Learning non-stationary and discontinuous functions using clustering, classification and Gaussian process modelling

TL;DR

This paper introduces a three-stage machine learning approach combining clustering, classification, and Gaussian process regression to effectively model non-smooth, discontinuous functions in engineering problems, improving surrogate modeling accuracy.

Contribution

It presents a novel three-stage method integrating DPMM, SVM, and GPR to handle non-smooth functions, addressing limitations of traditional smoothness assumptions in surrogate models.

Findings

Successfully modeled discontinuous functions with high accuracy.

Validated approach on analytical and finite element models.

Enhanced surrogate modeling for complex engineering systems.

Abstract

Surrogate models have shown to be an extremely efficient aid in solving engineering problems that require repeated evaluations of an expensive computational model. They are built by sparsely evaluating the costly original model and have provided a way to solve otherwise intractable problems. A crucial aspect in surrogate modelling is the assumption of smoothness and regularity of the model to approximate. This assumption is however not always met in reality. For instance in civil or mechanical engineering, some models may present discontinuities or non-smoothness, e.g., in case of instability patterns such as buckling or snap-through. Building a single surrogate model capable of accounting for these fundamentally different behaviors or discontinuities is not an easy task. In this paper, we propose a three-stage approach for the approximation of non-smooth functions which combines clustering, classification and regression. The idea is to split the space following the localized behaviors or regimes of the system and build local surrogates that are eventually assembled. A sequence of well-known machine learning techniques are used: Dirichlet process mixtures models (DPMM), support vector machines and Gaussian process modelling. The approach is tested and validated on two analytical functions and a finite element model of a tensile membrane structure.