Modeling Multivariate Time Series Dynamics with Differential Equations
Document Type
Thesis
Degree Name
Master of Science (MS)
Department
Computer Science and Info Sys
Date of Award
Summer 8-18-2025
Abstract
Time series data from the domains such as weather forecasting, financial markets and healthcare often involve nonlinear dynamics that are not easily captured by the traditional modeling approaches. While recurrent neural networks (RNNs) and their variants have made their strides in sequential learning, they often fall short when faced with irregular sampling. Differential equations provide a powerful framework for modeling multivariate time series by capturing nonlinear temporal behavior in a continuous-time representation that surpasses traditional discrete-time approaches. In this research, we leverage data-driven differential equation modeling to compactly represent complex time-dependent processes with nonlinear interactions, while naturally handling irregularly sampled data. We discover symbolic partial differential equations from multivariate time series data, yielding interpretable representations of the underlying dynamics using the PDE-FIND algorithm. The coefficients of these learned equations are then extracted as structured features for downstream machine learning models. Three model architectures, an RNN, a Neural Controlled Differential Equation (NCDE) and hypernetwork, are trained on these PDE-derived features and evaluated on diverse real-world datasets (weather measurements, influenza trends, and stock market indices). The hypernetwork model, which dynamically generates its parameters from the PDE coefficients, achieves the lowest root mean square error (RMSE) on forecasts, especially when input variables are highly correlated, indicating that it effectively exploits shared patterns across variables. Meanwhile, NCDE model excels at preserving sharp temporal transitions in the data, owing to its continuous-time formulation of latent dynamics. All continuous-time, differential equation driven models outperform a conventional RNN baseline, underscoring the advantages of this approach over traditional sequence modeling. These results demonstrate that combining symbolic structure from the discovered PDEs with deep neural networks enhances predictive performance, offering a robust framework for forecasting in complex dynamic systems.
Advisor
Dongeun Lee
Subject Categories
Computer Sciences | Physical Sciences and Mathematics
Recommended Citation
Chalamalasetty, Sri Lokesh, "Modeling Multivariate Time Series Dynamics with Differential Equations" (2025). Electronic Theses & Dissertations. 1300.
https://lair.etamu.edu/etd/1300
