Two-Stage Metamodel Training for Anisotropic Kernel Parameter Selection
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Solution Overview
Problem
Conventional engineering design optimization methods, such as finite element analysis, are computationally expensive and inefficient for complex products like automobiles, especially when using anisotropic kernels, due to the high cost of training and the exponential increase in training time with the number of design variables.
Innovation Solution
A two-stage metamodel training procedure using a Kriging model to determine kernel width parameters for an anisotropic Gaussian kernel, where the first metamodel outputs these parameters, which are then used in a second metamodel, such as support vector regression, to reduce training effort and improve approximation accuracy.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If anisotropic kernel parameters are selected using conventional cross-validation with full grid search, then parameter selection accuracy is improved, but training time increases exponentially with the number of design variables
Solution Approach 1:
The patent divides the parameter selection process into two stages: first selecting a reduced subset of kernel width parameters, then selecting the remaining parameters. This segmentation reduces the combinatorial complexity from O(Nσ^d) to O(Nσ^2 + Nσ^(d-1)), breaking the exponential time complexity into manageable segments that can be processed sequentially.
Solution Approach 2:
The patent performs preliminary selection of a subset of kernel width parameters before selecting the remaining parameters. This preliminary action reduces the search space for subsequent parameter selection, preventing the exponential growth of training time while maintaining adequate accuracy for the final metamodel.
2Measurement precision
If anisotropic kernel function is used to improve approximation accuracy, then metamodel accuracy is improved, but the number of kernel width parameters increases with dimensionality
Solution Approach 1:
The patent segments the set of d kernel width parameters into two groups: a small subset of 2 parameters selected with high accuracy, and the remaining d-2 parameters selected with reduced effort. This segmentation maintains metamodel accuracy while managing parameter complexity.
Solution Approach 2:
The patent applies partial action by selecting only a subset of parameters with high precision rather than all parameters equally. The two most influential kernel width parameters are selected with rigorous cross-validation, while the remaining parameters are selected with less computational effort, recognizing that not all parameters contribute equally to accuracy.
3Measurement precision
If a larger number of DOE samples are used to improve metamodel accuracy, then approximation quality is improved, but computational cost increases
Solution Approach 1:
The patent performs preliminary parameter selection for a subset of kernel width parameters before final metamodel construction. This preliminary action enables accurate parameter selection with fewer DOE samples, reducing the computational cost while maintaining approximation quality.
Solution Approach 2:
The patent changes the approach from selecting all parameters simultaneously to selecting parameters in two stages with different accuracy requirements. This parameter change in the selection strategy reduces the effective dimensionality of the optimization problem, allowing accurate metamodels with fewer training samples.
Data Source
AI summary
Methods of conducting design optimization of a product using multiple metamodels are described. First and the second metamodels are configured with common kernel function. Kernel width parameter is the output or result of the first metamodel while the second metamodel requires a set of substantially similar kernel width parameters defined a priori. Further, the second metamodel is configured with an anisotropic kernel. First and second metamodels are trained in two stages. In the first stage, kernel width parameters are obtained by fitting known responses (obtained in numerical simulations) into the first metamodel with one or more prediction trends. Additional kernel width parameter set is derived by algebraically combining the obtained kernel width parameters. The second metamodel is then trained by cross-validating with known responses using N trial sets of metamodel parameter values including the kernel width parameter values determined in the first stage along with various combinations of other parameters.