标题： 从偏微分方程和变分方法设计旋转不变的神经网络 显示英文标题

标题： Designing Rotationally Invariant Neural Networks from PDEs and Variational Methods

摘要： 偏微分方程（PDE）模型及其相关的变分能量公式通常在设计时具有旋转不变性。 这确保了输入的旋转会导致输出的相应旋转，这在图像分析等应用中是理想的。 卷积神经网络（CNN）不具有这一特性，现有的解决方法通常较为复杂。 我们论文的目标是研究扩散和变分模型如何实现旋转不变性，并将这些思想转移到神经网络中。 作为核心创新，我们提出了激活函数，这些函数通过结合多个方向滤波器的信息来耦合网络通道。 这保证了在网络的基本构建块内具有旋转不变性，同时仍允许方向滤波。 由此产生的神经架构本质上是旋转不变的。 仅使用少量小滤波器，它们就能实现与现有技术相同的效果，而现有技术需要对方向进行精细采样。 我们的发现有助于将扩散和变分模型转化为数学上合理的网络架构，并为基于模型的CNN设计提供新的概念。 显示英文摘要

摘要： Partial differential equation (PDE) models and their associated variational energy formulations are often rotationally invariant by design. This ensures that a rotation of the input results in a corresponding rotation of the output, which is desirable in applications such as image analysis. Convolutional neural networks (CNNs) do not share this property, and existing remedies are often complex. The goal of our paper is to investigate how diffusion and variational models achieve rotation invariance and transfer these ideas to neural networks. As a core novelty we propose activation functions which couple network channels by combining information from several oriented filters. This guarantees rotation invariance within the basic building blocks of the networks while still allowing for directional filtering. The resulting neural architectures are inherently rotationally invariant. With only a few small filters, they can achieve the same invariance as existing techniques which require a fine-grained sampling of orientations. Our findings help to translate diffusion and variational models into mathematically well-founded network architectures, and provide novel concepts for model-based CNN design.