Continuous-Time Machine Learning: A Unified Mathematical Perspective

Abstract

Continuous-time (CT) machine learning has emerged as a principled framework for modeling temporal dynamics as a continuous process, particularly when observations are sampled at arbitrary time points or span long-range horizons. However, major branches of CT machine learning have matured in separate research communities, leaving their mathematical relationships and design trade-offs insufficiently characterized. In this survey, we develop a unified, concept-driven view of major CT machine learning branches through a taxonomy that organizes families according to their underlying base mathematical formulations. We present a canonical mathematical formulation that relates these families through different architectural choices of vector-field parameterization, stochasticity, memory mechanisms, and discretization. We compare training algorithms, optimization strategies, and failure modes, highlighting the trade-offs across families. We further provide a comparative analysis of theoretical computational complexity alongside an illustrative architecture-controlled benchmark analysis on representative architectures from each family. We also review software ecosystems supporting their implementation. Finally, we identify open challenges in approximation theory, training stability, hardware-efficient implementations, benchmarking, foundation models, and scientific machine learning, and discuss an agenda for future research.