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.
This paper proposes a new data-driven method for the reliable prediction of power system post-fault trajectories. The proposed method is based on the fundamentally new concept of Deep Operator Networks (DeepONets). Compared to traditional neural networks that learn to approximate functions, DeepONets are designed to approximate nonlinear operators. Under this operator framework, we design a DeepONet to (1) take as inputs the fault-on trajectories collected, for example, via simulation or phasor measurement units, and (2) provide as outputs the predicted post-fault trajectories. In addition, we…
A new data-driven method for operator learning of stochastic differential equations(SDE) is proposed in this paper. The central goal is to solve forward and inverse stochastic problems more effectively using limited data. Deep operator network(DeepONet) has been proposed recently for operator learning. Compared to other neural networks to learn functions, it aims at the problem of learning nonlinear operators. However, it can be challenging by using the original model to learn nonlinear operators for high-dimensional stochastic problems. We propose a new multi-resolution autoencoder DeepONet m…
Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantification
Department of Applied Mechanics（应用力学系）；Indian Institute of Technology Delhi（印度理工学院德里）；Department of Mechanical Engineering（机械工程系）；Northern Arizona University（北亚利桑那大学）
Time dependent reliability analysis and uncertainty quantification of structural system subjected to stochastic forcing function is a challenging endeavour as it necessitates considerable computational time. We investigate the efficacy of recently proposed DeepONet in solving time dependent reliability analysis and uncertainty quantification of systems subjected to stochastic loading. Unlike conventional machine learning and deep learning algorithms, DeepONet learns is a operator network and learns a function to function mapping and hence, is ideally suited to propagate the uncertainty from th…
We present a simple and effective approach for posterior uncertainty quantification in deep operator networks (DeepONets); an emerging paradigm for supervised learning in function spaces. We adopt a frequentist approach based on randomized prior ensembles, and put forth an efficient vectorized implementation for fast parallel inference on accelerated hardware. Through a collection of representative examples in computational mechanics and climate modeling, we show that the merits of the proposed approach are fourfold. (1) It can provide more robust and accurate predictions when compared against…
Deep set based operator learning with uncertainty quantification
Department of Mathematics, Shanghai Normal University（上海师范大学数学系）；School of Mathematical Sciences, Institute of Natural Sciences（自然科学院数学系）；MOE-LSC, Shanghai Jiao Tong University（教育部重点实验室，上海交通大学）；Institute of Computational Mathematics（计算数学研究所）；Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences（科学与工程计算，中国科学院数学研究所）
IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learning
Shanghai Normal University（上海师范大学）；Shanghai Jiaotong University（上海交通大学）；Tongji University（同济大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Diffusion-Based Stochastic Operator Networks for Uncertainty Quantification in Stochastic Partial Differential Equations
Department of Mathematics, Florida State University（佛罗里达州立大学数学系）；Computer Science and Mathematics Division, Oak Ridge National Laboratory（橡树岭国家实验室计算机科学与数学 division）
The Charles Stark Draper Laboratory, Inc.（查尔斯·斯塔克·德雷珀实验室公司）
Deep learning has the potential to dramatically impact navigation and tracking state estimation problems critical to autonomous vehicles and robotics. Measurement uncertainties in state estimation systems based on Kalman and other Bayes filters are typically assumed to be a fixed covariance matrix. This assumption is risky, particularly for "black box" deep learning models, in which uncertainty can vary dramatically and unexpectedly. Accurate quantification of multivariate uncertainty will allow for the full potential of deep learning to be used more safely and reliably in these applications. …
Direct Interval Propagation Methods using Neural-Network Surrogates for Uncertainty Quantification in Physical Systems Surrogate Model
LMSD, Department of Mechanical Engineering, KU Leuven（KU莱顿机械工程系LMSD）；RAM, Department of Mechanical Engineering, KU Leuven（KU莱顿机械工程系RAM）；DistriNet, Department of Computer Science, KU Leuven（KU莱顿计算机科学系DistriNet）；Visual Artificial Intelligence Laboratory, Oxford Brookes University（奥克斯福德布鲁克斯大学视觉人工智能实验室）
