Deep Neural Networks Motivated by Partial Differential Equations (1804.04272v2)

Abstract: Partial differential equations (PDEs) are indispensable for modeling many physical phenomena and also commonly used for solving image processing tasks. In the latter area, PDE-based approaches interpret image data as discretizations of multivariate functions and the output of image processing algorithms as solutions to certain PDEs. Posing image processing problems in the infinite dimensional setting provides powerful tools for their analysis and solution. Over the last few decades, the reinterpretation of classical image processing problems through the PDE lens has been creating multiple celebrated approaches that benefit a vast area of tasks including image segmentation, denoising, registration, and reconstruction. In this paper, we establish a new PDE-interpretation of a class of deep convolutional neural networks (CNN) that are commonly used to learn from speech, image, and video data. Our interpretation includes convolution residual neural networks (ResNet), which are among the most promising approaches for tasks such as image classification having improved the state-of-the-art performance in prestigious benchmark challenges. Despite their recent successes, deep ResNets still face some critical challenges associated with their design, immense computational costs and memory requirements, and lack of understanding of their reasoning. Guided by well-established PDE theory, we derive three new ResNet architectures that fall into two new classes: parabolic and hyperbolic CNNs. We demonstrate how PDE theory can provide new insights and algorithms for deep learning and demonstrate the competitiveness of three new CNN architectures using numerical experiments.

• The paper introduces a PDE-inspired framework that refines CNN design by developing parabolic, Hamiltonian, and second-order hyperbolic architectures.
• The paper demonstrates competitive classification performance on datasets like STL-10 and CIFAR while ensuring forward-backward stability and efficiency.
• The paper advocates for integrating PDE theory into deep learning to create more robust, interpretable, and mathematically grounded neural networks.

Deep Neural Networks and Partial Differential Equations: A Technical Summary

The paper “Deep Neural Networks Motivated by Partial Differential Equations” by Lars Ruthotto and Eldad Haber explores the synthesis of deep learning and partial differential equations (PDEs), presenting a mathematical framework to enhance the understanding and design of deep convolutional neural networks (CNNs). The authors propose new network architectures inspired by the well-established mathematical theories of PDEs to address existing challenges in deep learning.

PDEs and Their Role in Image Processing

PDEs have long been integral to modeling physical phenomena and solving image processing problems. By interpreting image data as discretizations of multivariate functions, PDE-based approaches have led to significant advancements in tasks such as image segmentation, denoising, registration, and reconstruction. The paper leverages this established foundation to offer a novel PDE interpretation of deep learning tasks, particularly those involving image, speech, and video data.

Neural Networks and PDE Interpretations

The authors focus on convolutional ResNets, a type of neural network renowned for its success in image classification challenges. ResNets, however, are computationally intensive and often enigmatic in their workings. By interpreting these networks through a PDE lens, the paper aims to derive new insights, improve design, and enhance computational efficiency.

New Architectures Inspired by PDEs

Using PDE theory, the authors develop three new ResNet architectures that fall into two main classes: parabolic and hyperbolic CNNs:

1. Parabolic CNNs: These networks are motivated by the heat equation and are designed to smooth image features. The stability of parabolic equations ensures robustness to input perturbations.
2. Hamiltonian CNNs: Inspired by Hamiltonian systems, these networks preserve energy within the system, making them well-suited for tasks requiring reversibility. They offer memory efficiency by avoiding intermediate state storage.
3. Second-order Hyperbolic CNNs: These networks draw on the properties of the telegraph equation, mirroring biological networks' signal propagation. They provide forward-backward stability, which is advantageous for dynamic learning scenarios.

Theoretical and Practical Implications

The approaches outlined in the paper suggest that by constraining CNNs to specific PDE types, practitioners can achieve forward and backward propagation stability and improve the interpretability of deep networks. These insights pave the way for new, theoretically grounded architectures that potentially offer better generalization on unseen data.

Numerical Results and Competitiveness

The paper showcases the competitiveness of the proposed CNN architectures through experiments using standard datasets like STL-10, CIFAR-10, and CIFAR-100. Despite constraints on architecture design, the networks achieved competitive classification accuracies compared to larger, more complex models, highlighting the potential efficiency and efficacy advantages of PDE-based designs.

Future Directions

The integration of PDEs into the design of deep learning models opens pathways for further exploration. Theories and practices from PDE-domain applications might inform architectural decisions in neural networks, possibly spawning advances in training efficiency and model robustness. Further empirical and theoretical research is warranted to fully realize these benefits and to discover optimal architectures for diverse applications.

In conclusion, the paper presents a methodical approach to harnessing mathematical frameworks from PDEs to inform and innovate in the field of deep learning. By narrowing the gap between neural network models and established PDE theory, the paper provides a promising foundation for the development of more robust, stable, and interpretable deep learning systems.