Continuous-Time Machine Learning: A Unified Mathematical Perspective
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.
Code (1)
Similar Papers 제목 키워드 기반
Sinkhorn Flow: A Continuous-Time Framework for Understanding and Generalizing the Sinkhorn Algorithm
Many problems in machine learning can be formulated as solving entropy-regularized optimal transport on the space of probability measures. The canonical approach involves the Sinkhorn iterates, renowned for their rich ma…
On the Generalization Behavior of Deep Residual Networks From a Dynamical System Perspective
Deep neural networks (DNNs) have significantly advanced machine learning, with model depth playing a central role in their successes. The dynamical system modeling approach has recently emerged as a powerful framework, o…
Attacks in Adversarial Machine Learning: A Systematic Survey from the Life-cycle Perspective
Adversarial machine learning (AML) studies the adversarial phenomenon of machine learning, which may make inconsistent or unexpected predictions with humans. Some paradigms have been recently developed to explore this ad…
Adversarial AttackBackdoor AttackA Mathematical Explanation of Transformers
The Transformer architecture has revolutionized the field of sequence modeling and underpins the recent breakthroughs in large language models (LLMs). However, a comprehensive mathematical theory that explains its struct…
A Differential Manifold Perspective and Universality Analysis of Continuous Attractors in Artificial Neural Networks
Continuous attractors are critical for information processing in both biological and artificial neural systems, with implications for spatial navigation, memory, and deep learning optimization. However, existing research…