Title: Uncertainty quantification for deeponets with ensemble kalman inversion Show Chinese title

Title: 深度网络的不确定性量化与集合卡尔曼反演

Abstract: 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. Show Chinese abstract

Abstract: 近年来，算子学习，特别是DeepONet，在高效学习输入和输出函数之间的复杂映射方面引起了广泛关注，尤其是在多个领域中。然而，在有限和噪声数据的实际场景中，获取DeepONet预测的不确定性变得至关重要，特别是在任务关键或安全关键的应用中。现有的方法，要么计算密集，要么产生不令人满意的不确定性量化，因此需要开发针对DeepONets的高效且信息丰富的不确定性量化（UQ）技术。在本工作中，我们提出了一种新颖的推理方法，通过利用集成卡尔曼反演（EKI）方法的力量，实现了算子学习的高效UQ。EKI以其无需导数、抗噪声和高度并行化的特性而闻名，已在物理信息神经网络的UQ中展示了其优势[28]。我们创新地应用EKI，使我们能够高效地训练DeepONet的集合，同时获得感兴趣输出的信息性不确定性估计。我们部署了EKI的小批量变体以适应更大的数据集，从而减轻训练阶段由于大数据集带来的计算需求。此外，我们引入了一种启发式方法来估计人工动态协方差，从而改进我们的不确定性估计。最后，我们在各种基准问题上展示了所提出方法的有效性和通用性，展示了其在解决DeepONets中不确定性量化紧迫挑战方面的潜力，特别是在有限和噪声数据的实际应用中。