In this paper, a novel framework is established for uncertainty quantification via information bottleneck (IB-UQ) for scientific machine learning tasks, including deep neural network (DNN) regression and neural operator learning (DeepONet). Specifically, we first employ the General Incompressible-Flow Networks (GIN) model to learn a "wide" distribution fromnoisy observation data. Then, following the information bottleneck objective, we learn a stochastic map from input to some latent representation that can be used to predict the output. A tractable variational bound on the IB objective is constructed with a normalizing flow reparameterization. Hence, we can optimize the objective using the stochastic gradient descent method. IB-UQ can provide both mean and variance in the label prediction by explicitly modeling the representation variables. Compared to most DNN regression methods and the deterministic DeepONet, the proposed model can be trained on noisy data and provide accurate predictions with reliable uncertainty estimates on unseen noisy data. We demonstrate the capability of the proposed IB-UQ framework via several representative examples, including discontinuous function regression, real-world dataset regression and learning nonlinear operators for diffusion-reaction partial differential equation.

翻译：本文针对科学机器学习任务（包括深度神经网络回归与神经算子学习（DeepONet）），建立了一种基于信息瓶颈的不确定性量化新框架（IB-UQ）。具体而言，我们首先采用广义不可压缩流网络（GIN）模型从含噪声观测数据中学习"宽泛"分布；随后依据信息瓶颈目标，学习从输入到可预测输出的潜在表示的随机映射。通过归一化流重参数化技术构建IB目标的可处理变分界，从而利用随机梯度下降法优化目标函数。IB-UQ通过对表示变量显式建模，可同时提供标签预测的均值与方差。相较于大多数深度神经网络回归方法与确定性DeepONet，所提模型可在含噪数据上训练，并对未见含噪数据提供带有可靠不确定性估计的精确预测。我们通过若干代表性算例（包括不连续函数回归、真实数据集回归及扩散反应偏微分方程非线性算子学习）验证了所提IB-UQ框架的能力。