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A Streaming Sparse Cholesky Method for Derivative-Informed Gaussian Process Surrogates Within Digital Twin Applications

Krishna Prasath Logakannan, Shridhar Vashishtha, Jacob Hochhalter, Shandian Zhe, Robert M. Kirby

TL;DR

This work extends Gaussian process (GP) models to include derivative data, for improved accuracy, with dynamic updating to ingest physical twin data during service, using a sparse GP approximation, for which extensions to incorporate derivatives are developed.

Abstract

Digital twins are developed to model the behavior of a specific physical asset (or twin), and they can consist of high-fidelity physics-based models or surrogates. A highly accurate surrogate is often preferred over multi-physics models as they enable forecasting the physical twin future state in real-time. To adapt to a specific physical twin, the digital twin model must be updated using in-service data from that physical twin. Here, we extend Gaussian process (GP) models to include derivative data, for improved accuracy, with dynamic updating to ingest physical twin data during service. Including derivative data, however, comes at a prohibitive cost of increased covariance matrix dimension. We circumvent this issue by using a sparse GP approximation, for which we develop extensions to incorporate derivatives. Numerical experiments demonstrate that the prediction accuracy of the derivative-enhanced sparse GP method produces improved models upon dynamic data additions. Lastly, we apply the developed algorithm within a DT framework to model fatigue crack growth in an aerospace vehicle.

A Streaming Sparse Cholesky Method for Derivative-Informed Gaussian Process Surrogates Within Digital Twin Applications

TL;DR

This work extends Gaussian process (GP) models to include derivative data, for improved accuracy, with dynamic updating to ingest physical twin data during service, using a sparse GP approximation, for which extensions to incorporate derivatives are developed.

Abstract

Digital twins are developed to model the behavior of a specific physical asset (or twin), and they can consist of high-fidelity physics-based models or surrogates. A highly accurate surrogate is often preferred over multi-physics models as they enable forecasting the physical twin future state in real-time. To adapt to a specific physical twin, the digital twin model must be updated using in-service data from that physical twin. Here, we extend Gaussian process (GP) models to include derivative data, for improved accuracy, with dynamic updating to ingest physical twin data during service. Including derivative data, however, comes at a prohibitive cost of increased covariance matrix dimension. We circumvent this issue by using a sparse GP approximation, for which we develop extensions to incorporate derivatives. Numerical experiments demonstrate that the prediction accuracy of the derivative-enhanced sparse GP method produces improved models upon dynamic data additions. Lastly, we apply the developed algorithm within a DT framework to model fatigue crack growth in an aerospace vehicle.

Paper Structure

This paper contains 25 sections, 3 theorems, 91 equations, 24 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. GP Surrogate Modeling for DT Applications
  3. Gaussian Process (GP) Modeling
  4. Derivative-Enhanced Gaussian Process (GP) Surrogate Modeling for Noisy Data
  5. Noise-Free Derivative-Enhanced Gaussian Process (GP) Surrogate Modeling
  6. Numerical Verification Experiments
  7. Sparse Cholesky for Derivative-Enhanced GP
  8. Sparse GP using Cholesky Factorization
  9. Derivative-Enhanced Sparse GP for Noise-Free Data
  10. Ordering Derivatives and Supernodes with Derivatives
  11. Derivative-Enhanced Sparse GP for Noisy Data
  12. Numerical Verification Experiments: Derivative-Enhanced Sparse GP with Noise-Free Data
  13. Numerical Experiments: Derivative-Enhanced Sparse GP with Noisy Data
  14. Dynamic Sparse GP Cholesky For Use In Digital Twin Application
  15. Dynamic Derivative-Enhanced Sparse GP
  16. ...and 10 more sections

Key Result

Lemma D.1

Let there be two fixed training points $\mathbf{x}^{(i)}$ and $\mathbf{x}^{(j)}$ and let $\mathbf{B}_{ij}$ denote the covariance block coupling any finite collection of derivative components at $\mathbf{x}^{(i)}$ with any finite collection at $\mathbf{x}^{(j)}$, then there exits constants $C, \gamma Essentially, block coupling decays exponentially with the square of the distance.

Figures (24)

  • Figure 1: Schematic of workflow of a typical DT system with real-time tethering and dynamic update capability.
  • Figure 2: Results of numerical experiments for varying number of training points and derivative order (a) 1D Griewank function, (b) 2D Griewank function, and (c) 3D Griewank function.
  • Figure 3: Results of numerical experiments on 3D Rosenbrock function for varying number of training points and order of derivatives. Note that the order of MSE errors is larger due to the steeper and higher magnitude nature of the Rosenbrock function. Although the magnitude of MSE errors is larger, the GP model performs equally well in relative terms when compared with the predictive performance of the Griewank function in Fig. \ref{['fig:E_GP_error']}.
  • Figure 4: Results of numerical experiments for matrix conditioning on 1D Griewank function for varying number of training points, order of derivatives, and varying noise. (a) No noise, (b) Increasing noise, and (c) Decreasing noise.
  • Figure 5: Results of numerical experiments for the predictive performance (measured by MSE) on 1D Griewank function for varying number of training points, order of derivatives, and varying noise. (a) No noise, (b) Increasing noise, and (c) Decreasing noise.
  • ...and 19 more figures

Theorems & Definitions (11)

  • proof
  • proof
  • proof
  • proof
  • proof
  • Lemma D.1: Localization of block covariances
  • proof
  • Lemma D.2: Supernode aggregation and computational cost
  • proof
  • Lemma D.3: Number of supernodes in "point-wise ordering algorithm 1" and "measurement-wise ordering algorithm 1"
  • ...and 1 more