提出量化物理信息反问题中系数估计不确定性的方法，揭示科学模型选择的深层风险。
Uncertainties in Physics-informed Inverse Problems: The Hidden Risk in Scientific AI
- 基于几何约束构建不确定性量化框架，解决系数函数估计的多解性问题。
- 在磁流体简化模型中验证：无约束时存在多重解，引入几何约束后可唯一确定系数。
- 适用于追求物理可解释性的科研场景，尤其适合需要精确建模的科学计算领域。
物理信息机器学习（PIML）将偏微分方程（PDE）融入机器学习模型以求解反问题，如估计表征物理系统的系数函数（例如哈密顿量）。该框架支持复杂物理现象的数据驱动理解与预测。然而，传统PIML仅依赖预测性能评估系数，而物理学评价模型不仅看预测精度——如开普勒的日心说因行星运动偏差小而被青睐，尽管其预测能力与地心说相近。这揭示了数据驱动模型推断中的内在不确定性及选择物理上合理解的重要性。本文提出一个框架，用于量化和分析PIML中系数函数估计的不确定性。我们将其应用于磁流体简化模型，发现存在不确定性，且在引入几何约束后可实现唯一识别。最终证实，通过加入这些约束，可唯一估计简化模型。
原文摘要 · Abstract (English)
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.
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