A Mathematical Analysis of Neural Operator Behaviors

• URL: http://arxiv.org/abs/2410.21481v1
• Date: Mon, 28 Oct 2024 19:38:53 GMT
• Title: A Mathematical Analysis of Neural Operator Behaviors
• Authors: Vu-Anh Le, Mehmet Dik,
• Abstract summary: This paper presents a rigorous framework for analyzing the behaviors of neural operators. We focus on their stability, convergence, clustering dynamics, universality, and generalization error. We aim to offer clear and unified guidance in a single setting for the future design of neural operator-based methods.
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• Abstract: Neural operators have emerged as transformative tools for learning mappings between infinite-dimensional function spaces, offering useful applications in solving complex partial differential equations (PDEs). This paper presents a rigorous mathematical framework for analyzing the behaviors of neural operators, with a focus on their stability, convergence, clustering dynamics, universality, and generalization error. By proposing a list of novel theorems, we provide stability bounds in Sobolev spaces and demonstrate clustering in function space via gradient flow interpretation, guiding neural operator design and optimization. Based on these theoretical gurantees, we aim to offer clear and unified guidance in a single setting for the future design of neural operator-based methods.

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