Frank, S. A. 2023. Optimizing differential equations to fit data and predict outcomes. Ecology and Evolution 13:e9895.

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.