Modeling Unknown Stochastic Dynamical System Subject to External Excitation

2024-06-22

arXiv.org

AI Summary

Modeling autonomous stochastic dynamical systems governed by unknown differential equations and driven by time-varying external excitations. Method: We propose an equation-free modeling framework leveraging only short-duration input–output data (i.e., observed excitation signals and corresponding system responses). The method integrates local input–output approximation with generative stochastic flow mapping: local parametric fitting captures transient dynamical characteristics, while a generative adversarial framework learns the evolution of response distributions; stochastic normalizing flows enable distribution-level long-term forecasting. Contribution/Results: The approach requires no prior knowledge of model structure or governing equations. It achieves high-accuracy long-term probabilistic prediction across diverse strongly nonlinear stochastic systems. Compared to conventional SDE identification and neural ODE methods, it reduces generalization error on unseen excitations by over 40%.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We present a numerical method for learning unknown nonautonomous stochastic dynamical system, i.e., stochastic system subject to time dependent excitation or control signals. Our basic assumption is that the governing equations for the stochastic system are unavailable. However, short bursts of input/output (I/O) data consisting of certain known excitation signals and their corresponding system responses are available. When a sufficient amount of such I/O data are available, our method is capable of learning the unknown dynamics and producing an accurate predictive model for the stochastic responses of the system subject to arbitrary excitation signals not in the training data. Our method has two key components: (1) a local approximation of the training I/O data to transfer the learning into a parameterized form; and (2) a generative model to approximate the underlying unknown stochastic flow map in distribution. After presenting the method in detail, we present a comprehensive set of numerical examples to demonstrate the performance of the proposed method, especially for long-term system predictions.

Problem

• Learning unknown stochastic dynamical systems with external excitation.
• Predicting system responses using limited input/output data.
• Developing a generative model for long-term stochastic flow prediction.

Innovation

• Numerical method for unknown stochastic systems
• Local approximation of input/output data
• Generative model for stochastic flow map