Appendix B proves universal approximation of the Neural CDE model, and is substantially more technical than the rest of this paper. Appendix C proves that the Neural CDE model subsumes alternative ODE models which depend directly and nonlinearly on the data. Appendix D gives the full details of every experiment, such as choice of optimiser, hyperparameter searches, and so on. To evaluate the model as discussed in Section 3.2, X must be at least continuous and piecewise differentiable. A.1 Differentiating with respect to the time points However, there is a technical caveat in the specific case that derivatives with respect to the initial time t A.2 Adaptive step size solvers There is one further caveat that must be considered.

Appendix B proves universal approximation of the Neural CDE model, and is substantially more technical than the rest of this paper. Appendix C proves that the Neural CDE model subsumes alternative ODE models which depend directly and nonlinearly on the data. Appendix D gives the full details of every experiment, such as choice of optimiser, hyperparameter searches, and so on. To evaluate the model as discussed in Section 3.2, X must be at least continuous and piecewise differentiable. A.1 Differentiating with respect to the time points However, there is a technical caveat in the specific case that derivatives with respect to the initial time t A.2 Adaptive step size solvers There is one further caveat that must be considered.

Neural ordinary differential equations are an attractive option for modelling temporal dynamics. However, a fundamental issue is that the solution to an ordinary differential equation is determined by its initial condition, and there is no mechanism for adjusting the trajectory based on subsequent observations. Here, we demonstrate how this may be resolved through the well-understood mathematics of \emph{controlled differential equations}. The resulting \emph{neural controlled differential equation} model is directly applicable to the general setting of partially-observed irregularly-sampled multivariate time series, and (unlike previous work on this problem) it may utilise memory-efficient adjoint-based backpropagation even across observations. We demonstrate that our model achieves state-of-the-art performance against similar (ODE or RNN based) models in empirical studies on a range of datasets. Finally we provide theoretical results demonstrating universal approximation, and that our model subsumes alternative ODE models.