Robust design optimization for a nonlinear system via Bayesian neural network enhanced polynomial dimensional decomposition

TL;DR

The paper tackles robust design optimization under substantial uncertainty and strong nonlinearity by fusing Bayesian neural networks with polynomial dimensional decomposition to enable rapid, reliable moment estimation. It introduces transformed random variables to fix PDD bases, an uncertainty-driven active learning loop to improve surrogate fidelity, and a multi-point single-step strategy to partition the design space and manage nonlinearities. The approach yields near-global optima with far fewer function evaluations than GP-based or Monte Carlo methods, demonstrated on a 10D Rastrigin benchmark (99.97% mean reduction) and a high-dimensional ERPMSM cogging torque design (94.75% mean and 88.56% std reduction) with only 6644 FE evaluations. This framework offers a practical, scalable path for robust design in complex engineering systems, albeit with additional computational overhead for BNN training and hyperparameter tuning that warrants future automation.

Abstract

Uncertainties such as manufacturing tolerances cause performance variations in complex engineering systems, making robust design optimization (RDO) essential. However, simulation-based RDO faces high computational cost for statistical moment estimation, and strong nonlinearity limits the accuracy of conventional surrogate models. This study proposes a novel RDO method that integrates Bayesian neural networks (BNN) with polynomial dimensional decomposition (PDD). The method employs uncertainty-based active learning to enhance BNN surrogate accuracy and a multi-point single-step strategy that partitions the design space into dynamically adjusted subregions, within which PDD analytically estimates statistical moments from BNN predictions. Validation through a mathematical benchmark and an electric motor shape optimization demonstrates that the method converges to robust optimal solutions with significantly fewer function evaluations. In the ten-dimensional benchmark, the proposed method achieved a 99.97% mean reduction, while Gaussian process-based and Monte Carlo approaches failed to locate the global optimum. In the motor design problem, the method reduced cogging torque by 94.75% with only 6644 finite element evaluations, confirming its computational efficiency for high-dimensional, strongly nonlinear engineering problems.

Figures (25)

• Figure 2: The overall framework of the proposed BNN-PDD multi-point single-step optimization.
• Figure 3: Flowchart of the active learning algorithm using BNN.
• Figure 4: Flowchart of the statistical moment analysis using the proposed BNN-PDD method.
• Figure 5: Schematic description of the multi-point single-step process.
• Figure 6: Flowchart of the multi-point single-step BNN-PDD RDO algorithm.