Automatic differentiation of ODE integration

TL;DR

This paper explores how to compute derivatives of ODE solutions using automatic differentiation, demonstrating methods with examples like Lotka-Volterra equations and ode23, including Hessian evaluation.

Contribution

It presents a detailed approach for differentiating ODE solvers with ADiMat, including manual construction of substitution functions and Hessian computation techniques.

Findings

Demonstrates differentiation of ODE solutions with ADiMat

Shows how to differentiate functions calling ode23

Provides methods for Hessian evaluation using reverse mode

Abstract

We discuss the calculation of the derivatives of ODE systems with the automatic differentiation tool ADiMat. Using the well-known Lotka-Volterra equations and the ode23 ODE solver as examples we show the analytic derivatives and detail how to differentiate a top-level function that calls ode23 somewhere with ADiMat. This involves the manual construction of substitution function to propagate the derivatives in forward and reverse mode. We also show how to use the reverse mode code to evaluate the Hessian in forward-over-reverse mode.