论文标题
来自高保真综合数据的数据驱动的鲁棒性非本地物理学的学习
Data-driven learning of robust nonlocal physics from high-fidelity synthetic data
论文作者
论文摘要
非局部模型的一个关键挑战是从第一原则中得出它们的分析复杂性，并且经常使用它们是合理的后验。在这项工作中，我们从数据中提取非本地模型，规避这些挑战并为所得模型形式提供数据驱动的理由。由于非线性和缺乏凸性，提取可证明的可证明具有数据驱动的替代物是机器学习方法（ML）方法的主要挑战。我们的方案允许提取可证明的可逆非局部模型的内核可能部分为负。为了实现这一目标，基于既定的非局部理论，我们将足够的条件嵌入了内核的非阳性部分，保证了学识渊博的操作员的良好性。这些条件被施加为不平等约束，并确保模型甚至在小型数据制度中也是可靠的。我们为一系列应用程序展示了此工作流程，包括生产非本地核的复制；与异质周期性微观结构相关的Darcy流量的数值均质化；非局部近似与高阶局部运输现象；以及通过截短的内核对全球支持的分数扩散算子的近似。
A key challenge to nonlocal models is the analytical complexity of deriving them from first principles, and frequently their use is justified a posteriori. In this work we extract nonlocal models from data, circumventing these challenges and providing data-driven justification for the resulting model form. Extracting provably robust data-driven surrogates is a major challenge for machine learning (ML) approaches, due to nonlinearities and lack of convexity. Our scheme allows extraction of provably invertible nonlocal models whose kernels may be partially negative. 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 and ensure that models are robust, even in small-data regimes. 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.
