How Analysis Can Teach Us the Optimal Way to Design Neural Operators
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Abstract (arXiv)
This paper presents a mathematics-informed approach to neural operator design, building upon the theoretical framework established in our prior work. By integrating rigorous mathematical analysis with practical design strategies, we aim to enhance the stability, convergence, generalization, and computational efficiency of neural operators. We revisit key theoretical insights, including stability in high dimensions, exponential convergence, and universality of neural operators. Based on these insights, we provide detailed design recommendations, each supported by mathematical proofs and citations. Our contributions offer a systematic methodology for developing next-gen neural operators with improved performance and reliability.
Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 9 of its 32 references that are in the map. First posted on arXiv 4 Nov 2024.
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9 of 32 references.
References · 9
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
- 2018Spectral Normalization for Generative Adversarial Networkscited by 1,636
- 2017Spectrally-normalized margin bounds for neural networkscited by 686
- 2021Neural Operator: Learning Maps Between Function Spacescited by 414
- 2014On the Number of Linear Regions of Deep Neural Networkscited by 381
- 2016On the Expressive Power of Deep Neural Networkscited by 306
- 2017Bounding and Counting Linear Regions of Deep Neural Networkscited by 161
- 2015On the Expressive Power of Deep Learning: A Tensor Analysiscited by 97
- 2024A Mathematical Analysis of Neural Operator Behaviorscited by 8
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