Learning nonlocal regularization operators

从计划到映射：最优传输的非局部正则化

From plans to maps: Nonlocal regularization of optimal transport

该研究提出最优传输的非局部正则化方法，连接Kantorovich与Monge形式，分析其极限行为并推导对偶形式与数值格式，实现两类问题的等价性及相关测度的存在性结果。

受局部到非局部耦合约束的界面识别

Interface Identification constrained by Local-to-Nonlocal Coupling

本文针对界面识别问题，提出将基于能量的局部到非局部耦合作为约束条件，结合偏微分与非局部算子以兼顾模型适用性与计算成本。

An elementary note about the G-convergence for a type of nonlocal p-laplacian operator

Simple applications of a principle of minimum energy and the property of monotonicity for the corresponding non-local operator, have allowed a direct proof of the G-compactness in a weak sense, as well as in the strong sense. The definitions of H-convergence and G-convergence, weak or strong, have been proved to be equivalent.

基于算子理论的非局部条件分数阶发展方程的近似能控性

Approximate Controllability of Fractional Evolution Equations with Nonlocal Conditions via Operator Theory

本文研究了Hilbert空间中非局部条件分数阶发展方程的温和解存在性、唯一性与近似能控性，通过半线性算子方程近似可解性新方法建立充分条件，并基于Gramian算子构造控制函数。

nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications

Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integral equations with sparse, noisy, unstructured, and multi-fidelity data. PINNs incorporate all available information into a loss function, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal ope…