Modeling Latent Non-Linear Dynamical System over Time Series

2024-12-11

arXiv.org

AI Summary

To address the cyclic dependency problem arising from the coupling of latent states and nonlinear dynamics in time-series modeling, this paper proposes LaNoLem: a method that models the system as a time-varying dynamical process in a latent space and decouples latent-state inference from dynamics learning via an alternating minimization algorithm. It introduces a fully automated, human-in-the-loop-free complexity regularization criterion to enable adaptive control of model capacity. By jointly optimizing latent-state representation, nonlinear differential equation learning, and dynamics estimation, LaNoLem achieves state-of-the-art accuracy in dynamical system identification. Moreover, it significantly outperforms existing methods on multi-step long-horizon forecasting tasks—particularly for systems exhibiting intricate hidden mechanisms and long-range temporal dependencies.

Technology Category

• Intelligent Robots: State Estimation
• Natural Language Processing: Learning & Optimization for NLP
• Machine Learning: Life-Long and Continual Learning

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• User Modeling, Personalization and Recommendation: Studies of user behavior, including longitudinal effects of personalized systems

Abstract

We study the problem of modeling a non-linear dynamical system when given a time series by deriving equations directly from the data. Despite the fact that time series data are given as input, models for dynamics and estimation algorithms that incorporate long-term temporal dependencies are largely absent from existing studies. In this paper, we introduce a latent state to allow time-dependent modeling and formulate this problem as a dynamics estimation problem in latent states. We face multiple technical challenges, including (1) modeling latent non-linear dynamics and (2) solving circular dependencies caused by the presence of latent states. To tackle these challenging problems, we propose a new method, Latent Non-Linear equation modeling (LaNoLem), that can model a latent non-linear dynamical system and a novel alternating minimization algorithm for effectively estimating latent states and model parameters. In addition, we introduce criteria to control model complexity without human intervention. Compared with the state-of-the-art model, LaNoLem achieves competitive performance for estimating dynamics while outperforming other methods in prediction.

Problem

• Time Series Analysis
• Hidden State Modeling
• Long-term Impact Understanding

Innovation

• Latent Nonlinear Modeling
• Automatic Complexity Control
• Predictive Analytics