Neural networks and machine learning models for uncertainty quantification suffer from limited scalability and poor reliability compared to their deterministic counterparts. In industry-scale active learning settings, where generating a single high-fidelity simulation may require days or weeks of computation and produce data volumes on the order of gigabytes, they quickly become impractical. This paper proposes a scalable and reliable Bayesian surrogate model, termed the Bayesian Interpolating Neural Network (B-INN). The B-INN combines high-order interpolation theory with tensor decomposition and alternating direction algorithm to enable effective dimensionality reduction without compromising predictive accuracy. We theoretically show that the function space of a B-INN is a subset of that of Gaussian processes, while its Bayesian inference exhibits linear complexity, $\mathcal{O}(N)$, with respect to the number of training samples. Numerical experiments demonstrate that B-INNs can be from 20 times to 10,000 times faster with a robust uncertainty estimation compared to Bayesian neural networks and Gaussian processes. These capabilities make B-INN a practical foundation for uncertainty-driven active learning in large-scale industrial simulations, where computational efficiency and robust uncertainty calibration are paramount.

翻译：与确定性模型相比，用于不确定性量化的神经网络和机器学习模型存在可扩展性有限和可靠性不足的问题。在工业级主动学习场景中，生成单个高保真仿真可能需要数天或数周的计算时间并产生吉字节量级的数据，这使得传统方法迅速变得不可行。本文提出了一种可扩展且可靠的贝叶斯代理模型，称为贝叶斯插值神经网络（B-INN）。B-INN 将高阶插值理论与张量分解及交替方向算法相结合，在不牺牲预测精度的前提下实现有效的降维。我们从理论上证明 B-INN 的函数空间是高斯过程函数空间的子集，而其贝叶斯推断具有相对于训练样本数量的线性复杂度 $\mathcal{O}(N)$。数值实验表明，与贝叶斯神经网络和高斯过程相比，B-INN 在具备稳健不确定性估计的同时，速度可提升 20 倍至 10,000 倍。这些能力使得 B-INN 成为大规模工业仿真中不确定性驱动主动学习的实用基础，其中计算效率和稳健的不确定性校准至关重要。