A Practical Tutorial on Physics-Informed Networks for Pharmacometrics and Quantitative Systems Pharmacology

Inverse problems in pharmacometrics and quantitative systems pharmacology (QSP) often involve sparse, noisy data, limited measurable states, and complex dynamical systems. Traditional parameter estimation methods can struggle with ill-posed problems, stiffness, discontinuities, and gray-box scenarios where only part of the system dynamics is known or observable. This tutorial provides an end-to-end, reproducible workflow for applying physics-informed neural networks (PINNs) to PK/PD/QSP inverse problems and gray-box discovery tasks. We present three worked case studies: (1) constant parameter recovery in a three-state compartmental system, (2) gray-box discovery of an unknown right-hand-side function, and (3) inference of a time-varying chemotherapy efficacy function under partial observation. For each, we provide implementation guidance, validation checks, and practical advice on collocation design, feature expansion, optimizer selection, numerical precision, constraint enforcement, loss weighting, and residual-based attention. We also discuss when to consider newer architectures such as Kolmogorov-Arnold networks (KANs) as drop-in alternatives to standard multilayer perceptrons. All code and notebooks are provided through our PhINs library, short for Pharmacometrics-Informed Networks, and are publicly available at https://github.com/NazAhmadi/PhINs/.