Reliable mechanistic operator recovery with biologically-informed neural networks: principles for architecture and optimisation design

Abstract

Many biological processes are governed by complex dynamical mechanisms that remain incompletely understood despite increasing volumes of experimental data. Biologically-informed neural networks (BINNs) seek to address this challenge by embedding differential equations into neural network training, enabling constitutive operators to be recovered directly from sparse and noisy observations. However, the extent to which operator recovery depends on architectural design, optimisation strategy and the information within the data is not yet well understood. We present an empirical study of how these factors influence mechanistic inference using BINNs applied to one-dimensional advection-diffusion-reaction partial differential equations. Across a suite of problems, we investigate how network expressivity, learning rate, loss weighting and batch size influence optimisation behaviour, reconstruction accuracy and operator recovery. We show that mechanistic inference is governed by balancing competing objectives rather than maximising any single aspect. Moderately expressive architectures outperform complex networks, intermediate learning rates balance efficient exploration with optimisation stability, accurate operator recovery requires a balance between data-fitting and PDE residual losses and intermediate batch sizes provide the best compromise between efficient parameter space exploration, computational efficiency and reproducibility. We further identify practical diagnostics for recognising common failure modes, including over-fitting, unstable optimisation and poor mechanistic recovery. These findings establish guidelines for deploying BINNs as credible tools for biological model discovery and demonstrate that reliable mechanistic inference is achieved by appropriately balancing model expressivity, optimisation, physical consistency and data informativeness.