[论文解读] Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons

Apostolos F Psaros, Xuhui Meng+3 more|arXiv (Cornell University)|Jan 19, 2022

Model Reduction and Neural Networks被引用 24

一句话总结

本文提出一个用于在 SciML 中建模和评估总不确定性的综合框架，整合贝叶斯方法、集成、函数先验和事后校准，覆盖 PINN 和 DeepONet 语境，并有广泛的比较研究。

ABSTRACT

Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. This is because in addition to aleatoric uncertainty associated with noisy data, there is also uncertainty due to limited data, but also due to NN hyperparameters, overparametrization, optimization and sampling errors as well as model misspecification. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this work, we present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. To demonstrate the applicability and reliability of our framework, we present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions. In the Appendix, we include a comprehensive description of all the UQ methods employed, which we will make available as open-source library of all codes included in this framework.

研究动机与目标

• 动机并形式化在 SciML 中不确定性量化的挑战，包括数据、模型和优化源造成的误差。
• 提出一个将物理信息神经网络和神经算子与概率方法相结合的统一不确定性建模框架。
• 在 SciML 内部对常见的不确定性量化方法（贝叶斯、集成、函数先验）和评估指标进行综述与整合。
• 通过对具有异构数据的神经偏微分方程和神经算子进行广泛比较研究来演示该框架。
• 提供开源资源和对 SciML 的后训练改进与校准的指导。

提出的方法

• 描述 SciML 中的不确定性来源，并将总不确定性表述为泛然误差（aleatoric）与知识不确定性（epistemic）分量的组合。
• 将 PINN 与 DeepONet 的基础作为求解神经 PDEs 和神经算子的问题的组成模块，分别。
• 使用基于似然的后验、贝叶斯模型平均和蒙特卡罗抽样来建模预测分布，以量化总不确定性。
• 引入函数先验以及集成与贝叶斯后验推断方法，以估计神经网络参数的后验分布。
• 纳入后训练校准和评估指标，以评估准确性和不确定性质量。

实验结果

研究问题

• RQ1 如何在神经 PDEs 和神经算子中对 SciML 的总不确定性进行建模和量化？
• RQ2 在多样化数据和模型错配下，哪种贝叶斯、集成和函数先验方法的组合能产生可靠的预测分布？
• RQ3 如何在一个统一的 UQ 框架中处理异方差噪声、混合确定性/随机问题以及算子学习？
• RQ4 哪些指标和校准技术最能评估 SciML 应用中不确定性估计的质量？
• RQ5 在前向、混合和随机 PDE 问题及算子学习任务的对比研究中，不同方法的表现如何？

主要发现

• 通过在 PINN 和 DeepONet 上整合先验、贝叶斯推断、集成和校准，构建一个适用于 SciML 的统一不确定性量化框架是可行的。
• 该框架支持异质数据设置，包括带噪声和缺失数据，以及混合确定性/随机问题。
• 后验推断方法产生的预测分布可分解为泛然误差和知识不确定性分量，从而实现不确定性分解。
• 校准和后训练技术提高不确定性估计与模型预测的可靠性。
• 对函数近似、混合 PDE、随机偏微分方程（SPDE）、前向问题和算子学习的广泛适用性得到验证。
• 提供了随框架附带的开源代码库（附录引用）。