我们提出了一类新的贝叶斯神经网络（BNN），可以使用噪声数据的变量保真度进行训练，我们将它们应用于学习函数逼近以及解决基于偏微分方程（PDE）的逆问题。这些多保真度BNN由三个神经网络组成：第一个是全连接的神经网络，它是按照最大后验概率（MAP）方法训练的，以拟合低保真度数据：第二个是贝叶斯神经网络，用于捕获低和高保真数据之间的交叉相关性和不确定性量化;最后一个是物理信息神经网络，它对偏微分方程描述的物理规律进行编码。对于后两个神经网络的训练，我们使用Hamilton Monte Carlo方法精确估计相应超参数的后验分布。我们使用合成数据以及实际测量证明了本方法的准确性。具体来说，我们首先近似一个一维和四维函数，然后推断在一维和二维扩散反应系统的反应速率。此外，我们推断海表面温度（SST）在马萨诸塞州和科德角湾使用卫星图像和现场测量。两者合计，我们的研究结果表明，本方法可以捕捉线性和非线性相关性的低和高保真度的数据自适应，识别未知参数的偏微分方程，并量化预测的不确定性，给出了一些分散的噪声高保真度数据。最后，我们证明，我们可以有效地和高效地减少不确定性，从而提高预测精度的主动学习方法，作为一个例子，使用特定的一维函数逼近和逆偏微分方程问题。
英文摘要：
We propose a new class of Bayesian neural networks (BNNs) that can be trained using noisy data of variable fidelity, and we apply them to learn function approximations as well as to solve inverse problems based on partial differential equations (PDEs). These multi-fidelity BNNs consist of three neural networks: The first is a fully connected neural network, which is trained following the maximum a posteriori probability (MAP) method to fit the low-fidelity data; the second is a Bayesian neural network employed to capture the cross-correlation with uncertainty quantification between the low- and high-fidelity data; and the last one is the physics-informed neural network, which encodes the physical laws described by PDEs. For the training of the last two neural networks, we use the Hamiltonian Monte Carlo method to estimate accurately the posterior distributions for the corresponding hyperparameters. We demonstrate the accuracy of the present method using synthetic data as well as real measurements. Specifically, we first approximate a one- and four-dimensional function, and then infer the reaction rates in one- and two-dimensional diffusion-reaction systems. Moreover, we infer the sea surface temperature (SST) in the Massachusetts and Cape Cod Bays using satellite images and in-situ measurements. Taken together, our results demonstrate that the present method can capture both linear and nonlinear correlation between the low- and high-fideilty data adaptively, identify unknown parameters in PDEs, and quantify uncertainties in predictions, given a few scattered noisy high-fidelity data. Finally, we demonstrate that we can effectively and efficiently reduce the uncertainties and hence enhance the prediction accuracy with an active learning approach, using as examples a specific one-dimensional function approximation and an inverse PDE problem.
Multi-Fidelity Bayesian Neural Network for Uncertainty Quantification in Transonic Aerodynamic Loads
School of Engineering, Zurich University of Applied Sciences ZHAW（苏黎世应用科技大学工程学院）；Faculty of Engineering and Physical Sciences, University of Southampton（南安普顿大学工程与物理科学学院）；Federal Institute of Technology Zurich ETHZ（苏黎世联邦理工学院）
Multi-Objective Multi-Fidelity Bayesian Optimization with Causal Priors
Department of Computer Science；Engineering, University of South Carolina, SC, USA；Computer Science Department, Appalachian State University, NC, USA；Engineering, Texas A\&M University, TX, USA
On Some Tunable Multi-fidelity Bayesian Optimization Frameworks
Department of Chemical and Biomolecular Engineering, Johns Hopkins University, U.S.A.（化学与生物分子工程系，约翰霍普金斯大学，美国）；Department of Civil and Systems Engineering, Johns Hopkins University, U.S.A.（土木与系统工程系，约翰霍普金斯大学，美国）；Department of Mechanical Engineering, Massachusetts Institute of Technology, U.S.A.（机械工程系，麻省理工学院，美国）
Multi-fidelity Bayesian Optimization with Max-value Entropy Search and its parallelization
Gifu University（岐阜大学）；Nagoya Institute of Technology（名古屋工业大学）；Nagoya University（名古屋大学）；Japan Science and Technology Agency（日本科学技术振兴机构）；National Institute for Material Science（国立材料科学研究所）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）
TensorBNN: Bayesian Inference for Neural Networks using Tensorflow
Davidson College（戴维森学院）；University of Maryland（马里兰大学）；Florida State University（佛罗里达州立大学）
TensorBNN是一个基于TensorFlow的新软件包，它为现代神经网络模型实现了贝叶斯推断。神经网络模型参数的后验密度表示为使用Hamilton Monte Carlo采样的点云。TensorBNN包利用了TensorFlow的架构和训练功能，以及在训练和预测阶段使用现代图形处理单元（GPU）的能力。
Department of Aerospace and Mechanical Engineering, University of Southern California（航空航天与机械工程系，南加州大学）；School of Mathematics, University of Birmingham（数学学院，伯明翰大学）；Computing and Mathematical Sciences, California Institute of Technology（计算与数学科学系，加州理工学院）
Bayesian Optimization of Genetic Algorithm Hyperparameters in a Multi-Fidelity Framework for Efficient Lattice Material Design
IMDEA Materials Institute（IMDEA材料研究所）；Department of Materials Science and Engineering and Chemical Engineering（材料科学与工程及化学工程系）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
Towards accelerating physical discovery via non-interactive and interactive multi-fidelity Bayesian Optimization: Current challenges and future opportunities
UT-Battelle, LLC（UT-Battelle有限责任公司）；U.S. Department of Energy（美国能源部）
贝叶斯神经网络利用概率层来捕获权重和激活的不确定性，并使用贝叶斯推理进行训练。由于这些概率层被设计成其确定性对应部分的直接替代，贝叶斯神经网络提供了一种直接和自然的方式来扩展传统的深度神经网络以支持概率深度学习。然而，由于贝叶斯神经网络的复杂性，理解、设计和训练贝叶斯神经网络是非常重要的。我们讨论了贝叶斯神经网络的本质，包括对偶（深度神经网络，概率模型），近似贝叶斯推理，贝叶斯先验，贝叶斯后验和深度变分学习。我们使用TensorFlow Probability API和代码示例进行说明。贝叶斯神经网络的主要问题是，深度神经网络的架构使得考虑大量连续层的不确定性变得非常冗余和昂贵。混合贝叶斯神经网络，它使用很少的概率层，公正地定位在网络中，提供了一个实用的解决方案。
Multi-fidelity Bayesian Data-Driven Design of Energy Absorbing Spinodoid Cellular Structures
Department of Electronic Components, Technology and Materials, Delft University of Technology（电子组件、技术和材料系，代尔夫特理工大学）；School of Engineering and Materials Science, Queen Mary University of London（工程与材料科学学院，伦敦女王学院）
