Modeling Unknown Stochastic Dynamical System via Autoencoder

TL;DR

This work presents an autoencoder-based stochastic flow-map learning (sFML) framework to model unknown stochastic dynamical systems from trajectory data. By encoding hidden stochasticity into latent Gaussian variables and decoding to future states, the method yields a robust predictive model that can produce long-horizon forecasts from short data bursts, even under non-Gaussian noise. Key contributions include a novel loss design combining distributional and moment penalties, a sub-sampling strategy to ensure latent independence from current states, and automatic identification of the latent dimension. The approach is validated across linear, nonlinear, and multi-dimensional SDEs, demonstrating accurate drift/diffusion recovery and faithful long-term statistics, with practical implications for data-driven modeling of stochastic systems.

Abstract

We present a numerical method to learn an accurate predictive model for an unknown stochastic dynamical system from its trajectory data. The method seeks to approximate the unknown flow map of the underlying system. It employs the idea of autoencoder to identify the unobserved latent random variables. In our approach, we design an encoding function to discover the latent variables, which are modeled as unit Gaussian, and a decoding function to reconstruct the future states of the system. Both the encoder and decoder are expressed as deep neural networks (DNNs). Once the DNNs are trained by the trajectory data, the decoder serves as a predictive model for the unknown stochastic system. Through an extensive set of numerical examples, we demonstrate that the method is able to produce long-term system predictions by using short bursts of trajectory data. It is also applicable to systems driven by non-Gaussian noises.

Figures (25)

• Figure 1: An illustration of the network structure and training loss for the proposed autoencoder sFML method.
• Figure 1: OU process \ref{['eq: OUP']}: mean and standard deviation by the learned sFML model against the ground truth.
• Figure 2: An illustration of the proposed batch sampling: Left: histograms of 6 batches of sampled $\mathbf{x}_0$'s; Right: histograms of the corresponding $\mathbf{x}_1$'s.
• Figure 2: OU process \ref{['eq: OUP']}: Learned and reference effective drift and diffusion of the sFML model. Left: drift $a(x)=1.2-x$; Right: diffusion $b(x)=0.3$.
• Figure 3: OU process \ref{['eq: OUP']}: Left: Distribution of the latent variable from the trained encoder $\mathbf{E}_\Delta(x_0,x_1)$, $x_0 = 1.5$, against unit Gaussian $\mathcal{N}(0,1)$; Right: one-step conditional distribution of the trained decoder $\mathbf{D}_\Delta(1.5,z)$, $z\sim \mathcal{N}(0,1)$, against the ground truth.

Theorems & Definitions (1)

• Remark 3.1