Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network

TL;DR

The paper introduces the nonlocal kernel network (NKN), a deep, resolution-independent neural operator capable of learning PDE solution maps and classifying images, with enhanced stability and long-range dependency modeling.

Contribution

It proposes a novel nonlocal neural operator that is deep, resolution independent, and capable of handling diverse tasks, with stability analysis and shallow-to-deep generalization techniques.

Findings

NKN outperforms baseline methods in learning PDEs and image classification.

NKN generalizes well across different resolutions and depths.

The stability of NKN is analyzed via nonlocal vector calculus.

Abstract

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.