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 first employ the mean-field variational inference (VI) to maximize the evidence lower bound (ELBO) to obtain informative prior distributions for the hyperparameters in the BNNs, and subsequently we use the Hamiltonian Monte Carlo (HMC) 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-fidelity 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.

我们提出了一种新的贝叶斯神经网络（BNN），可以使用可变保真度的噪声数据进行训练，并将其用于学习函数逼近以及基于偏微分方程（PDE）求解逆问题。这些多保真BNN由三个神经网络组成：第一个是完全连接的神经网络，它遵循最大后验概率（MAP）方法进行训练以适合低保真数据；第二个是贝叶斯神经网络，用于捕获低保真和高保真数据之间的互相关和不确定性量化。最后一个是物理信息神经网络，该网络编码PDE所描述的物理定律。对于最后两个神经网络的训练，我们首先采用均值场变分推理（VI）来最大化证据下界（ELBO），以获得BNN中超参数的信息性先验分布，然后我们使用哈密顿量蒙特卡罗（HMC）方法准确估算后验相应超参数的分布。我们使用合成数据以及实际测量值证明了本方法的准确性。具体来说，我们首先近似一维和四维函数，然后推断一维和二维扩散反应系统中的反应速率。此外，我们使用卫星图像和原位测量来推断马萨诸塞州和科德角湾的海表温度（SST）。在一起 我们的结果表明，在一些散布有噪声的高保真度数据的情况下，本方法可以自适应地捕获低保真度和高保真度数据之间的线性和非线性相关性，识别PDE中的未知参数，并量化预测中的不确定性。最后，我们证明了我们可以通过使用主动学习方法（例如使用特定的一维函数逼近和逆PDE问题）来有效，有效地减少不确定性，从而提高预测精度。