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Gaussian Process Model with Tensorial Inputs and Its Application to the Design of 3D Printed Antennas

Xi Chen, Yashika Sharma, Hao Helen Zhang, Xin Hao, Qiang Zhou

TL;DR

This work introduces an IMED-based Gaussian process kernel to handle tensor-valued, spatially structured inputs from 3D printed designs, enabling efficient surrogate modeling for antenna design. By embedding the Image Euclidean Distance into the kernel and transforming inputs to a Euclidean space, the method builds spatially aware surrogates with ARD capabilities, while maintaining tractable inference through iterative solvers and GPU acceleration. The authors demonstrate improved predictive performance and uncertainty quantification on 2D and 3D monopole antenna datasets, using B-spline dimensionality reduction for functional outputs and comparing against several baselines, including convolutional and multi-linear kernels. The work offers a practical, interpretable approach to incorporate spatial structure into GP surrogates for time-consuming simulations, with potential extensions to more flexible nonlinear transformations and scalable inference techniques.

Abstract

In simulation-based engineering design with time-consuming simulators, Gaussian process (GP) models are widely used as fast emulators to speed up the design optimization process. In its most commonly used form, the input of GP is a simple list of design parameters. With rapid development of additive manufacturing (also known as 3D printing), design inputs with 2D/3D spatial information become prevalent in some applications, for example, neighboring relations between pixels/voxels and material distributions in heterogeneous materials. Such spatial information, vital to 3D printed designs, is hard to incorporate into existing GP models with common kernels such as squared exponential or Matérn. In this work, we propose to embed a generalized distance measure into a GP kernel, offering a novel and convenient technique to incorporate spatial information from freeform 3D printed designs into the GP framework. The proposed method allows complex design problems for 3D printed objects to take advantage of a plethora of tools available from the GP surrogate-based simulation optimization such as designed experiments and GP-based optimizations including Bayesian optimization. We investigate the properties of the proposed method and illustrate its performance by several numerical examples of 3D printed antennas. The dataset is publicly available at: https://github.com/xichennn/GP_dataset.

Gaussian Process Model with Tensorial Inputs and Its Application to the Design of 3D Printed Antennas

TL;DR

This work introduces an IMED-based Gaussian process kernel to handle tensor-valued, spatially structured inputs from 3D printed designs, enabling efficient surrogate modeling for antenna design. By embedding the Image Euclidean Distance into the kernel and transforming inputs to a Euclidean space, the method builds spatially aware surrogates with ARD capabilities, while maintaining tractable inference through iterative solvers and GPU acceleration. The authors demonstrate improved predictive performance and uncertainty quantification on 2D and 3D monopole antenna datasets, using B-spline dimensionality reduction for functional outputs and comparing against several baselines, including convolutional and multi-linear kernels. The work offers a practical, interpretable approach to incorporate spatial structure into GP surrogates for time-consuming simulations, with potential extensions to more flexible nonlinear transformations and scalable inference techniques.

Abstract

In simulation-based engineering design with time-consuming simulators, Gaussian process (GP) models are widely used as fast emulators to speed up the design optimization process. In its most commonly used form, the input of GP is a simple list of design parameters. With rapid development of additive manufacturing (also known as 3D printing), design inputs with 2D/3D spatial information become prevalent in some applications, for example, neighboring relations between pixels/voxels and material distributions in heterogeneous materials. Such spatial information, vital to 3D printed designs, is hard to incorporate into existing GP models with common kernels such as squared exponential or Matérn. In this work, we propose to embed a generalized distance measure into a GP kernel, offering a novel and convenient technique to incorporate spatial information from freeform 3D printed designs into the GP framework. The proposed method allows complex design problems for 3D printed objects to take advantage of a plethora of tools available from the GP surrogate-based simulation optimization such as designed experiments and GP-based optimizations including Bayesian optimization. We investigate the properties of the proposed method and illustrate its performance by several numerical examples of 3D printed antennas. The dataset is publicly available at: https://github.com/xichennn/GP_dataset.
Paper Structure (21 sections, 22 equations, 4 figures, 3 tables)

This paper contains 21 sections, 22 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Primer on Gaussian processes
  5. IMED: Image Euclidean Distance
  6. Methodology
  7. IMED kernel
  8. Computational issues
  9. Simulations and Model Setup
  10. 2D monopole antenna
  11. 3D monopole antenna
  12. Functional output
  13. Model setup and implementations
  14. Estimation of $G$
  15. Results Analysis
  16. ...and 6 more sections

Figures (4)

  • Figure 1: 2D monopole antenna. (a) a monopole antenna with a 3D-printed dielectric loading, left and right half planes are symmetric. (b) layout of the HFSS model of a monopole antenna surrounded by a grid (6 by 6) of dielectric cells with vertical symmetry. (c) a radiation pattern on a linear scale spanning from 0 to 360 degrees, used as monopole antenna performance measurement.
  • Figure 2: 3D monopole antenna. Layout of the HFSS model of a monopole antenna surrounded by $6\times6\times3$ dielectric unit cells
  • Figure 3: (a) one randomly selected estimated matrix $\hat{G}$ for 2D monopole, off-diagonal elements account for correlations among pixels whose indices are annotated. (b) an estimated $\hat{G}$ in ARD-IMED for 2D monopole with the pixel value annotated. (c) an estimated $\hat{K}=\hat{K}_2\otimes\hat{K}_1$ in M-Lin for 2D monopole with the pixel value annotated. (d) an estimated $\hat{K}=\hat{K}_3\otimes\hat{K}_2\otimes\hat{K}_1$ in M-Lin for 3D monopole.
  • Figure 4: Predicted mean for radiation pattern at the main lobe of randomly selected 2D monopole designs, ARD-IMED outperforms ARD-RBF and WConv2.