Surrogate models of parametric dynamical systems are essential for many-query and real-time predictions in engineering applications such as design optimization and digital twins. However, generating high-fidelity (HF) training data over a broad range of parameters and operating conditions remains computationally expensive. To address this challenge, we propose a Bayesian multi-fidelity Laplace neural operator (MF-LNO) for uncertainty-aware active learning of oscillatory parametric PDEs. Specifically, the proposed Bayesian MF-LNO iteratively calibrates the discrepancy between low- and high-fidelity data, where predictive uncertainty guides the adaptive acquisition of informative HF trajectories. Such predictive uncertainty is quantified via replica-exchange stochastic gradient Langevin dynamics (reSGLD), whose broad posterior exploration enables uncertainty to serve as an error indicator for adaptive HF sample acquisition. Numerical experiments on the Lorenz system, Duffing oscillator, and beam dynamics demonstrate that uncertainty-guided HF sample acquisition consistently outperforms random sampling, while the proposed Bayesian MF-LNO achieves higher prediction accuracy than MF-DeepONet with predictive uncertainty quantification. These results demonstrate that Bayesian multi-fidelity LNOs, combined with uncertainty-guided active learning, provide a data-efficient framework for operator learning in engineering dynamical systems.
Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domains
Department of Mathematics, Purdue University（普渡大学数学系）；Department of Mathematics and Department of Computer Science, University of Maryland College Park（马里兰大学学院市数学系和计算机科学系）
Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations
Department of Mathematics, Pohang University of Science and Technology (POSTECH)（POSTECH数学系）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院机械工程系）
MiNO: Cotangent-bundle propagator learning for PDEs
Axiom Research Group（公理研究集团）；The Nelson Mandela African Institution of Science and Technology（纳尔逊·曼德拉非洲科学技术研究院）；African Institute for Mathematical Sciences（非洲数学科学研究所）；The University of Tokyo（东京大学）
Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs
School of Mathematics and Statistics（数学与统计学学院）；Department of Engineering Mechanics（工程力学系）；Department of Aeronautical Engineering（航空工程系）；MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University（复杂系统动力学与控制国家重点实验室，西北工业大学）；College of Computer Science and Technology（计算机科学与技术学院）；National University of Defense Technology（国防科技大学）；Department of Aerospace Engineering（航空航天工程系）
IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients
Oden Institute for Computational Science and Engineering, The University of Texas at Austin（计算科学与工程院，德克萨斯大学奥斯汀分校）；Walker Department of Mechanical Engineering, The University of Texas at Austin（机械工程系，德克萨斯大学奥斯汀分校）
Post-Hoc Uncertainty-Aware Explanations for Deployed Power Quality Disturbance Classifiers via Laplace Approximation
School of Engineering, Deakin University（德肯大学工程学院）；ARC Training Centre in Energy Technologies for Future Grids, School of Engineering, University of Wollongong（未来电网能源技术培训中心，沃尔灵宗大学工程学院）
Solving Partial Differential Equations in Different Domains by Operator Learning method Based on Boundary Integral Equations
School of Computer Science and Engineering, Sun Yat-sen University（中山大学计算机科学与工程学院）；Guangdong Province Key Laboratory of Computational Science（广东省计算科学重点实验室）
