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

Derivatives of differential equation solutions are commonly for parameter estimation, 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 to be the most effective on their particular application. In this manuscript we investigate the performance characteristics of Discrete Local Sensitivity Analysis implemented via Automatic Differentiation (DSAAD) against continuous adjoint sensitivity analysis. Non-stiff and stiff biological and pharmacometric models, including a PDE discretization, are used to quantify the performance of sensitivity analysis methods. Our benchmarks show that on small systems of ODEs (approximately $<100$ parameters+ODEs), forward-mode DSAAD is more efficient than both reverse-mode and continuous forward/adjoint sensitivity analysis. The scalability of continuous adjoint methods is shown to be more efficient than discrete adjoints and forward methods after crossing this size range. These comparative studies demonstrate a trade-off between memory usage and performance in the continuous adjoint methods that should be considered when choosing the technique, while numerically unstable backsolve techniques from the machine learning literature are demonstrated as unsuitable for most scientific models. The performance of adjoint methods is shown to be heavily tied to the reverse-mode AD method, with tape-based AD methods shown to be 2 orders of magnitude slower on nonlinear partial differential equations than static AD techniques. These results also demonstrate the applicability of DSAAD to differential-algebraic equations, delay differential equations, and hybrid differential equation systems, showcasing an ease of implementation advantage for DSAAD approaches.