In recent years, operator learning, particularly the DeepONet, has received much attention for efficiently learning complex mappings between input and output functions across diverse fields. However, in practical scenarios with limited and noisy data, accessing the uncertainty in DeepONet predictions becomes essential, especially in mission-critical or safety-critical applications. Existing methods, either computationally intensive or yielding unsatisfactory uncertainty quantification, leave room for developing efficient and informative uncertainty quantification (UQ) techniques tailored for DeepONets. In this work, we proposed a novel inference approach for efficient UQ for operator learning by harnessing the power of the Ensemble Kalman Inversion (EKI) approach. EKI, known for its derivative-free, noise-robust, and highly parallelizable feature, has demonstrated its advantages for UQ for physics-informed neural networks [28]. Our innovative application of EKI enables us to efficiently train ensembles of DeepONets while obtaining informative uncertainty estimates for the output of interest. We deploy a mini-batch variant of EKI to accommodate larger datasets, mitigating the computational demand due to large datasets during the training stage. Furthermore, we introduce a heuristic method to estimate the artificial dynamics covariance, thereby improving our uncertainty estimates. Finally, we demonstrate the effectiveness and versatility of our proposed methodology across various benchmark problems, showcasing its potential to address the pressing challenges of uncertainty quantification in DeepONets, especially for practical applications with limited and noisy data.

翻译：近年来，算子学习（尤其是DeepONet）因其能够高效学习不同领域输入与输出函数之间的复杂映射而备受关注。然而，在数据有限且存在噪声的实际场景中，评估DeepONet预测结果的不确定性变得至关重要，特别是在关键任务或安全关键型应用中。现有方法或计算成本过高，或无法提供令人满意的不确定性量化，这为开发针对DeepONet的高效且信息丰富的UQ技术留下了空间。本研究提出一种新型推理方法，通过利用集成卡尔曼反演（EKI）的强大能力实现算子学习的高效不确定性量化。EKI以其无需导数、抗噪性强和高度并行化的特点著称，已在物理信息神经网络的不确定性量化中展现出优势[28]。我们创新性地将EKI应用于DeepONet的集成训练，在高效训练模型的同时为目标输出提供信息丰富的后验不确定性估计。为适应大规模数据集，我们采用EKI的小批量变体，从而降低训练阶段因大数据带来的计算需求。此外，我们引入一种启发式方法来估计人工动力学协方差，进而提升不确定性估计质量。最后，通过多个基准问题的验证，本研究方法展现了其在解决DeepONet不确定性量化关键挑战中的有效性与普适性，尤其适用于数据有限且存在噪声的实际应用场景。