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Sequential Experimental Designs for Kriging Model

Ruonan Zheng, Min-Qian Liu, Yongdao Zhou, Xuan Chen

TL;DR

The paper addresses the efficiency of designing Kriging surrogates by tackling limitations of one-point sequential designs and the scarcity of observation-based batch strategies. It introduces two criteria—gradient-based and variance-based—for sequential point selection to improve global fitting, and provides a batch-extension framework (cluster-based top-\\(b\\)) that preserves criterion behavior while reducing iteration overhead. The authors formulate the criteria with explicit GP-based expressions, supply a practical clustering algorithm, and validate the approach via extensive simulations on diverse test functions, showing improvements in RMSE and MAE. The work enables more resource-efficient experimental design in computer experiments and offers practical guidance on selecting gradient- or variance-focused strategies depending on the error objective.

Abstract

Computer experiments have become an indispensable alternative to complex physical and engineering experiments. The Kriging model is the most widely used surrogate model, with the core goal of minimizing the discrepancy between the surrogate and true models across the entire experimental domain. However, existing sequential design methods have critical limitations: observation-based batch sequential designs are rarely studied, while one-point sequential designs have insufficient information utilization and suffer from inefficient resource utilization -- they require numerous repeated observation rounds to accumulate sufficient points, leading to prolonged experimental cycles. To address these gaps, this paper proposes two novel one-point sequential design criteria and a general batch sequential design framework. Moreover, the batch sequential design framework solves the inherent point clustering problem in naive batch selection, enabling efficient extension of any sequential criterion to batch scenarios. Simulations on some test functions demonstrate that the proposed methods outperform existing approaches in terms of fitting accuracy in most cases.

Sequential Experimental Designs for Kriging Model

TL;DR

The paper addresses the efficiency of designing Kriging surrogates by tackling limitations of one-point sequential designs and the scarcity of observation-based batch strategies. It introduces two criteria—gradient-based and variance-based—for sequential point selection to improve global fitting, and provides a batch-extension framework (cluster-based top-\) that preserves criterion behavior while reducing iteration overhead. The authors formulate the criteria with explicit GP-based expressions, supply a practical clustering algorithm, and validate the approach via extensive simulations on diverse test functions, showing improvements in RMSE and MAE. The work enables more resource-efficient experimental design in computer experiments and offers practical guidance on selecting gradient- or variance-focused strategies depending on the error objective.

Abstract

Computer experiments have become an indispensable alternative to complex physical and engineering experiments. The Kriging model is the most widely used surrogate model, with the core goal of minimizing the discrepancy between the surrogate and true models across the entire experimental domain. However, existing sequential design methods have critical limitations: observation-based batch sequential designs are rarely studied, while one-point sequential designs have insufficient information utilization and suffer from inefficient resource utilization -- they require numerous repeated observation rounds to accumulate sufficient points, leading to prolonged experimental cycles. To address these gaps, this paper proposes two novel one-point sequential design criteria and a general batch sequential design framework. Moreover, the batch sequential design framework solves the inherent point clustering problem in naive batch selection, enabling efficient extension of any sequential criterion to batch scenarios. Simulations on some test functions demonstrate that the proposed methods outperform existing approaches in terms of fitting accuracy in most cases.
Paper Structure (16 sections, 4 theorems, 34 equations, 5 figures, 6 tables, 2 algorithms)

This paper contains 16 sections, 4 theorems, 34 equations, 5 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Kriging model
  4. Sequential design framework
  5. Review of sequential design criteria
  6. Two novel criteria for sequential design
  7. Gradient-based criterion
  8. Motivation
  9. Specific expression of the gradient-based criterion
  10. Variance-based criterion
  11. Cluster-based top-$b$ sequential design
  12. Simulations
  13. Test functions
  14. Comparison of one-point sequential designs
  15. Comparison of batch sequential design
  16. ...and 1 more sections

Key Result

Lemma 1

For a Gaussian process with correlation function $k(\boldsymbol{x},\boldsymbol{y})$ and variance $\tau^2$, we have where $\nabla\hat{y}_{\boldsymbol{X}}(\boldsymbol{x})=\nabla r^T_{\boldsymbol{X}}(\boldsymbol{x})\boldsymbol{K}^{-1}(\boldsymbol{X})\boldsymbol{Y}_{\boldsymbol{X}}$.

Figures (5)

  • Figure 1: The distribution of design points in the uniform design and gradient-based sequential design.
  • Figure 2: The RMSE of the uniform design and the gradient-based sequential design with different number of design points.
  • Figure 3: The order of design points in $\boldsymbol{D}_2$.
  • Figure 4: The one-point sequential design and the order of values of $\phi(\boldsymbol{x})$ on $\boldsymbol{D}_{all}$.
  • Figure 5: The order of points in a cluster-based top-$b$ sequential design.

Theorems & Definitions (6)

  • Lemma 1: Erickson:2021
  • Theorem 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['the:1']}
  • Lemma 4
  • proof : Proof of Theorem \ref{['the:2']}