  Sequential deep learning models such as RNN, causal CNN and attention
mechanism do not readily consume continuous-time information. Discretizing the
temporal data, as we show, causes inconsistency even for simple continuous-time
processes. Current approaches often handle time in a heuristic manner to be
consistent with the existing deep learning architectures and implementations.
In this paper, we provide a principled way to characterize continuous-time
systems using deep learning tools. Notably, the proposed approach applies to
all the major deep learning architectures and requires little modifications to
the implementation. The critical insight is to represent the continuous-time
system by composing neural networks with a temporal kernel, where we gain our
intuition from the recent advancements in understanding deep learning with
Gaussian process and neural tangent kernel. To represent the temporal kernel,
we introduce the random feature approach and convert the kernel learning
problem to spectral density estimation under reparameterization. We further
prove the convergence and consistency results even when the temporal kernel is
non-stationary, and the spectral density is misspecified. The simulations and
real-data experiments demonstrate the empirical effectiveness of our temporal
kernel approach in a broad range of settings.
