用误差界提升物理神经网络的不确定性估计，助力宇宙学参数反演
Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles
- 基于误差界构建异方差方差，改进解的不确定性建模
- 在宇宙学逆问题中实现前向求解与参数估计的联合不确定性分析
- 适合关注物理模型可信度与科学推断不确定性的研究者
物理信息神经网络（PINNs）被广泛用于求解各类由微分方程建模的物理现象。由于PINNs本身缺乏不确定性量化机制，已有研究尝试对其不确定性进行建模。本文提出一种两步训练策略，构建贝叶斯神经网络以提供对微分方程系统解的不确定性估计。利用现有PINNs的误差界，构建异方差方差，从而提升不确定性估计精度。此外，通过求解前向问题并利用所得不确定性，在宇宙学逆问题中实现了参数估计。该方法在真实数据场景下验证了其有效性。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks (PINNs) have been widely used to obtain solutions to various physical phenomena modeled as Differential Equations. As PINNs are not naturally equipped with mechanisms for Uncertainty Quantification, some work has been done to quantify the different uncertainties that arise when dealing with PINNs. In this paper, we use a two-step procedure to train Bayesian Neural Networks that provide uncertainties over the solutions to differential equation systems provided by PINNs. We use available error bounds over PINNs to formulate a heteroscedastic variance that improves the uncertainty estimation. Furthermore, we solve forward problems and utilize the obtained uncertainties when doing parameter estimation in inverse problems in cosmology.
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