Active operator learning with predictive uncertainty quantification for partial differential equations
Sandia National Laboratories（桑迪亚国家实验室）；Department of Statistics（统计学系）；Data Science, Yale University（数据科学，耶鲁大学）；Environmental Engineering, Yale University（环境工程，耶鲁大学）；Department of Mathematics（数学系）；Department of Mechanical Engineering（机械工程系）
Online machine-learning forecast uncertainty estimation for sequential data assimilation
Servicio Meteorológico Nacional（国家气象局）；Universidad de Buenos Aires（布宜诺斯艾利斯大学）；Instituto Franco-Argentino para el Estudio del Clima y sus Impactos (IRL 3351 IFAECI)（法阿气候及其影响研究所（IRL 3351 IFAECI））；Universidad Nacional del Nordeste（东北国立大学）；Centro de Investigaciones del Mar y la Atmósfera（海洋与大气研究中心）；IMT Atlantique（大西洋国立高等工业技术学院）；Inria（法国国家信息与自动化研究所）
Standard neural networks can approximate general nonlinear operators, represented either explicitly by a combination of mathematical operators, e.g., in an advection-diffusion-reaction partial differential equation, or simply as a black box, e.g., a system-of-systems. The first neural operator was the Deep Operator Network (DeepONet), proposed in 2019 based on rigorous approximation theory. Since then, a few other less general operators have been published, e.g., based on graph neural networks or Fourier transforms. For black box systems, training of neural operators is data-driven only but if…
Bayesian Physics Informed Neural Networks (B-PINNs) have gained significant attention for inferring physical parameters and learning the forward solutions for problems based on partial differential equations. However, the overparameterized nature of neural networks poses a computational challenge for high-dimensional posterior inference. Existing inference approaches, such as particle-based or variance inference methods, are either computationally expensive for high-dimensional posterior inference or provide unsatisfactory uncertainty estimates. In this paper, we present a new efficient infere…
e:fs TechHub GmbH（e:fs科技中心有限公司）；Ruhr West University of Applied Sciences（鲁尔西区应用科学大学）
Reliable spatial uncertainty evaluation of object detection models is of special interest and has been subject of recent work. In this work, we review the existing definitions for uncertainty calibration of probabilistic regression tasks. We inspect the calibration properties of common detection networks and extend state-of-the-art recalibration methods. Our methods use a Gaussian process (GP) recalibration scheme that yields parametric distributions as output (e.g. Gaussian or Cauchy). The usage of GP recalibration allows for a local (conditional) uncertainty calibration by capturing dependen…
Geosteering workflows are increasingly based on the quantification of subsurface uncertainties during real-time operations. As a consequence operational decision making is becoming both better informed and more complex. This paper presents an experimental web-based decision support system, which can be used to both aid expert decisions under uncertainty or further develop decision optimization algorithms in controlled environment. A user of the system (either human or AI) controls the decisions to steer the well or stop drilling. Whenever a user drills ahead, the system produces simulated meas…
Probabilistic Predictions of Process-Induced Deformation in Carbon/Epoxy Composites Using a Deep Operator Network
Department of Applied Mathematics, Brown University, USA（布朗大学应用数学系）；Center for Composite Materials, University of Delaware, Newark, Delaware 19716, USA（德雷塞尔大学复合材料中心）；Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716, USA（德雷塞尔大学机械工程系）；Department of Material Science and Engineering, University of Delaware, Newark, Delaware 19716, USA（德雷塞尔大学材料科学与工程系）；Department of Chemical and Biomolecular Engineering, University of Delaware, Newark, Delaware 19716, USA（德雷塞尔大学化学与生物分子工程系）；Department of Computer and Information Sciences, University of Delaware, Newark, Delaware 19716, USA（德雷塞尔大学计算机与信息科学系）；Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA（克莱姆森大学机械工程系）
Distribution-Free Uncertainty-Aware Virtual Sensing via Conformalized Neural Operators
The Grainger College of Engineering, Nuclear, Plasma & Radiological Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格雷格尔工程学院）；National Center for Supercomputing Applications（国家超级计算应用中心）；Department of Applied Mechanics, Indian Institute of Technology Delhi（印度理工学院德里应用力学系）；Yardi School of Artificial Intelligence, Indian Institute of Technology Delhi（印度理工学院德里人工智能学院）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
提出CMCO框架，通过在DeepONet架构中统一Monte Carlo dropout与split conformal prediction，为神经算子提供无分布、免重训练的校准预测区间，实现跨异构域的高效不确定性量化，在多个应用中达到接近标称的覆盖率。
