Nardini

William Lavery, Jodie A. Cochrane, Christian Olesen et al.

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.

John T. Nardini, D. M. Bortz

The inverse problem methodology is a commonly-used framework in the sciences for parameter estimation and inference. It is typically performed by fitting a mathematical model to noisy experimental data. There are two significant sources of error in the process: 1.\ Noise from the measurement and collection of experimental data and 2.\ numerical error in approximating the true solution to the mathematical model. Little attention has been paid to how this second source of error alters the results of an inverse problem. As a first step towards a better understanding of this problem, we present a modeling and simulation study using a simple advection-driven PDE model. We present both analytical and computational results concerning how the different sources of error impact the least squares cost function as well as parameter estimation and uncertainty quantification. We investigate residual patterns to derive an autocorrelative statistical model that can improve parameter estimation and confidence interval computation for first order methods. Building on the results of our investigation, we provide guidelines for practitioners to determine when numerical or experimental error is the main source of error in their inference, along with suggestions of how to efficiently improve their results.

Maria-Veronica Ciocanel, John T. Nardini, Kevin B. Flores et al.

Agent-based modeling (ABM) is a powerful tool for understanding self-organizing biological systems, but it is computationally intensive and often not analytically tractable. Equation learning (EQL) methods can derive continuum models from ABM data, but they typically require extensive simulations for each parameter set, raising concerns about generalizability. In this work, we extend EQL to Multi-experiment equation learning (ME-EQL) by introducing two methods: one-at-a-time ME-EQL (OAT ME-EQL), which learns individual models for each parameter set and connects them via interpolation, and embedded structure ME-EQL (ES ME-EQL), which builds a unified model library across parameters. We demonstrate these methods using a birth--death mean-field model and an on-lattice agent-based model of birth, death, and migration with spatial structure. Our results show that both methods significantly reduce the relative error in recovering parameters from agent-based simulations, with OAT ME-EQL offering better generalizability across parameter space. Our findings highlight the potential of equation learning from multiple experiments to enhance the generalizability and interpretability of learned models for complex biological systems.

John T. Nardini, Charles W. J. Pugh, Helen M. Byrne

Disease complications can alter vascular network morphology and disrupt tissue functioning. Diabetic retinopathy, for example, is a complication of types 1 and 2 diabetes mellitus that can cause blindness. Microvascular diseases are assessed by visual inspection of retinal images, but this can be challenging when diseases exhibit silent symptoms or patients cannot attend in-person meetings. We examine the performance of machine learning algorithms in detecting microvascular disease when trained on statistical and topological summaries of segmented retinal vascular images. We apply our methods to three publicly-available datasets and find that, among the 13 total descriptor vectors we consider, either a statistical Box-counting descriptor vector or a topological Flooding descriptor vector achieves the highest accuracy levels on these datasets. We then created a fourth dataset by merging several datasets: the Box-counting vector outperforms all descriptors on this dataset, including the topological Flooding vector which is sensitive to differences in the annotation styles within the combined dataset. Our work represents a first step to establishing which computational methods are most suitable for identifying microvascular disease as well as some of their current limitations. In the longer term, these methods could be incorporated into automated disease assessment tools.