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Causal operators (CO), such as various solution operators to stochasticdifferential equations, play a central role in contemporary stochasticanalysis; however, there is still no canonical framework for designing DeepLearning (DL) models capable of approximating COs. This paper proposes a"geometry-aware'" solution to this open problem by introducing a DLmodel-design framework that takes suitable infinite-dimensional linear metricspaces as inputs and returns a universal sequential DL model adapted to theselinear geometries. We call these models Causal Neural Operators (CNOs). Ourmain result states that the models produced by our framework can uniformlyapproximate on compact sets and across arbitrarily finite-time horizonsH\"older or smooth trace class operators, which causally map sequences betweengiven linear metric spaces. Our analysis uncovers new quantitativerelationships on the latent state-space dimension of CNOs which even have newimplications for (classical) finite-dimensional Recurrent Neural Networks(RNNs). We find that a linear increase of the CNO's (or RNN's) latent parameterspace's dimension and of its width, and a logarithmic increase of its depthimply an exponential increase in the number of time steps for which itsapproximation remains valid. A direct consequence of our analysis shows thatRNNs can approximate causal functions using exponentially fewer parameters thanReLU networks.

Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis