Uncertainty quantification and reduction using Jacobian and Hessian information

Abstract

Robust design methods have expanded from experimental techniques to include sampling methods, sensitivity analysis and probabilistic optimisation. Such methods typically require many evaluations. We study design and noise variable cross-term second derivatives of a response to quickly identify design variables that reduce response variability. We first compute the response uncertainty and variance decomposition to determine contributing noise variables of an initial design. Then we compute the Hessian second-derivative matrix cross-terms between the variance-contributing noise variables and proposed design change variables. Design variable with large Hessian terms are those that can reduce response variability. We relate the Hessian coefficients to reduction in Sobol indices and response variance change. Next, the first derivative Jacobian terms indicate which design variable can shift the mean to maintain a desired nominal target value. Thereby, design changes can be proposed to reduce variability while maintaining a targeted nominal value. This workflow finds changes that improve robustness with a minimal four runs per design change. We also explore further computation reductions achieved through compounding variables. An example is shown on a Stirling engine where the top four variance-contributing tolerances and design changes identified through 16 Hessian terms generated a design with 20% less variance.

A reinforcement learning approach to system modularization under constraints

“…In this paper, we propose using machine learning to ensure optimality of the solutions obtained by the clustering algorithm. Modern machine learning techniques have been employed for a wide range of tasks in engineering design ranging from surrogate modeling, 10 design exploration and optimization, 11 to uncertainty quantification and robust design, 12 to design synthesis, 13 and to extracting human preferences 14 . Unsupervised learning is used with input data without labeling information.…”