Automatic differentiation and the optimization of differential equation models in biology
This work addresses the challenge of efficiently optimizing complex biological models for researchers in computational biology and related fields, representing an incremental advancement by extending automatic differentiation techniques to differential equations.
The paper tackles the problem of optimizing differential equation models in biology by applying automatic differentiation through numerical algorithms like Runge-Kutta to compute derivatives of performance measures over trajectories, enabling improved parameter estimation and model fitting. It highlights how this computational breakthrough can advance theoretical and statistical studies by unifying optimization across learning, evolutionary, and information landscapes.
A computational revolution unleashed the power of artificial neural networks. At the heart of that revolution is automatic differentiation, which calculates the derivative of a performance measure relative to a large number of parameters. Differentiation enhances the discovery of improved performance in large models, an achievement that was previously difficult or impossible. Recently, a second computational advance optimizes the temporal trajectories traced by differential equations. Optimization requires differentiating a measure of performance over a trajectory, such as the closeness of tracking the environment, with respect to the parameters of the differential equations. Because model trajectories are usually calculated numerically by multistep algorithms, such as Runge-Kutta, the automatic differentiation must be passed through the numerical algorithm. This article explains how such automatic differentiation of trajectories is achieved. It also discusses why such computational breakthroughs are likely to advance theoretical and statistical studies of biological problems, in which one can consider variables as dynamic paths over time and space. Many common problems arise between improving success in computational learning models over performance landscapes, improving evolutionary fitness over adaptive landscapes, and improving statistical fits to data over information landscapes.