A Unified Framework for U-Net Design and Analysis

TL;DR

This paper introduces a comprehensive theoretical framework for designing and analyzing U-Net architectures, revealing their properties, proposing simplified variants, and demonstrating their effectiveness across various tasks.

Contribution

It provides the first unified theoretical analysis of U-Nets, introduces Multi-ResNets with wavelet-based encoders, and guides the design of architectures encoding data constraints and geometry.

Findings

Multi-ResNets achieve competitive performance in image segmentation.

U-Nets with average pooling exploit noise characteristics in diffusion models.

The framework enables designing U-Nets that encode function constraints and data geometry.

Abstract

U-Nets are a go-to, state-of-the-art neural architecture across numerous tasks for continuous signals on a square such as images and Partial Differential Equations (PDE), however their design and architecture is understudied. In this paper, we provide a framework for designing and analysing general U-Net architectures. We present theoretical results which characterise the role of the encoder and decoder in a U-Net, their high-resolution scaling limits and their conjugacy to ResNets via preconditioning. We propose Multi-ResNets, U-Nets with a simplified, wavelet-based encoder without learnable parameters. Further, we show how to design novel U-Net architectures which encode function constraints, natural bases, or the geometry of the data. In diffusion models, our framework enables us to identify that high-frequency information is dominated by noise exponentially faster, and show how U-Nets with average pooling exploit this. In our experiments, we demonstrate how Multi-ResNets achieve competitive and often superior performance compared to classical U-Nets in image segmentation, PDE surrogate modelling, and generative modelling with diffusion models. Our U-Net framework paves the way to study the theoretical properties of U-Nets and design natural, scalable neural architectures for a multitude of problems beyond the square.