Data-driven learning of nonlocal physics from high-fidelity synthetic data
Data-driven learning of nonlocal physics from high-fidelity synthetic data
复制标题
从高保真合成数据中进行数据驱动的非局域物理学习
DOI：
10.1016/j.cma.2020.113553
复制
发表时间：
2021
影响因子：
7.2
通讯作者：
D’Elia, Marta.
中科院分区：
文献类型：
--
作者：
You, Huaiqian;Yu, Yue;Trask, Nathaniel;Gulian, Mamikon;D’Elia, Marta.
非局部模型的一个关键挑战是从第一原理推导它们的分析复杂性，并且它们的使用经常是事后证明的。在这项工作中，我们从数据中提取非局部模型，规避这些挑战，并为结果模型形式提供数据驱动的理由。由于非线性和缺乏凸性，提取数据驱动的代理是机器学习（ML）方法的主要挑战——提取可证明是适定的和数值稳定的代理尤其具有挑战性。我们的方案不仅产生一个凸优化问题，而且还允许提取非局部模型，这些模型的核可能是部分负的，同时即使在小数据区域也能保持适定性。为了实现这一点，我们基于已建立的非局部理论，在我们的算法中嵌入核的非正部分的充分条件，以保证学习算子的适定性。这些条件作为不等式约束施加，以满足非定域理论的必要条件。我们为一系列应用程序演示了该工作流，包括制造的非本地内核的复制；与非均质周期性微观结构相关的达西流动数值均匀化；高阶局部输运现象的非局部近似用截断核逼近全局支持的分数扩散算子。
A key challenge to nonlocal models is the analytical complexity of deriving them from first principles, and frequently their use is justifieda posteriori. In this work we extract nonlocal models from data, circumventing these challenges and providing data-driven justification for the resulting model form. Extracting data-driven surrogates is a major challenge for machine learning (ML) approaches, due to nonlinearities and lack of convexity — it is particularly challenging to extract surrogates which are provably well-posed and numerically stable. Our scheme not only yields a convex optimization problem, but also allows extraction of nonlocal models whose kernels may be partially negative while maintaining well-posedness even in small-data regimes. To achieve this, based on established nonlocal theory, we embed in our algorithm sufficient conditions on the non-positive part of the kernel that guarantee well-posedness of the learnt operator. These conditions are imposed as inequality constraints to meet the requisite conditions of the nonlocal theory. We demonstrate this workflow for a range of applications, including reproduction of manufactured nonlocal kernels; numerical homogenization of Darcy flow associated with a heterogeneous periodic microstructure; nonlocal approximation to high-order local transport phenomena; and approximation of globally supported fractional diffusion operators by truncated kernels.