PDE-based Group Equivariant Convolutional Neural Networks

Abstract

We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where the equation's geometrically meaningful coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these networks to be designed with built-in symmetries such as rotation equivariance instead of being restricted to just translation equivariance as in traditional CNNs. Having all the desired symmetries included in the design obviates the need to include them by means of costly techniques such as data augmentation. Roto-translation equivariance for image analysis applications is the example we will be using throughout the paper.Our default PDE is solved by a combination of linear group convolutions and non-linear morphological group convolutions. Just like for linear convolution a morphological convolution is specified by a kernel and this kernel is what is being optimized during the training process. We demonstrate how the common CNN operations of max/min-pooling and ReLUs arise naturally from solving a PDE and how they are subsumed by morphological convolutions.We present a proof-of-concept experiment to demonstrate the potential of this framework in increasing the performance of deep learning based imaging applications.

Steerable Partial Differential Operators for Equivariant Neural Networks

“…A different approach to equivariant PDO-based networks was taken by Smets et al [23]. Instead of applying a differential operator to input features, they use layers that map an initial condition for a PDE to its solution at a fixed later time.…”

“…However, one remaining difference is that physics uses equivariant partial differential operators (PDOs) to define maps between fields, such as the gradient or Laplacian. Therefore, using PDOs instead of convolutions in deep learning would complete the analogy to physics and could lead to even more transfer of ideas between subjects.Equivariant PDO-based networks have already been designed in prior work [21][22][23]. Most relevant for our work are PDO-eConvs [22], which can be seen as the PDO-analogon of group convolutions.…”

“…Equivariant PDO-based networks have already been designed in prior work [21][22][23]. Most relevant for our work are PDO-eConvs [22], which can be seen as the PDO-analogon of group convolutions.…”

ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups

“…Expressing tangent vectors in such left-invariant vector fields allows us to reason in terms of the generators of the group and even design learnable equivariant differential operators for the construction of deep NNs [Smets et al, 2020]. Consider the G = SE(2) case.…”

Translating Numerical Concepts for PDEs into Neural Architectures

“…One strategy consists of interpreting networks as approximations of evolution equations; see e.g. [3,22,23]. Then training a network comes down to parameter identification of ordinary or partial differential equations (PDEs).…”