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Scalable multitask Gaussian processes for complex mechanical systems with functional covariates

Razak Christophe Sabi Gninkou, Andrés F. López-Lopera, Franck Massa, Rodolphe Le Riche

TL;DR

This work extends the framework of GPs with functional covariates to multitask problems by introducing a fully separable kernel structure that captures dependencies across tasks and functional inputs and takes advantage of the Kronecker structure of the covariance matrix.

Abstract

Functional covariates arise in many scientific and engineering applications when model inputs take the form of time-dependent or spatially distributed profiles, such as varying boundary conditions or changing material behaviours. In addition, new practices in digital simulation require predictions accompanied by confidence intervals. Models based on Gaussian processes (GPs) provide principled uncertainty quantification. However, GPs capable of jointly handling functional covariates and multiple correlated functional tasks remain largely under-explored. In this work, we extend the framework of GPs with functional covariates to multitask problems by introducing a fully separable kernel structure that captures dependencies across tasks and functional inputs. By taking advantage of the Kronecker structure of the covariance matrix, the model is made scalable. The proposed model is validated on a synthetic benchmark and applied to a realistic structure, a riveted assembly with functional descriptions of the material behaviour and response forces. The proposed functional multitask GP significantly improves over single task GPs. For the riveted assembly, it requires less than 100 samples to produce an accurate mean and confidence interval prediction. Despite its larger number of parameters, the multitask GP is computationally easier to learn than its single task pendant.

Scalable multitask Gaussian processes for complex mechanical systems with functional covariates

TL;DR

This work extends the framework of GPs with functional covariates to multitask problems by introducing a fully separable kernel structure that captures dependencies across tasks and functional inputs and takes advantage of the Kronecker structure of the covariance matrix.

Abstract

Functional covariates arise in many scientific and engineering applications when model inputs take the form of time-dependent or spatially distributed profiles, such as varying boundary conditions or changing material behaviours. In addition, new practices in digital simulation require predictions accompanied by confidence intervals. Models based on Gaussian processes (GPs) provide principled uncertainty quantification. However, GPs capable of jointly handling functional covariates and multiple correlated functional tasks remain largely under-explored. In this work, we extend the framework of GPs with functional covariates to multitask problems by introducing a fully separable kernel structure that captures dependencies across tasks and functional inputs. By taking advantage of the Kronecker structure of the covariance matrix, the model is made scalable. The proposed model is validated on a synthetic benchmark and applied to a realistic structure, a riveted assembly with functional descriptions of the material behaviour and response forces. The proposed functional multitask GP significantly improves over single task GPs. For the riveted assembly, it requires less than 100 samples to produce an accurate mean and confidence interval prediction. Despite its larger number of parameters, the multitask GP is computationally easier to learn than its single task pendant.
Paper Structure (36 sections, 60 equations, 15 figures, 1 table)

This paper contains 36 sections, 60 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Gaussian processes with functional covariates
  4. Construction of valid kernel
  5. Dimensionality reduction for functional covariates
  6. Multitask Gaussian processes with functional covariates
  7. Modeling with a separable kernel structure
  8. Hyperparameter estimation
  9. Efficient Kronecker-based inference
  10. Posterior prediction
  11. Numerical experiments
  12. Experimental setup
  13. Performance criteria
  14. The Rayleigh-based synthetic benchmark
  15. Dataset generation
  16. ...and 21 more sections

Figures (15)

  • Figure 1: Rayleigh-based dataset. The panels show: (top) Samples of the three functional inputs $\bm{\mathcal{F}} = (f_1, f_2, f_3)$, randomly generated as Rayleigh-shaped functions $h_\rho$, and (bottom) the corresponding sampled outputs $y_1$ and $y_2$ generated by the baseline MTGP.
  • Figure 2: LOO MTGP predictions for the test scenarios 45, 140, and 429 (displayed by columns) of the Rayleigh-based dataset. These scenarios correspond to the "best", "average", and "worst" predictive cases, respectively. The ground truth is shown in black, while predictions are shown in blue with $95\%$ confidence intervals.
  • Figure 3: $Q^2$ boxplots computed over the 500 LOO cross-validation replicates used in the Rayleigh-based dataset. Results are shown for each output $y_s$ for $s \in \{1, 2\}$.
  • Figure 4: Runtime for solving the linear system $\mathbf{L} \boldsymbol{\alpha} = \boldsymbol{y}$ in the Rayleigh-based dataset. The time is measured for different numbers of functional dimensions $n_f$.
  • Figure 5: Numerical model of the assembly. The panels show: (a) the finite element mesh and simulation conditions, and (b) the deformed shape at the end of the simulation.
  • ...and 10 more figures