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A Unified Hierarchical Multi-Task Multi-Fidelity Framework for Data-Efficient Surrogate Modeling in Manufacturing

Manan Mehta, Zhiqiao Dong, Yuhang Yang, Chenhui Shao

TL;DR

A novel hierarchical multi-task multi-fidelity framework for Gaussian process-based surrogate modeling that decomposes each task's response into a task-specific global trend and a residual local variability component that is jointly learned across tasks using a hierarchical Bayesian formulation.

Abstract

Surrogate modeling is an essential data-driven technique for quantifying relationships between input variables and system responses in manufacturing and engineering systems. Two major challenges limit its effectiveness: (1) large data requirements for learning complex nonlinear relationships, and (2) heterogeneous data collected from sources with varying fidelity levels. Multi-task learning (MTL) addresses the first challenge by enabling information sharing across related processes, while multi-fidelity modeling addresses the second by accounting for fidelity-dependent uncertainty. However, existing approaches typically address these challenges separately, and no unified framework simultaneously leverages inter-task similarity and fidelity-dependent data characteristics. This paper develops a novel hierarchical multi-task multi-fidelity (H-MT-MF) framework for Gaussian process-based surrogate modeling. The proposed framework decomposes each task's response into a task-specific global trend and a residual local variability component that is jointly learned across tasks using a hierarchical Bayesian formulation. The framework accommodates an arbitrary number of tasks, design points, and fidelity levels while providing predictive uncertainty quantification. We demonstrate the effectiveness of the proposed method using a 1D synthetic example and a real-world engine surface shape prediction case study. Compared to (1) a state-of-the-art MTL model that does not account for fidelity information and (2) a stochastic kriging model that learns tasks independently, the proposed approach improves prediction accuracy by up to 19% and 23%, respectively. The H-MT-MF framework provides a general and extensible solution for surrogate modeling in manufacturing systems characterized by heterogeneous data sources.

A Unified Hierarchical Multi-Task Multi-Fidelity Framework for Data-Efficient Surrogate Modeling in Manufacturing

TL;DR

A novel hierarchical multi-task multi-fidelity framework for Gaussian process-based surrogate modeling that decomposes each task's response into a task-specific global trend and a residual local variability component that is jointly learned across tasks using a hierarchical Bayesian formulation.

Abstract

Surrogate modeling is an essential data-driven technique for quantifying relationships between input variables and system responses in manufacturing and engineering systems. Two major challenges limit its effectiveness: (1) large data requirements for learning complex nonlinear relationships, and (2) heterogeneous data collected from sources with varying fidelity levels. Multi-task learning (MTL) addresses the first challenge by enabling information sharing across related processes, while multi-fidelity modeling addresses the second by accounting for fidelity-dependent uncertainty. However, existing approaches typically address these challenges separately, and no unified framework simultaneously leverages inter-task similarity and fidelity-dependent data characteristics. This paper develops a novel hierarchical multi-task multi-fidelity (H-MT-MF) framework for Gaussian process-based surrogate modeling. The proposed framework decomposes each task's response into a task-specific global trend and a residual local variability component that is jointly learned across tasks using a hierarchical Bayesian formulation. The framework accommodates an arbitrary number of tasks, design points, and fidelity levels while providing predictive uncertainty quantification. We demonstrate the effectiveness of the proposed method using a 1D synthetic example and a real-world engine surface shape prediction case study. Compared to (1) a state-of-the-art MTL model that does not account for fidelity information and (2) a stochastic kriging model that learns tasks independently, the proposed approach improves prediction accuracy by up to 19% and 23%, respectively. The H-MT-MF framework provides a general and extensible solution for surrogate modeling in manufacturing systems characterized by heterogeneous data sources.
Paper Structure (19 sections, 2 theorems, 35 equations, 4 figures, 2 tables)

This paper contains 19 sections, 2 theorems, 35 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Multi-Task Learning of Gaussian Processes
  4. Multi-fidelity Gaussian Process Modeling
  5. Summary of Research Gaps
  6. Model Formulation
  7. Review of Stochastic Kriging
  8. Multi-Task Multi-Fidelity Modeling Framework
  9. Parameter Estimation
  10. Estimating the Intrinsic Uncertainty
  11. Estimating the MTL Parameters Using EM
  12. Estimating the Parameters of the Global Trend
  13. Estimating Hyperparameters
  14. Stopping Criteria and Algorithm Complexity
  15. Case Studies
  16. ...and 4 more sections

Key Result

Theorem 1.A

The mean $\bm{\upmu}$ and covariance $\bm{\Sigma}_\mathsf{M}$ for the GP prior on function values $\bm{\upeta} = [ \eta(\mathbf{x}_1),\dots,\eta(\mathbf{x}_n)]^\top$ are samples drawn from a Normal Inverse Wishart (NIW) distribution with scale matrix equal to a base kernel $\bm{\kappa} \succ 0$.

Figures (4)

  • Figure 1: Flowchart for the iterative model update procedure for parameter estimation.
  • Figure 2: A 1D RSM example illustrating the H-MT-MF framework.
  • Figure 3: An example of an engine surface used in the case study.
  • Figure 4: Results for the prediction comparison at different gauge resolutions. $\sigma_\text{high-res}^2$ and $\sigma_\text{low-res}^2$ are the repeatabilities of the high and low resolution gauges respectively, expressed as percentage of the average surface height.

Theorems & Definitions (2)

  • Theorem 1.A
  • Theorem 1.B