嵌入式非局部算子回归（ENOR）：量化学习非局部算子中的模型误差
Embedded Nonlocal Operator Regression (ENOR): Quantifying model error in learning nonlocal operators
AI总结：
本文提出嵌入式非局部算子回归 ENOR，在 NOR 核学习中引入结构化模型误差并结合 MLDA-MCMC 贝叶斯推断，以量化长期均匀化模拟中的模型误差与预测不确定性。
摘要：
非局部积分算子因其能够表示长程依赖和多尺度效应，已成为自下而上均匀化的一种高效替代模型。然而，非局部均匀化模型与微观尺度模型之间存在不可避免的差异。此类误差会在长期模拟中累积并传播，从而使所得预测不可靠。为了开发稳健且可靠的自下而上均匀化框架，我们提出了一个新框架，并将其命名为嵌入式非局部算子回归（ENOR），用于学习非局部均匀化替代模型及其结构化模型误差。该框架为长期模拟中的均匀化材料响应预测提供了差异自适应的不确定性量化。该方法建立在非局部算子回归（NOR）这一基于优化的非局部核学习方法之上，并在可训练核中加入嵌入式模型误差项。随后，采用贝叶斯推断同时推断模型误差项参数和核参数。为使该问题在计算上可行，我们使用多层延迟接受马尔可夫链蒙特卡洛（MLDA-MCMC）方法，从而实现高效的贝叶斯模型校准和模型误差估计。我们将该技术应用于预测非均匀一维杆中的长期波传播，并将其性能与加性噪声模型进行比较。由于具备捕获模型误差的能力，所学得的 ENOR 实现了更好的后验预测不确定性估计。
英文摘要：
Nonlocal, integral operators have become an efficient surrogate for bottom-up homogenization, due to their ability to represent long-range dependence and multiscale effects. However, the nonlocal homogenized model has unavoidable discrepancy from the microscale model. Such errors accumulate and propagate in long-term simulations, making the resultant prediction unreliable. To develop a robust and reliable bottom-up homogenization framework, we propose a new framework, which we coin Embedded Nonlocal Operator Regression (ENOR), to learn a nonlocal homogenized surrogate model and its structural model error. This framework provides discrepancy-adaptive uncertainty quantification for homogenized material response predictions in long-term simulations. The method is built on Nonlocal Operator Regression (NOR), an optimization-based nonlocal kernel learning approach, together with an embedded model error term in the trainable kernel. Then, Bayesian inference is employed to infer the model error term parameters together with the kernel parameters. To make the problem computationally feasible, we use a multilevel delayed acceptance Markov chain Monte Carlo (MLDA-MCMC) method, enabling efficient Bayesian model calibration and model error estimation. We apply this technique to predict long-term wave propagation in a heterogeneous one-dimensional bar, and compare its performance with additive noise models. Owing to its ability to capture model error, the learned ENOR achieves improved estimation of posterior predictive uncertainty.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
