Neural operators: Difference between revisions

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'''Neural operators''' are a class of [[Deep learning|deep learning]] architectures designed to learn maps between infinite-dimensional [[Function space|function spaces]]. Neural operators represent an extension of traditional [[Artificial neural network|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 [[Operator (mathematics)|operators]] between function spaces; they can receive input functions, and the output function can be evaluated at any discretization.<ref name="NO journal">{{cite journal |last1=Kovachki |first1=Nikola |last2=Li |first2=Zongyi |last3=Liu |first3=Burigede |last4=Azizzadenesheli |first4=Kamyar |last5=Bhattacharya |first5=Kaushik |last6=Stuart |first6=Andrew |last7=Anandkumar |first7=Anima |title=Neural operator: Learning maps between function spaces |journal=Journal of Machine Learning Research |date=2021 |volume=24 |page=1-97 |arxiv=2108.08481 |url=https://www.jmlr.org/papers/volume24/21-1524/21-1524.pdf}}</ref>

The primary application of neural operators is in learning surrogate maps for the solution operators of [[Partial differential equation|partial differential equations]] (PDEs)<ref name="NO journal" />, which are critical tools in modeling the natural environment.<ref name="Evans"> {{cite book |author-link=Lawrence C. Evans |first=L. C. |last=Evans |title=Partial Differential Equations |publisher=American Mathematical Society |___location=Providence |year=1998 |isbn=0-8218-0772-2 }}</ref> Standard PDE solvers can be time-consuming and computationally intensive, especially for complex systems. Neural operators have demonstrated improved performance in solving PDEs compared to existing machine learning methodologies while being significantly faster than numerical solvers.<ref name="FNO">{{cite arXiv |last1=Li |first1=Zongyi |last2=Kovachki |first2=Nikola |last3=Azizzadenesheli |first3=Kamyar |last4=Liu |first4=Burigede |last5=Bhattacharya |first5=Kaushik |last6=Stuart |first6=Andrew |last7=Anima |first7=Anandkumar |title=Fourier neural operator for parametric partial differential equations |journal=arXiv preprint arXiv:2010.08895 |date=2020 |class=cs.LG |eprint=2010.08895 }}</ref>. 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 <ref name="meshfreeflownet">{{cite journal | vauthors=((Esmaeilzadeh, S., Azizzadenesheli, K., Kashinath, K., Mustafa, M., Tchelepi, H. A., Marcus, P., Prabhat, M., Anandkumar, A., others)) | title=Meshfreeflownet: A physics-constrained deep continuous space-time super-resolution framework | pages=1–15 | publisher=IEEE | date=19 October 2020| arxiv=2005.01463 }}</ref><ref name="deeponet">{{cite journal | vauthors=((Lu, L., Jin, P., Pang, G., Zhang, Z., Karniadakis, G. E.)) | title=Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators | volume=3 | issue=3 | pages=218–229 | publisher=Nature Publishing Group UK London | date=19 October 2021}}</ref>, and subsumes these settings when limited to fixed input resolution.

== Operator learning ==