Operator Learning Meets Numerical Analysis: Improving Neural Networks through Iterative Methods

• URL: http://arxiv.org/abs/2310.01618v1
• Date: Mon, 2 Oct 2023 20:25:36 GMT
• Title: Operator Learning Meets Numerical Analysis: Improving Neural Networks through Iterative Methods
• Authors: Emanuele Zappala, Daniel Levine, Sizhuang He, Syed Rizvi, Sacha Levy and David van Dijk
• Abstract summary: We develop a theoretical framework grounded in iterative methods for operator equations. We demonstrate that popular architectures, such as diffusion models and AlphaFold, inherently employ iterative operator learning. Our work aims to enhance the understanding of deep learning by merging insights from numerical analysis.
• Score: 2.226971382808806
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Deep neural networks, despite their success in numerous applications, often function without established theoretical foundations. In this paper, we bridge this gap by drawing parallels between deep learning and classical numerical analysis. By framing neural networks as operators with fixed points representing desired solutions, we develop a theoretical framework grounded in iterative methods for operator equations. Under defined conditions, we present convergence proofs based on fixed point theory. We demonstrate that popular architectures, such as diffusion models and AlphaFold, inherently employ iterative operator learning. Empirical assessments highlight that performing iterations through network operators improves performance. We also introduce an iterative graph neural network, PIGN, that further demonstrates benefits of iterations. Our work aims to enhance the understanding of deep learning by merging insights from numerical analysis, potentially guiding the design of future networks with clearer theoretical underpinnings and improved performance.

Related papers

• Convergence Analysis for Deep Sparse Coding via Convolutional Neural Networks [7.956678963695681]
  We introduce a novel class of Deep Sparse Coding (DSC) models. We derive convergence rates for CNNs in their ability to extract sparse features. Inspired by the strong connection between sparse coding and CNNs, we explore training strategies to encourage neural networks to learn more sparse features.
  arXiv  Detail & Related papers  (2024-08-10T12:43:55Z)
• Improving Network Interpretability via Explanation Consistency Evaluation [56.14036428778861]
  We propose a framework that acquires more explainable activation heatmaps and simultaneously increase the model performance. Specifically, our framework introduces a new metric, i.e., explanation consistency, to reweight the training samples adaptively in model learning. Our framework then promotes the model learning by paying closer attention to those training samples with a high difference in explanations.
  arXiv  Detail & Related papers  (2024-08-08T17:20:08Z)
• Simple initialization and parametrization of sinusoidal networks via their kernel bandwidth [92.25666446274188]
  sinusoidal neural networks with activations have been proposed as an alternative to networks with traditional activation functions. We first propose a simplified version of such sinusoidal neural networks, which allows both for easier practical implementation and simpler theoretical analysis. We then analyze the behavior of these networks from the neural tangent kernel perspective and demonstrate that their kernel approximates a low-pass filter with an adjustable bandwidth.
  arXiv  Detail & Related papers  (2022-11-26T07:41:48Z)
• The Principles of Deep Learning Theory [19.33681537640272]
  This book develops an effective theory approach to understanding deep neural networks of practical relevance. We explain how these effectively-deep networks learn nontrivial representations from training. We show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks.
  arXiv  Detail & Related papers  (2021-06-18T15:00:00Z)
• What can linearized neural networks actually say about generalization? [67.83999394554621]
  In certain infinitely-wide neural networks, the neural tangent kernel (NTK) theory fully characterizes generalization. We show that the linear approximations can indeed rank the learning complexity of certain tasks for neural networks. Our work provides concrete examples of novel deep learning phenomena which can inspire future theoretical research.
  arXiv  Detail & Related papers  (2021-06-12T13:05:11Z)
• Explainability-aided Domain Generalization for Image Classification [0.0]
  We show that applying methods and architectures from the explainability literature can achieve state-of-the-art performance for the challenging task of domain generalization. We develop a set of novel algorithms including DivCAM, an approach where the network receives guidance during training via gradient based class activation maps to focus on a diverse set of discriminative features. Since these methods offer competitive performance on top of explainability, we argue that the proposed methods can be used as a tool to improve the robustness of deep neural network architectures.
  arXiv  Detail & Related papers  (2021-04-05T02:27:01Z)
• A Convergence Theory Towards Practical Over-parameterized Deep Neural Networks [56.084798078072396]
  We take a step towards closing the gap between theory and practice by significantly improving the known theoretical bounds on both the network width and the convergence time. We show that convergence to a global minimum is guaranteed for networks with quadratic widths in the sample size and linear in their depth at a time logarithmic in both. Our analysis and convergence bounds are derived via the construction of a surrogate network with fixed activation patterns that can be transformed at any time to an equivalent ReLU network of a reasonable size.
  arXiv  Detail & Related papers  (2021-01-12T00:40:45Z)
• How Neural Networks Extrapolate: From Feedforward to Graph Neural Networks [80.55378250013496]
  We study how neural networks trained by gradient descent extrapolate what they learn outside the support of the training distribution. Graph Neural Networks (GNNs) have shown some success in more complex tasks.
  arXiv  Detail & Related papers  (2020-09-24T17:48:59Z)
• Learning Connectivity of Neural Networks from a Topological Perspective [80.35103711638548]
  We propose a topological perspective to represent a network into a complete graph for analysis. By assigning learnable parameters to the edges which reflect the magnitude of connections, the learning process can be performed in a differentiable manner. This learning process is compatible with existing networks and owns adaptability to larger search spaces and different tasks.
  arXiv  Detail & Related papers  (2020-08-19T04:53:31Z)
• Optimizing Neural Networks via Koopman Operator Theory [6.09170287691728]
  Koopman operator theory was recently shown to be intimately connected with neural network theory. In this work we take the first steps in making use of this connection. We show that Koopman operator theory methods allow predictions of weights and biases of feed weights over a non-trivial range of training time.
  arXiv  Detail & Related papers  (2020-06-03T16:23:07Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.