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Neural operators are a class of deep learning architectures designed to learn maps between infinite-dimensional function spaces. Neural operators represent an extension of traditional artificial neural networks, marking a departure from the typical focus on learning mappings between finite-dimensional Euclidean spaces or finite sets. Neural operators directly learn operators between function spaces; they can receive input functions, and the output function can be evaluated at any discretization.[1]
The primary application of neural operators is in learning surrogate maps for the solution operators of partial differential equations (PDEs),[1] which are critical tools in modeling the natural environment.[2][3] Standard PDE solvers can be time-consuming and computationally intensive, especially for complex systems. Neural operators have demonstrated improved performance in solving PDEs[4] compared to existing machine learning methodologies while being significantly faster than numerical solvers.[5][6][7] Neural operators have also been applied to various scientific and engineering disciplines such as turbulent flow modeling, computational mechanics, graph-structured data,[8] and the geosciences.[9] In particular, they have been applied to learning stress-strain fields in materials, classifying complex data like spatial transcriptomics, predicting multiphase flow in porous media,[10] and carbon dioxide migration simulations. Finally, the operator learning paradigm allows learning maps between function spaces, and is different from parallel ideas of learning maps from finite-dimensional spaces to function spaces,[11][12] and subsumes these settings when limited to fixed input resolution.
^Evans, L. C. (1998). Partial Differential Equations. Providence: American Mathematical Society. ISBN 0-8218-0772-2.
^"How AI models are transforming weather forecasting: A showcase of data-driven systems". phys.org (Press release). European Centre for Medium-Range Weather Forecasts. 6 September 2023.
^Russ, Dan; Abinader, Sacha (23 August 2023). "Microsoft and Accenture partner to tackle methane emissions with AI technology". Microsoft Azure Blog.
^Hao, Karen (30 October 2020). "AI has cracked a key mathematical puzzle for understanding our world". MIT Technology Review.
^Ananthaswamy, Anil (19 April 2021). "Latest Neural Nets Solve World's Hardest Equations Faster Than Ever Before". Quanta Magazine.
^Sharma, Anuj; Singh, Sukhdeep; Ratna, S. (15 August 2023). "Graph Neural Network Operators: a Review". Multimedia Tools and Applications. 83 (8): 23413–23436. doi:10.1007/s11042-023-16440-4.
^Wen, Gege; Li, Zongyi; Azizzadenesheli, Kamyar; Anandkumar, Anima; Benson, Sally M. (May 2022). "U-FNO—An enhanced Fourier neural operator-based deep-learning model for multiphase flow". Advances in Water Resources. 163: 104180. arXiv:2109.03697. Bibcode:2022AdWR..16304180W. doi:10.1016/j.advwatres.2022.104180.
^Choubineh, Abouzar; Chen, Jie; Wood, David A.; Coenen, Frans; Ma, Fei (2023). "Fourier Neural Operator for Fluid Flow in Small-Shape 2D Simulated Porous Media Dataset". Algorithms. 16 (1): 24. doi:10.3390/a16010024.
^Jiang, Chiyu Lmaxr; Esmaeilzadeh, Soheil; Azizzadenesheli, Kamyar; Kashinath, Karthik; Mustafa, Mustafa; Tchelepi, Hamdi A.; Marcus, Philip; Prabhat, Mr; Anandkumar, Anima (2020). "MESHFREEFLOWNET: A Physics-Constrained Deep Continuous Space-Time Super-Resolution Framework". SC20: International Conference for High Performance Computing, Networking, Storage and Analysis. pp. 1–15. doi:10.1109/SC41405.2020.00013. ISBN 978-1-7281-9998-6.
^Lu, Lu; Jin, Pengzhan; Pang, Guofei; Zhang, Zhongqiang; Karniadakis, George Em (18 March 2021). "Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators". Nature Machine Intelligence. 3 (3): 218–229. arXiv:1910.03193. doi:10.1038/s42256-021-00302-5.
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