Modeling Unknown Stochastic Dynamical System Subject to External Excitation

TL;DR

The paper develops a data-driven framework to learn unknown nonautonomous stochastic dynamics from short input/output bursts. It converts the problem into a locally parameterized stochastic flow map by locally modeling the external excitation with polynomials and learning a one-step generator via conditional normalizing flow, producing distributional predictions for unseen excitations. Key contributions include extending stochastic flow map learning to non-autonomous systems and demonstrating long-horizon, non-Gaussian predictive accuracy across diverse examples, from OU to SPDEs. This approach enables flexible, scalable modeling of complex stochastic systems where governing equations are unavailable, with potential impact on predictive uncertainty quantification in engineering and sciences.

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.

Figures (22)

• Figure 1: The DNN model structure for the proposed normalizing flow sFML method \ref{['x1']}.
• Figure 1: Sample trajectories of Example \ref{['ex:OU1']} with initial condition $x_0=2.0$ and $\alpha(t)=\frac{1}{2}\sin(6t)$. Left: ground truth; Right: Simulation using the trained sFML model.
• Figure 2: Mean and standard deviation (STD) of Example \ref{['ex:OU1']} with initial condition $x_0=2.0$ and $\alpha(t)=\frac{1}{2}\sin(6t)$.
• Figure 3: Comparasion of distribution of Example \ref{['ex:OU1']} at $T=2,4,8$ with initial condition $x_0=2.0$ and $\alpha(t)=0.5\sin(6t)$.
• Figure 4: Sample trajectories of Example \ref{['ex:OU1']} with initial condition $x_0=-1.0$ and $\alpha(t)=\frac{1}{2}\sin(5t)+\frac{1}{5}\sin(1.5t)$. Left: ground truth; Right: Simulation using the trained sFML model.

Theorems & Definitions (2)

• Remark 3.1
• Remark 3.2