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

TL;DR

GPLaSDI addresses the challenge of rapid, uncertainty-aware reduced-order modeling for PDEs in a non-intrusive setting. It combines end-to-end autoencoder-based state compression with SINDy to identify latent-space ODEs and uses Gaussian Process regression to interpolate the ODE coefficients across parameter space, yielding predictive means and credible intervals without requiring the PDE residual. A variance-based greedy sampling strategy efficiently augments training data in regions of high predictive uncertainty, enabling accurate ROM predictions with substantial speed-ups on Burgers, Vlasov, and rising thermal bubble problems. The approach delivers reliable uncertainty quantification and competitive accuracy (typically under 7% relative error) with speed-ups ranging from hundreds to tens of thousands of times, making it a practical tool for non-intrusive, data-driven PDE surrogate modeling.

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.

Figures (15)

• Figure 1: GPLaSDI general framework. (1) Combined training of the autoencoder and the SINDy latent space identification. (2) Interpolation of the latent space governing sets of ODEs using Gaussian Processes. (3) ROM Prediction methodology using GPLaSDI. (4) FOM training data greedy sampling algorithm, using (1), (2) and (3).
• Figure 2: 1D Burgers Equation -- Predictive mean of each ODE coefficients ($m_{j,k}^{(*)}$) given $\pmb{\mu}^{(*)}$ at the end of the training. The black marks represent each sampled data point.
• Figure 3: 1D Burgers Equation -- Predictive standard deviation of each ODE coefficients ($s_{j,k}^{(*)}$) given $\pmb{\mu}^{(*)}$. The heatmaps are similar for each coefficients, which is expected because here the input point locations within $\mathcal{D}^h$ are all the same. Notice that the uncertainty is higher in regions with no training data points, as one might intuitively expect.
• Figure 4: 1D Burgers Equation -- Maximum relative error ($\%$) using GPLaSDI (left) and gLaSDI (right). The values in a red square correspond to the original FOM data at the beginning of the training (located at the four corners). The values in a black square correspond to parameters and FOM runs that were sampled during training.
• Figure 5: 1D Burgers Equation -- Maximum predictive standard deviation for GPLaSDI. The numbers inside each box are in scientific notations, scaled by the factor of 10 specified in the title. For example, the maximum value across the figure is $1.9\cdot10^{-1}$.