This work presents novel extensions for combining two frameworks for
quantifying both aleatoric (i.e., irreducible) and epistemic (i.e., reducible)
sources of uncertainties in the modeling of engineered systems. The
data-consistent (DC) framework poses an inverse problem and solution for
quantifying aleatoric uncertainties in terms of pullback and push-forward
measures for a given Quantity of Interest (QoI) map. Unfortunately, a
pre-specified QoI map is not always available a priori to the collection of
data associated with system outputs. The data themselves are often polluted
with measurement errors (i.e., epistemic uncertainties), which complicates the
process of specifying a useful QoI. The Learning Uncertain Quantities (LUQ)
framework defines a formal three-step machine-learning enabled process for
transforming noisy datasets into samples of a learned QoI map to enable
DC-based inversion. We develop a robust filtering step in LUQ that can learn
the most useful quantitative information present in spatio-temporal datasets.
The learned QoI map transforms simulated and observed datasets into
distributions to perform DC-based inversion. We also develop a DC-based
inversion scheme that iterates over time as new spatial datasets are obtained
and utilizes quantitative diagnostics to identify both the quality and impact
of inversion at each iteration. Reproducing Kernel Hilbert Space theory is
leveraged to mathematically analyze the learned QoI map and develop a
quantitative sufficiency test for evaluating the filtered data. An illustrative
example is utilized throughout while the final two examples involve the
manufacturing of shells of revolution to demonstrate various aspects of the
presented frameworks.

本文介绍了将两个框架相结合的新扩展，用于量化工程系统建模中的不可避免不确定性和可归约不确定性来源。

从位移到分布：一种机器学习支持的用于量化计算模型参数不确定性的框架

From Displacements to Distributions: A Machine-Learning Enabled  Framework for Quantifying Uncertainties in Parameters of Computational Models

Obtaining heteroscedastic predictive uncertainties from a Bayesian Neural
Network (BNN) is vital to many applications. Often, heteroscedastic aleatoric
uncertainties are learned as outputs of the BNN in addition to the predictive
means, however doing so may necessitate adding more learnable parameters to the
network. In this work, we demonstrate that both the heteroscedastic aleatoric
and epistemic variance can be embedded into the variances of learned BNN
parameters, improving predictive performance for lightweight networks. By
complementing this approach with a moment propagation approach to inference, we
introduce a relatively simple framework for sampling-free variational inference
suitable for lightweight BNNs.

本研究提出一种对轻量级贝叶斯神经网络（BNN）进行抽样无关变分推断的简单框架，通过将异方差预测不确定性和经验差异嵌入到学习得到的 BNN 参数的方差中，提高了预测性能。

轻量级贝叶斯神经网络具有异方差不确定性的变分推断框架

A Framework for Variational Inference of Lightweight Bayesian Neural  Networks with Heteroscedastic Uncertainties

In this research, we present SLYKLatent, a novel approach for enhancing gaze
estimation by addressing appearance instability challenges in datasets due to
aleatoric uncertainties, covariant shifts, and test domain generalization.
SLYKLatent utilizes Self-Supervised Learning for initial training with facial
expression datasets, followed by refinement with a patch-based tri-branch
network and an inverse explained variance-weighted training loss function. Our
evaluation on benchmark datasets achieves an 8.7% improvement on Gaze360,
rivals top MPIIFaceGaze results, and leads on a subset of ETH-XGaze by 13%,
surpassing existing methods by significant margins. Adaptability tests on
RAF-DB and Affectnet show 86.4% and 60.9% accuracies, respectively. Ablation
studies confirm the effectiveness of SLYKLatent's novel components. This
approach has strong potential in human-robot interaction.

我们提出了一种名为 SLYKLatent 的新颖方法，通过解决由于不确定性、协变量偏移和测试域泛化引起的数据集外观不稳定性挑战，来增强凝视估计。该方法利用自监督学习进行初始训练，使用基于图像块的三支网络和反向解释方差加权训练损失函数进行改进。我们在基准数据集上的评估结果在 Gaze360 上实现了 8.7% 的改进，在 MPIIFaceGaze 上与顶级结果不相上下，在 ETH-XGaze 的子集上超过现有方法的 13%。RAF-DB 和 Affectnet 上的适应性测试分别显示了 86.4% 和 60.9% 的准确率。消融研究证实了 SLYKLatent 的新颖组件的有效性。该方法在人机交互中具有巨大潜力。

SLYKLatent，面部特征估计的学习框架

SLYKLatent, a Learning Framework for Facial Features Estimation

The hybrid neural differentiable models mark a significant advancement in the
field of scientific machine learning. These models, integrating numerical
representations of known physics into deep neural networks, offer enhanced
predictive capabilities and show great potential for data-driven modeling of
complex physical systems. However, a critical and yet unaddressed challenge
lies in the quantification of inherent uncertainties stemming from multiple
sources. Addressing this gap, we introduce a novel method, DiffHybrid-UQ, for
effective and efficient uncertainty propagation and estimation in hybrid neural
differentiable models, leveraging the strengths of deep ensemble Bayesian
learning and nonlinear transformations. Specifically, our approach effectively
discerns and quantifies both aleatoric uncertainties, arising from data noise,
and epistemic uncertainties, resulting from model-form discrepancies and data
sparsity. This is achieved within a Bayesian model averaging framework, where
aleatoric uncertainties are modeled through hybrid neural models. The unscented
transformation plays a pivotal role in enabling the flow of these uncertainties
through the nonlinear functions within the hybrid model. In contrast, epistemic
uncertainties are estimated using an ensemble of stochastic gradient descent
(SGD) trajectories. This approach offers a practical approximation to the
posterior distribution of both the network parameters and the physical
parameters. Notably, the DiffHybrid-UQ framework is designed for simplicity in
implementation and high scalability, making it suitable for parallel computing
environments. The merits of the proposed method have been demonstrated through
problems governed by both ordinary and partial differentiable equations.

混合神经可微模型 (Hybrid Neural Differentiable Models) 在科学机器学习领域中具有重要的进展。这些模型将已知物理学的数值表示与深度神经网络相结合，提供了增强的预测能力，并展现了在数据驱动的复杂物理系统建模方面的巨大潜力。然而，一个至关重要但尚未解决的挑战在于量化来自多个来源的内在不确定性。针对这个问题，我们引入了一种新的方法 DiffHybrid-UQ，用于混合神经可微模型中的有效和高效的不确定性传播和估计，充分利用了深度集成贝叶斯学习和非线性变换的优势。具体地说，我们的方法能够有效地识别和量化同时来自数据噪声 (aleatoric uncertainties) 和模型形式偏差以及数据稀疏性引起的认知不确定性 (epistemic uncertainties)。这是在贝叶斯模型平均框架下实现的，其中通过混合神经模型来建模随机噪声。在混合模型中的非线性函数通过无损变换 (unscented transformation) 实现这些不确定性的传播。与此相反，我们使用一组随机梯度下降 (SGD) 轨迹来估计认知不确定性。这种方法为网络参数和物理参数的后验分布提供了实用的近似。值得注意的是，DiffHybrid-UQ 框架的设计考虑到了实施的简单性和高可扩展性，使其适用于并行计算环境。通过一些受常微分方程和偏微分方程影响的问题，我们展示了所提出方法的优势。