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Derivatives of differential equation solutions are commonly for parameterestimation, fitting neural differential equations, and as model diagnostics.However, with a litany of choices and a Cartesian product of potential methods,it can be difficult for practitioners to understand which method is likely tobe the most effective on their particular application. In this manuscript weinvestigate the performance characteristics of Discrete Local SensitivityAnalysis implemented via Automatic Differentiation (DSAAD) against continuousadjoint sensitivity analysis. Non-stiff and stiff biological and pharmacometricmodels, including a PDE discretization, are used to quantify the performance ofsensitivity analysis methods. Our benchmarks show that on small systems of ODEs(approximately $<100$ parameters+ODEs), forward-mode DSAAD is more efficientthan both reverse-mode and continuous forward/adjoint sensitivity analysis. Thescalability of continuous adjoint methods is shown to be more efficient thandiscrete adjoints and forward methods after crossing this size range. Thesecomparative studies demonstrate a trade-off between memory usage andperformance in the continuous adjoint methods that should be considered whenchoosing the technique, while numerically unstable backsolve techniques fromthe machine learning literature are demonstrated as unsuitable for mostscientific models. The performance of adjoint methods is shown to be heavilytied to the reverse-mode AD method, with tape-based AD methods shown to be 2orders of magnitude slower on nonlinear partial differential equations thanstatic AD techniques. These results also demonstrate the applicability of DSAADto differential-algebraic equations, delay differential equations, and hybriddifferential equation systems, showcasing an ease of implementation advantagefor DSAAD approaches.

A Comparison of Automatic Differentiation and Continuous Sensitivity Analysis for Derivatives of Differential Equation Solutions