Stably Accelerating Stiff Quantitative Systems Pharmacology Models: Continuous-Time Echo State Networks as Implicit Machine Learning

In this paper, we propose the use of neural network surrogates of stiff QsP models, which reduces and accelerates QsP models by training ML approximations on simulations. We describe how common neural network methodologies, such as residual neural networks, recurrent neural networks, and physics/biologically-informed neural networks, are fundamentally related to explicit solvers of ordinary differential equations (ODEs), and thus are ill-equipped to deal with stiffness. To address this issue, we showcase methods from scientific machine learning (SciML) which combine techniques from mechanistic modeling with traditional deep learning. We describe the continuous-time echo state network (CTESN) as the implicit analogue of ML architectures. We demonstrate the CTESN's ability to surrogatize a production QsP model, a >1,000 ODE chemical reaction system from the SBML Biomodels repository, and a reaction-diffusion partial differential equation. We showcase the ability to accelerate QsP simulations by up to 5.2x against the optimized Differential Equations. jl solvers while achieving <5% relative error. This shows how incorporating the numerical properties of QsP methods into ML can improve the intersection, and thus presents a potential method for accelerating repeated calculations such as global sensitivity analysis and virtual populations.Copyright (c) 2022 The Authors This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)