A clustering adaptive Gaussian process regression method: response patterns based real-time prediction for nonlinear solid mechanics problems

TL;DR

The paper addresses real-time prediction of nonlinear solid-mechanics responses, where high-fidelity simulations are too slow for online assessment. It introduces clustering adaptive Gaussian process regression (CAG), which partitions training data into distinct response patterns via offline adaptive sampling, trains pattern-specific GP emulators, and uses online KNN-based pattern classification to select the appropriate GP for prediction, followed by dimensionality reduction and field reconstruction. Six nonlinear problems demonstrate that CAG delivers real-time predictions (roughly $<1\,\text{s}$) with about $M\approx 20$ training samples and achieves $1$–$3$ orders of magnitude improvement in accuracy over traditional GPR with uniform sampling, while reducing online costs by $3$–$6$ orders of magnitude. This approach provides an efficient, interpretable framework for pattern-aware real-time prediction in nonlinear solid mechanics and offers pathways to applications such as online health monitoring and digital twins.

Abstract

Numerical simulation is powerful to study nonlinear solid mechanics problems. However, mesh-based or particle-based numerical methods suffer from the common shortcoming of being time-consuming, particularly for complex problems with real-time analysis requirements. This study presents a clustering adaptive Gaussian process regression (CAG) method aiming for real-time prediction for nonlinear structural responses in solid mechanics. It is a data-driven machine learning method featuring a small sample size, high accuracy, and high efficiency, leveraging nonlinear structural response patterns. Similar to the traditional Gaussian process regression (GPR) method, it operates in offline and online stages. In the offline stage, an adaptive sample generation technique is introduced to cluster datasets into distinct patterns for demand-driven sample allocation. This ensures comprehensive coverage of the critical samples for the solution space of interest. In the online stage, following the divide-and-conquer strategy, a pre-prediction classification categorizes problems into predefined patterns sequentially predicted by the trained multi-pattern Gaussian process regressor. In addition, dimension reduction and restoration techniques are employed in the proposed method to enhance its efficiency. A set of problems involving material, geometric, and boundary condition nonlinearities is presented to demonstrate the CAG method's abilities. The proposed method can offer predictions within a second and attain high precision with only about 20 samples within the context of this study, outperforming the traditional GPR using uniformly distributed samples for error reductions ranging from 1 to 3 orders of magnitude. The CAG method is expected to offer a powerful tool for real-time prediction of nonlinear solid mechanical problems and shed light on the complex nonlinear structural response pattern.

Figures (14)

• Figure 1: Implementation procedure of the proposed CAG method, where the operator $\mathcal{C}$ refers to the clustering process detailed in Eqs. \ref{['eq_initcent']}, \ref{['eq_edist']}, \ref{['eq_put']} and \ref{['eq_updatecent']}, the operator $\mathcal{A}$ for the adaptive sample addition according to Eq. \ref{['eq_xiadd']}, the operator $\mathcal{S}$ for the high-fidelity simulation solver, the operator $\mathcal{L}$ for the optimization process, the operator $\mathcal{V}$ for Eq. \ref{['eq_svd']}, the operator $\mathcal{D}$ for pre-prediction classification in Eq. \ref{['eq_knn']} and the operator $\mathcal{G}$ for the online predictions.
• Figure 2: Comparison between the proposed CAG method and the GPR with uniform sampling: (a) The sample distribution with 71 training samples; (b) Variation of maximum relative error with different training sample sizes.
• Figure 3: Regression of the wavelet-like function by the proposed CAG method and GPR using uniform sampling.
• Figure 4: Schematic diagram for free vibration of the mass-spring-damper system.
• Figure 5: Comparison of the proposed CAG method and the GPR with uniform sampling for sample distribution and vibration patterns using 16 samples.