Physics-informed machine learning (PIML) integrates partial differential equations (PDEs) into machine learning models to solve inverse problems, such as estimating coefficient functions (e.g., the Hamiltonian function) that characterize physical systems. This framework enables data-driven understanding and prediction of complex physical phenomena. While coefficient functions in PIML are typically estimated on the basis of predictive performance, physics as a discipline does not rely solely on prediction accuracy to evaluate models. For example, Kepler's heliocentric model was favored owing to small discrepancies in planetary motion, despite its similar predictive accuracy to the geocentric model. This highlights the inherent uncertainties in data-driven model inference and the scientific importance of selecting physically meaningful solutions. In this paper, we propose a framework to quantify and analyze such uncertainties in the estimation of coefficient functions in PIML. We apply our framework to reduced model of magnetohydrodynamics and our framework shows that there are uncertainties, and unique identification is possible with geometric constraints. Finally, we confirm that we can estimate the reduced model uniquely by incorporating these constraints.

翻译：物理信息机器学习（PIML）通过将偏微分方程（PDEs）融入机器学习模型，以解决逆向问题，例如估计表征物理系统的系数函数（如哈密顿函数）。该框架实现了对复杂物理现象的数据驱动理解和预测。尽管PIML中的系数函数通常基于预测性能进行估计，但物理学作为一门学科并不完全依赖预测准确性来评估模型。例如，开普勒的日心模型因行星运动中的微小差异而受到青睐，尽管其预测准确性与地心模型相似。这凸显了数据驱动模型推断中固有的不确定性，以及选择具有物理意义解的科学重要性。本文提出一个框架，用于量化和分析PIML中系数函数估计的此类不确定性。我们将该框架应用于磁流体动力学的简化模型，结果表明存在不确定性，但通过几何约束可实现唯一识别。最后，我们证实通过纳入这些约束，可以唯一地估计简化模型。