线性等式约束下物理场的多输出高斯过程预测

中文摘要 AI 辅助

我们解决受线性等式约束的多个高维物理场的同时预测问题，该场景出现在物理机器学习的诸多实际应用中。高斯过程（GP）回归是一种广泛使用的代理建模方法，因其在小样本场景下的有效性及提供不确定性量化的能力而备受青睐。然而，在该场景下应用GP模型会面临两大核心挑战：离散输出场的高维性，以及预测中物理约束的满足。针对后者，常用策略是通过约束关系从其他输出中推导出一个输出。通过基准测试，我们发现这种推导方法对需推导的输出的任意选择十分敏感，会同时影响预测精度与不确定性量化。因此，亟需一种能对称处理所有场且严格遵循 underlying physics（ underlying physics 译为底层物理规律）的方法。受这些局限的启发，我们提出了一种用于联合建模受约束多场数据的鲁棒框架。该方法首先利用一种针对多场数据的特定PCA流程，即行方向PCA，其具有在潜空间中保留约束的特性；由于标准多场数据PCA策略无法保留此类约束，我们从理论上研究了行方向选择的最优性。第二步，我们考虑一种基于特定核参数化的线性约束多输出GP方法，该方法在行方向PCA的潜空间上进行训练。所提框架在种群动力学问题及一项工业CFD应用中得到验证，该应用涉及不可压缩约束下雷诺应力张量分量的预测。

英文摘要

We address the simultaneous prediction of multiple high-dimensional physical fields governed by linear equality constraints, a setting that arises in many real-world applications in physics machine learning. Gaussian process (GP) regression is a widely used surrogate modeling approach due to its effectiveness in small-sample regimes and its ability to provide uncertainty quantification. However, applying GP models in this setting raises two major challenges: the high dimensionality of the discretized output fields and the enforcement of the physical constraint in predictions. For the latter, a common strategy consists in deducing one output from the others via the constraint relation. Through a benchmark, we show that this deductive approach is sensitive to the arbitrary choice of which output to deduce, affecting both predictive accuracy and uncertainty quantification. Consequently, there is a need for an approach that treats all fields symmetrically while strictly respecting the underlying physics. Motivated by these limitations, we propose a robust framework for jointly modeling constrained multi-field data. Our approach first leverages a specific PCA procedure for multi-field data, coined row-wise PCA, which has the interesting property of preserving the constraint in the latent space. Since standard PCA strategies for multi-field data do not preserve such constraints, we investigate theoretically the optimality of the row-wise choice. In a second step, we consider a linearly-constrained multi-output GP approach based on a specific kernel parametrization which is trained on the latent space of row-wise PCA. The proposed framework is validated on a population dynamics problem and on an industrial CFD application, which involves the prediction of Reynolds stress tensor components under the incompressibility constraint.

发表机构

• Université Paris-Saclay（巴黎-萨克雷大学）
• CEA（法国原子能和替代能源委员会）
• Univ Rennes（雷恩大学）
• Ensai（国立统计与信息分析学校）
• CNRS（法国国家科学研究中心）
• CREST - UMR 9194（统计、风险、经济与金融研究中心）

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