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{{Short description|Machine learning framework}}
'''Neural operators''' are a class of [[deep learning]] architectures designed to learn maps between infinite-dimensional [[Function space|function spaces]]
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> <ref> X, S. (2023, September 6). How ai models are transforming weather forecasting: A showcase of data-driven systems. Phys.org. https://phys.org/news/2023-09-ai-weather-showcase-data-driven.html </ref> Standard PDE solvers can be time-consuming and computationally intensive, especially for complex systems. Neural operators have demonstrated improved performance in solving PDEs <ref>Kadri Umay, Y. O. (2023, September 20). Microsoft and accenture partner to tackle methane emissions with AI technology. Microsoft Azure Blog. https://azure.microsoft.com/en-us/blog/microsoft-and-accenture-partner-to-tackle-methane-emissions-with-ai-technology/ </ref> compared to existing machine learning methodologies while being significantly faster than numerical solvers.
</ref> and the geosciences.<ref> Gege Wen, Zongyi Li, Kamyar Azizzadenesheli, Anima Anandkumar, Sally M. Benson,
U-FNO—An enhanced Fourier neural operator-based deep-learning model for multiphase flow,
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== Definition and formulation ==
Architecturally, neural operators are similar to feed-forward neural networks in the sense that they are composed of alternating [[Linear map|linear maps]] and non-linearities. Since neural operators act on and output functions, neural operators have been instead formulated as a sequence of alternating linear [[integral operators]] on function spaces and point-wise non-linearities.
Neural operators seek to approximate some operator <math>\mathcal{G} : \mathcal{A} \to \mathcal{U}</math> between function spaces <math>\mathcal{A}</math> and <math>\mathcal{U}</math> by building a parametric map <math>\mathcal{G}_\phi : \mathcal{A} \to \mathcal{U}</math>. Such parametric maps <math>\mathcal{G}_\phi</math> can generally be defined in the form
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The above approximation, along with parametrizing <math>\kappa_\phi</math> as an implicit neural network, results in the graph neural operator (GNO).<ref name="Graph NO">{{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=Neural operator: Graph kernel network for partial differential equations |date=2020 |class=cs.LG |eprint=2003.03485 }}</ref>
There have been various parameterizations of neural operators for different applications.
<math>(\mathcal{K}_\phi v_t)(x) = \mathcal{F}^{-1} (R_\phi \cdot (\mathcal{F}v_t))(x), </math>
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<math>\mathcal{L}_\mathcal{U}(\{(a_i, u_i)\}_{i=1}^N) := \sum_{i=1}^N \|u_i - \mathcal{G}_\theta (a_i) \|_\mathcal{U}^2</math>,
where <math>\|\cdot \|_\mathcal{U}</math> is a norm on the output function space <math>\mathcal{U}</math>. Neural operators can be trained directly using [[backpropagation]] and [[gradient descent]]-based methods
Another training paradigm is associated with physics-informed machine learning. In particular, [[physics-informed neural networks]] (PINNs) use complete physics laws to fit neural networks to solutions of PDEs. Extensions of this paradigm to operator learning are broadly called physics-informed neural operators (PINO),<ref name="PINO">{{cite arXiv |last1=Li |first1=Zongyi | last2=Hongkai| first2=Zheng |last3=Kovachki |first3=Nikola | last4=Jin | first4=David | last5=Chen | first5= Haoxuan |last6=Liu |first6=Burigede | last7=Azizzadenesheli |first7=Kamyar |last8=Anima |first8=Anandkumar |title=Physics-Informed Neural Operator for Learning Partial Differential Equations |date=2021 |class=cs.LG |eprint=2111.03794 }}</ref>, where loss functions can include full physics equations or partial physical laws. As opposed to standard PINNs, the PINO paradigm incorporates a data loss (as defined above) in addition to the physics loss <math>\mathcal{L}_{PDE}(a, \mathcal{G}_\theta (a))</math>. The physics loss <math>\mathcal{L}_{PDE}(a, \mathcal{G}_\theta (a))</math> quantifies how much the predicted solution of <math>\mathcal{G}_\theta (a)</math> violates the PDEs equation for the input <math>a</math>.
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