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

arXiv: https://arxiv.org/abs/2308.05882 · HTML: https://mlsanity.com/paper/2308.05882/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 73 references that are in the map.
- First posted on arXiv 10 Aug 2023.

## Builds on (most cited first, 17 of 17)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification](https://mlsanity.com/paper/2204.12005/) (2022)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants](https://mlsanity.com/paper/2211.10575/) (2022)
- [PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data](https://mlsanity.com/paper/2212.04971/) (2022)
- [Efficient nonlinear manifold reduced order model](https://mlsanity.com/paper/2011.07727/) (2020)
- [Local approximate Gaussian process regression for data-driven constitutive laws: Development and comparison with neural networks](https://mlsanity.com/paper/2105.04554/) (2021)
- [Component-wise reduced order model lattice-type structure design](https://mlsanity.com/paper/2010.10770/) (2020)
- [Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data](https://mlsanity.com/paper/2108.04085/) (2021)
- [MuyGPs: Scalable Gaussian Process Hyperparameter Estimation Using Local Cross-Validation](https://mlsanity.com/paper/2104.14581/) (2021)
- [Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators](https://mlsanity.com/paper/2106.09658/) (2021)
- [S-OPT: A Points Selection Algorithm for Hyper-Reduction in Reduced Order Models](https://mlsanity.com/paper/2203.16494/) (2022)
- [Sliced gradient-enhanced Kriging for high-dimensional function approximation](https://mlsanity.com/paper/2204.03562/) (2022)

## Built on by (most cited first, 30 of 30)

- [tLaSDI: Thermodynamics-informed latent space dynamics identification](https://mlsanity.com/paper/2403.05848/) (2024)
- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [Multi-fidelity reduced-order surrogate modeling](https://mlsanity.com/paper/2309.00325/) (2023)
- [Physics-informed active learning with simultaneous weak-form latent space dynamics identification](https://mlsanity.com/paper/2407.00337/) (2024)
- [Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbone](https://mlsanity.com/paper/2406.17119/) (2024)
- [Reduced-order modeling for parameterized PDEs via implicit neural representations](https://mlsanity.com/paper/2311.16410/) (2023)
- [Thermodynamically Consistent Latent Dynamics Identification for Parametric Systems](https://mlsanity.com/paper/2506.08475/) (2025)
- [A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEs](https://mlsanity.com/paper/2502.08683/) (2025)
- [Latent Twins](https://mlsanity.com/paper/2509.20615/) (2025)
- [mLaSDI: Multi-stage latent space dynamics identification](https://mlsanity.com/paper/2506.09207/) (2025)
- [Rollout-LaSDI: Enhancing the long-term accuracy of Latent Space Dynamics](https://mlsanity.com/paper/2509.08191/) (2025)
- [Sequential decoder training for improved latent space dynamics identification](https://mlsanity.com/paper/2510.03535/) (2025)
- [Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives](https://mlsanity.com/paper/2512.15997/) (2025)
- [Accurate Data-Driven Surrogates of Dynamical Systems for Forward Propagation of Uncertainty](https://mlsanity.com/paper/2310.10831/) (2023)
- [Parametric Taylor series based latent dynamics identification neural networks](https://mlsanity.com/paper/2410.04193/) (2024)
- [Understanding the role of autoencoders for stiff dynamical systems using information theory](https://mlsanity.com/paper/2503.06325/) (2025)
- [Spatially-Enhanced Temporal Fusion Transformer: Interpretable Multi-Output Prediction for Parametric Dynamical Systems with Time-Varying Inputs](https://mlsanity.com/paper/2505.00473/) (2025)
- [Extrapolating Phase-Field Simulations in Space and Time with Purely Convolutional Architectures](https://mlsanity.com/paper/2509.20770/) (2025)
- [Latent Space Element Method](https://mlsanity.com/paper/2601.01741/) (2026)
- [GeoQ: Geometry-Aware Conditional Quantile Error Estimation for Scientific Surrogate Models](https://mlsanity.com/paper/2608.21652/) (2026)
- [Data-Driven Modeling of an Unsaturated Bentonite Buffer Model Test Under High Temperatures Using an Enhanced Axisymmetric Reproducing Kernel Particle Method](https://mlsanity.com/paper/2309.13519/) (2023)
- [Latent Representation Learning of Multi-scale Thermophysics: Application to Dynamics in Shocked Porous Energetic Material](https://mlsanity.com/paper/2506.12996/) (2025)
- [Revealing Low-Dimensional Structure in 2D Richtmyer-Meshkov Instabilities via Parametric Reduced-Order Modeling](https://mlsanity.com/paper/2510.16197/) (2025)
- [ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marching](https://mlsanity.com/paper/2512.19643/) (2025)
- [Towards Spatio-Temporal Extrapolation of Phase-Field Simulations with Convolution-Only Neural Networks](https://mlsanity.com/paper/2601.04510/) (2026)
- [Tackling multiphysics problems via finite element-guided physics-informed operator learning](https://mlsanity.com/paper/2603.01420/) (2026)
- [WGFINNs: Weak formulation-based GENERIC formalism informed neural networks](https://mlsanity.com/paper/2604.02601/) (2026)
- [Time-Dependent PDE-Constrained Optimization via Weak-Form Latent Dynamics](https://mlsanity.com/paper/2605.20639/) (2026)
- [Continual-Learning Physics-Informed Neural Networks for Parameterized Partial Differential Equations](https://mlsanity.com/paper/2608.04778/) (2026)

## Abstract (arXiv)

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.