Multi-fidelity wavelet neural operator surrogate model for time-independent and time-dependent reliability analysis时不变与时变可靠性分析的多保真小波神经算子代理模型
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Tripura T, Thakur A, Chakraborty S, 2024. Multi-fidelity wavelet neural operator surrogate model for time-independent and time-dependent reliability analysis. Probabilistic Engineering Mechanics, 77: 103672.DOI: 10.1016/j.probengmech.2024.103672
算子学习框架最近成为了一一类有效的科学机器学习工具，用于学习复杂非线性微分方程算子。由于神经算子可学习无限维函数映射，因此适用于需快速预测大量输入函数求解的问题。在不确定性量化的许多应用中，包括可靠性估计和不确定性下的设计，有许多类似问题，这些应用需在大量输入条件下进行数千次抽样，而神经算子专门针对这一问题。尽管神经算子能学习复杂非线性求解算子，但需要大量数据训练。与计算机视觉应用不同，数值模拟的计算复杂性以及合成和真实训练数据的物理试验成本影响了训练神经算子模型的表现，从而直接影响不确定性量化结果的精度。我们在神经算子中采用多保真学习来突破数据瓶颈，其中神经算子通过采用大量低成本的低保真数据和少量高成本的高保真数据进行训练。我们提出了多保真小波神经算子，可从多保真数据集中学习求解算子，以实现动力系统的高效和有效数据驱动可靠性分析。我们在粗网格和细网格上模拟的双保真数据来说明所提框架的性能。关键词: 可靠性分析, 多保真, 算子学习, 小波神经算子, 不确定性量化Operator learning frameworks have recently emerged as an effective scientific machine learning tool for learning complex nonlinear operators of differential equations. Since neural operators learn an infinite-dimensional functional mapping, it is useful in applications requiring rapid prediction of solutions for a wide range of input functions. A task of a similar nature arises in many applications of uncertainty quantification, including reliability estimation and design under uncertainty, each of which demands thousands of samples subjected to a wide range of possible input conditions, an aspect to which neural operators are specialized. Although the neural operators are capable of learning complex nonlinear solution operators, they require an extensive amount of data for successful training. Unlike the applications in computer vision, the computational complexity of the numerical simulations and the cost of physical experiments contributing to the synthetic and real training data compromise the performance of the trained neural operator model, thereby directly impacting the accuracy of uncertainty quantification results. We aim to alleviate the data bottleneck by using multi-fidelity learning in neural operators, where a neural operator is trained by using a large amount of inexpensive low-fidelity data along with a small amount of expensive high-fidelity data. We propose the multi-fidelity wavelet neural operator, capable of learning solution operators from a multi-fidelity dataset, for efficient and effective data-driven reliability analysis of dynamical systems. We illustrate the performance of the proposed framework on bi-fidelity data simulated on coarse and refined grids for spatial and spatiotemporal systems.
Keywords: Reliability analysis; Multi-fidelity; Operator learning; Wavelet neural operator; Uncertainty quantification.提出了学习动力系统算子的多保真小波神经算子
- 作为代理模型，为各类初始和边界条件提供精确可靠性估计
采用多保真数据克服了神经算子的数据瓶颈
Fig. 1. Schematic diagram of the multi-fidelity wavelet neural operator
Fig. 2. Poisson’s equation: Comparison of the predictive performance of different neural operators against true HF solution for one representative forcing function
Fig. 3. Darcy flow in an irregular domain: Comparison of the predictive performance of different neural operators for one representative boundary conditions
Fig. 4. Allen-Cahn equation: Comparison of the spatiotemporal solutions of different neural operators against HF ground truth for a representative initial condition
图 5: 时变 Allen-Cahn 方程首次超越时间的概率分布函数Fig. 5. Probability distribution function of the first passage failure time for time-dependent Allen–Cahn equation
作者信息 | Authors
Tapas Tripura, 共同通讯作者 (Corresp.)印度理工学院德里分校 (Indian Institute of Technology Delhi) 应用力学系Email: tapas.t@am.iitd.ac.in
美国圣母大学 (University of Notre Dame) 航空航天与机械工程系
Souvik Chakraborty, 共同通讯作者 (Corresp.)印度理工学院德里分校 (Indian Institute of Technology Delhi) 应用力学系Email: souvik@am.iitd.ac.in
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
