Adaptive Learning for Surrogate Models in Active Subspace for High Dimensional Problems

Abstract

Surrogate models are often employed in engineering analysis to replace a detailed model with complicated geometry, loading, material properties and boundary conditions, in order to achieve computational efficiency in iterative calculations such as model calibration or design optimization. The accuracy of the surrogate model depends on the quality and quantity of data collected from the expensive physics-based model. This paper presents a novel approach to efficiently construct and improve surrogate models for high dimensional problems in both the input and output spaces.

In the proposed method, the principal components and corresponding features in the output field quantity are first identified. Mapping between inputs and each feature is then considered, and the active subspace methodology is used to capture the relationship in a low-dimensional subspace in the input domain. Thus dimension reduction is accomplished in both the input and output spaces, and surrogate models are built within the reduced spaces. A new low-dimensional adaptive learning strategy is proposed in this work to improve the surrogate model. With multiple iterations of this adaptive learning procedure, the optimal surrogate is achieved without intensive model simulations.

In contrast to existing adaptive learning methods which focus on scalar output or a limited number of output quantities, this paper addresses adaptive learning for both high-dimensional input and output, with a novel learning function balancing exploration and exploitation. The adaptive learning is based on the active variables in the low-dimensional space and once the newly-added training sample is selected, it can be easily mapped back to the original space for running the physics-based model. The proposed method is demonstrated on an additively manufactured component, with a high-dimensional field output quantity of interest, namely the residual stress in the component that has spatial variability due to the stochastic nature of multiple input variables (including process variables and material properties).