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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]]s. Neural operators represent an extension of traditional [[artificial neural network]]s, 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 |pages=1–97 |arxiv=2108.08481 |url=https://www.jmlr.org/papers/volume24/21-1524/21-1524.pdf}}</ref><ref name="NO Nature">{{cite journal |last1=Azizzadenesheli |first1=Kamyar |last2=Kovachki |first2=Nikola |last3=Li |first3=Zongyi |last4=Liu-Schiaffini |first4=Miguel |last5=Kossaifi |first5=Jean |last6=Anandkumar |first6=Anima |title=Neural operators for accelerating scientific simulations and design |journal=Nature Reviews Physics |date=2024 |volume=6 |pages=320–328 |arxiv=2309.15325 |url=https://www.nature.com/articles/s42254-024-00712-5}}</ref>
The primary application of neural operators is in learning surrogate maps for the solution operators of [[partial differential equation]]s (PDEs),<ref name="NO journal" /><ref
== Operator learning ==
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<math>\mathcal{G}_\phi := \mathcal{Q} \circ \sigma(W_T + \mathcal{K}_T + b_T) \circ \cdots \circ \sigma(W_1 + \mathcal{K}_1 + b_1) \circ \mathcal{P},</math>
where <math>\mathcal{P}, \mathcal{Q}</math> are the lifting (lifting the codomain of the input function to a higher dimensional space) and projection (projecting the codomain of the intermediate function to the output
<math>(\mathcal{K}_\phi v_t)(x) := \int_D \kappa_\phi(x, y, v_t(x), v_t(y))v_t(y)dy, </math>
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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>.
== See also ==
* [[Neural network (machine learning)|Neural network]]
* [[Physics-informed neural networks]]
* [[Neural field]]
== References ==
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