Stochastic parameter reduced-order model based on hybrid machine learning approaches

TL;DR

The paper tackles the challenge of building efficient surrogates for high-dimensional PDEs by proposing a stochastic reduced-order framework that blends a Convolutional Autoencoder (CAE) for latent-space compression with a parametric Reservoir Computing-Normalizing Flow (RC-NF) to evolve and refine latent dynamics. Applied to the viscous Burgers equation, the approach yields a two-dimensional latent representation and a stochastic evolution equation for the latent state, enabling interpolation and extrapolation across Reynolds numbers $Re$ beyond the training set. The Normalizing Flow component provides distributional refinement and log-likelihood training, while Bayesian optimization selects RC hyperparameters, resulting in a mesh-free, fast-to-train surrogate that captures key features such as advection shocks. Overall, the method demonstrates reliable reconstruction and forecasting in both training and unseen scenarios, indicating substantial potential for scalable, data-driven surrogates in more complex PDEs and multi-parameter settings.

Abstract

Establishing appropriate mathematical models for complex systems in natural phenomena not only helps deepen our understanding of nature but can also be used for state estimation and prediction. However, the extreme complexity of natural phenomena makes it extremely challenging to develop full-order models (FOMs) and apply them to studying many quantities of interest. In contrast, appropriate reduced-order models (ROMs) are favored due to their high computational efficiency and ability to describe the key dynamics and statistical characteristics of natural phenomena. Taking the viscous Burgers equation as an example, this paper constructs a Convolutional Autoencoder-Reservoir Computing-Normalizing Flow algorithm framework, where the Convolutional Autoencoder is used to construct latent space representations, and the Reservoir Computing-Normalizing Flow framework is used to characterize the evolution of latent state variables. In this way, a data-driven stochastic parameter reduced-order model is constructed to describe the complex system and its dynamic behavior.

Figures (5)

• Figure 1: The flowchart of the Convolutional Autoencoder-Reservoir Computing-Normalizing Flow model. The high-dimensional state variable $\bm{u}$ is first compressed into a low-dimensional representation $\bm{Y}_t$ using an encoder. The parametric Reservoir Computing-Normalizing Flow model then characterizes the evolution of the low-dimensional variable in the latent state space. Finally, the latent state variable $\bm{Y}_t$ and its predicted value $\tilde{\bm{Y}}_t$ are reconstructed back to the original high-dimensional space through a decoder.
• Figure 2: Reconstruction Error. The blue solid line represents the results of the Convolutional Autoencoder model, while the red dashed line represents the results of the Convolutional Autoencoder-Reservoir Computing-Normalizing Flow model. Left: Reconstruction error on the training set. Right: Reconstruction error on the test set.
• Figure 3: Prediction error of the Reservoir Computing-Normalizing Flow Model. The blue solid line represents the prediction results of the model on the training set, while the red dashed line represents the prediction results on the test set. Left: Prediction error for latent state space dimension 1. Right: Prediction error for latent state space dimension 2.
• Figure 4: Results for parameter $Re=1050$. Top Left: Data (left) and reconstruction results using the Convolutional Autoencoder-Reservoir Computing-Normalizing Flow model (right). Top Right: Data and reconstruction results at time $t = 2$, with the reference (blue solid line) and the prediction from the Convolutional Autoencoder-Reservoir Computing-Normalizing Flow model (red dashed line). Bottom: Latent state variables constructed by the encoder (blue solid line) and prediction results from the Reservoir Computing-Normalizing Flow model (red dashed line).
• Figure 5: Results for parameter $Re=2250$. Top Left: Data (left) and reconstruction results using the Convolutional Autoencoder-Reservoir Computing-Normalizing Flow (right). Top Right: Data and reconstruction results at time $t = 2$, with the reference (blue solid line) and the Convolutional Autoencoder-Reservoir Computing-Normalizing Flow (red dashed line). Bottom: Latent state variables constructed by the encoder (blue solid line) and prediction results from the Reservoir Computing-Normalizing Flow (red dashed line).