非局部核网络（NKN）：一种稳定且与分辨率无关的深度神经网络
Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network
摘要：
神经算子最近已经成为设计神经网络形式的函数空间之间的解映射的流行工具。与经典的科学机器学习方法不同的是，神经操作符近似一族偏微分方程（PDE）的解映射，所述经典的科学机器学习方法以固定的分辨率针对输入参数的单个实例来学习已知偏微分方程（PDE）的参数。尽管它们取得了成功，但神经运算符的使用迄今为止仅限于相对较浅的神经网络，并限于学习隐藏的支配定律。在这项工作中，我们提出了一种新的非局部神经算子，我们称之为非局部核网络（NKN），这是分辨率无关的，其特征在于深度神经网络，并能够处理各种任务，如学习控制方程和分类图像。我们的NKN源于解释的神经网络作为一个离散的非局部扩散反应方程，在无限层的限制，相当于一个抛物型非局部方程，其稳定性通过非局部向量微积分分析。与神经操作符的积分形式的相似性允许NKNs捕获特征空间中的长程依赖关系，而节点到节点的相互作用的连续处理使得NKNs分辨率独立。在非局部意义上重新解释的与神经ODE的相似性，以及层之间的稳定网络动态允许NKN的最佳参数从浅网络推广到深网络。这一事实使得能够使用从浅到深的初始化技术。我们的测试表明，NKN在学习控制方程和图像分类任务方面都优于基线方法，并且可以很好地推广到不同的分辨率和深度。
英文摘要：
Neural operators have recently become popular tools for designing solution maps between function spaces in the form of neural networks. Differently from classical scientific machine learning approaches that learn parameters of a known partial differential equation (PDE) for a single instance of the input parameters at a fixed resolution, neural operators approximate the solution map of a family of PDEs. Despite their success, the uses of neural operators are so far restricted to relatively shallow neural networks and confined to learning hidden governing laws. In this work, we propose a novel nonlocal neural operator, which we refer to as nonlocal kernel network (NKN), that is resolution independent, characterized by deep neural networks, and capable of handling a variety of tasks such as learning governing equations and classifying images. Our NKN stems from the interpretation of the neural network as a discrete nonlocal diffusion reaction equation that, in the limit of infinite layers, is equivalent to a parabolic nonlocal equation, whose stability is analyzed via nonlocal vector calculus. The resemblance with integral forms of neural operators allows NKNs to capture long-range dependencies in the feature space, while the continuous treatment of node-to-node interactions makes NKNs resolution independent. The resemblance with neural ODEs, reinterpreted in a nonlocal sense, and the stable network dynamics between layers allow for generalization of NKN's optimal parameters from shallow to deep networks. This fact enables the use of shallow-to-deep initialization techniques. Our tests show that NKNs outperform baseline methods in both learning governing equations and image classification tasks and generalize well to different resolutions and depths.
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