GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder

TL;DR

GPLaSDI introduces a Gaussian process-based latent space dynamics identification framework that enhances reduced-order modeling of PDEs by providing uncertainty quantification and adaptive training, achieving significant speed-ups with maintained accuracy.

Contribution

It presents a novel non-intrusive framework combining Gaussian processes with autoencoder-based latent space dynamics for PDEs, enabling uncertainty quantification and adaptive training without prior PDE knowledge.

Findings

Achieves 200 to 100,000 times speed-up over traditional methods.

Maintains up to 7% relative error in predictions.

Demonstrates effectiveness on multiple complex PDE problems.

Abstract

Numerically solving partial differential equations (PDEs) can be challenging and computationally expensive. This has led to the development of reduced-order models (ROMs) that are accurate but faster than full order models (FOMs). Recently, machine learning advances have enabled the creation of non-linear projection methods, such as Latent Space Dynamics Identification (LaSDI). LaSDI maps full-order PDE solutions to a latent space using autoencoders and learns the system of ODEs governing the latent space dynamics. By interpolating and solving the ODE system in the reduced latent space, fast and accurate ROM predictions can be made by feeding the predicted latent space dynamics into the decoder. In this paper, we introduce GPLaSDI, a novel LaSDI-based framework that relies on Gaussian process (GP) for latent space ODE interpolations. Using GPs offers two significant advantages. First, it enables the quantification of uncertainty over the ROM predictions. Second, leveraging this prediction uncertainty allows for efficient adaptive training through a greedy selection of additional training data points. This approach does not require prior knowledge of the underlying PDEs. Consequently, GPLaSDI is inherently non-intrusive and can be applied to problems without a known PDE or its residual. We demonstrate the effectiveness of our approach on the Burgers equation, Vlasov equation for plasma physics, and a rising thermal bubble problem. Our proposed method achieves between 200 and 100,000 times speed-up, with up to 7% relative error.