Operator Learning Meets Numerical Analysis: Improving Neural Networks through Iterative Methods
Keywords: Fixed Point, Picard Iterations, Operator Learning, Nonlinear Operators, Banach Space
TL;DR: We introduce a general framework for iterative methods applicable to several machine learning models, and show that these approaches based on numerical analysis improve operator learning tasks
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.
Primary Area: general machine learning (i.e., none of the above)
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Submission Number: 6564
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