Automatic differentiation and the optimization of differential equation models in biology

TL;DR

This paper discusses how automatic differentiation combined with numerical methods for differential equations can enhance optimization in biological models, enabling better analysis of dynamic systems and trajectories.

Contribution

It explains how to perform automatic differentiation through numerical algorithms for differential equations, advancing biological modeling and analysis.

Findings

Method for differentiating trajectories computed by Runge-Kutta

Potential to improve biological model optimization

Facilitates analysis of dynamic biological systems

Abstract

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.