用聚类自适应方法，20个样本内实现非线性固体力学实时高精度预测。
A clustering adaptive Gaussian process regression method: response patterns based real-time prediction for nonlinear solid mechanics problems
- 基于响应模式聚类，动态生成关键样本以覆盖解空间
- 在线预测时分而治之，1秒内完成且误差降低1-3个数量级
- 适合需要实时反馈的复杂结构仿真场景，如工程设计优化
数值模拟在研究非线性固体力学问题中非常强大，但基于网格或粒子的传统方法通常耗时较长，尤其在需实时分析的复杂问题中。本文提出一种聚类自适应高斯过程回归（CAG）方法，用于非线性结构响应的实时预测。该数据驱动方法具有小样本、高精度、高效率的特点，利用非线性结构响应模式。与传统高斯过程回归（GPR）类似，分为离线和在线阶段。离线阶段引入自适应采样生成技术，将数据集按模式聚类，实现按需分配样本，确保对目标解空间的关键样本充分覆盖。在线阶段采用分治策略，通过预分类将问题划分为预定义模式，并由训练好的多模式高斯过程回归器依次预测。此外，还结合降维与重构技术提升效率。针对材料、几何及边界条件非线性的多组问题验证了该方法性能。结果表明，在本研究设定下，仅需约20个样本即可实现1秒内预测，精度显著优于使用均匀分布样本的传统GPR，误差降低达1至3个数量级。该方法有望成为非线性固体力学实时预测的强大工具，并揭示复杂非线性结构响应模式。
原文摘要 · Abstract (English)
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.
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