  Many scientific problems focus on observed patterns of change or on how to
design a system to achieve particular dynamics. Those problems often require
fitting differential equation models to target trajectories. Fitting such
models can be difficult because each evaluation of the fit must calculate the
distance between the model and target patterns at numerous points along a
trajectory. The gradient of the fit with respect to the model parameters can be
challenging. Recent technical advances in automatic differentiation through
numerical differential equation solvers potentially change the fitting process
into a relatively easy problem, opening up new possibilities to study dynamics.
However, application of the new tools to real data may fail to achieve a good
fit. This article illustrates how to overcome a variety of common challenges,
using the classic ecological data for oscillations in hare and lynx
populations. Models include simple ordinary differential equations (ODEs) and
neural ordinary differential equations (NODEs), which use artificial neural
networks to estimate the derivatives of differential equation systems.
Comparing the fits obtained with ODEs versus NODEs, representing small and
large parameter spaces, and changing the number of variable dimensions provide
insight into the geometry of the observed and model trajectories. To analyze
the quality of the models for predicting future observations, a
Bayesian-inspired preconditioned stochastic gradient Langevin dynamics (pSGLD)
calculation of the posterior distribution of predicted model trajectories
clarifies the tendency for various models to underfit or overfit the data.
Coupling fitted differential equation systems with pSGLD sampling provides a
powerful way to study the properties of optimization surfaces, raising an
analogy with mutation-selection dynamics on fitness landscapes.
