Comprehensive Review of Neural Differential Equations for Time Series Analysis

Oh, Y., Kam, S., Lee, J., Lim, D.-Y., Kim, S., & Bui, A. A. T. (2025). Comprehensive Review of Neural Differential Equations for Time Series Analysis. In J. Kwok (Ed.), Proceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence, IJCAI-25 (pp. 10621–10631). International Joint Conferences on Artificial Intelligence Organization. https://doi.org/10.24963/ijcai.2025/1179 [paper]

Overview

Time series modeling and analysis have become critical in various domains. Conventional methods such as RNNs and Transformers, while effective for discrete-time and regularly sampled data, face significant challenges in capturing the continuous dynamics and irregular sampling patterns inherent in real-world scenarios. Neural Differential Equations (NDEs) represent a paradigm shift by combining the flexibility of neural networks with the mathematical rigor of differential equations. This paper presents a comprehensive review of NDE-based methods for time series analysis, including neural ordinary differential equations, neural controlled differential equations, and neural stochastic differential equations. We provide a detailed discussion of their mathematical formulations, numerical methods, and applications, highlighting their ability to model continuous-time dynamics. Furthermore, we address key challenges and future research directions. This survey serves as a foundation for researchers and practitioners seeking to leverage NDEs for advanced time series analysis.

This survey provides a comprehensive review of NDE-based methods for time series analysis, covering the three core families:

Neural Ordinary Differential Equations (NODEs)
Neural Controlled Differential Equations (NCDEs)
Neural Stochastic Differential Equations (NSDEs)

We offer a detailed discussion of their mathematical formulations, numerical methods, and key applications, highlighting their unique ability to model continuous-time dynamics. This survey serves as a foundational guide for researchers and practitioners aiming to leverage NDEs for advanced time series analysis.

Key Contributions of This Survey

This paper provides a structured synthesis of the NDE field for time series analysis, with a focus on:

Foundational Models: A detailed breakdown of the formulations and extensions for NODEs, NCDEs, and NSDEs.
Theoretical Underpinnings: An analysis of universal approximation properties, conditions for the existence and uniqueness of solutions, and the stability of different NDE families.
Practical Implementation: A guide to practical considerations, including optimization with the adjoint method, regularization techniques, and the selection of numerical solvers.
Empirical Comparison: A consolidated summary of benchmark results across a wide range of tasks, including interpolation, forecasting, and classification.