Modeling Unknown Stochastic Dynamical System Subject to External Excitation

Abstract. We present a numerical method for learning an unknown nonautonomous stochastic dynamical systems, i.e., stochastic systems 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 signal-to-state data consisting of certain known excitation signals and their corresponding system responses are available. When a sufficient amount of such signal-to-state 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 signal-to-state 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.