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

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.

References in corpus (18)

• Adam: A Method for Stochastic Optimization
• Scikit-learn: Machine Learning in Python
• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Weak SINDy For Partial Differential Equations
• Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations
• Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)
• A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder
• Local approximate Gaussian process regression for data-driven constitutive laws: Development and comparison with neural networks
• LaSDI: Parametric Latent Space Dynamics Identification
• Thoughts on Massively Scalable Gaussian Processes
• Component-wise reduced order model lattice-type structure design
• gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification
• Efficient space-time reduced order model for linear dynamical systems in Python using less than 120 lines of code
• Accelerating design optimization using reduced order models
• Efficient nonlinear manifold reduced order model
• MuyGPs: Scalable Gaussian Process Hyperparameter Estimation Using Local Cross-Validation
• Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants
• PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data

Cited by in corpus (7)

• Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbone
• Gappy AE: A Nonlinear Approach for Gappy Data Reconstruction using Auto-Encoder
• Physics-informed active learning with simultaneous weak-form latent space dynamics identification
• Data-Driven Modeling of an Unsaturated Bentonite Buffer Model Test Under High Temperatures Using an Enhanced Axisymmetric Reproducing Kernel Particle Method
• A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEs
• Interpretable and flexible non-intrusive reduced-order models using reproducing kernel Hilbert spaces
• Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives