Neural operators: Difference between revisions

Revision as of 21:40, 4 October 2023

Revision as of 05:16, 6 October 2023

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in some norm <math>\|\cdot \|_\mathcal{U}.</math> Neural operators can be trained directly using [[Backpropagation|backpropagation]] and [[Gradient descent|gradient descent]]-based methods.

When dealing with modeling natural phenomena, often physics equations, mostly in the form of PDEs, drive the physical world around us.<ref name="Evans"> {{cite journal |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>. Based on this idea, physics-informed neural networks[[Physics-informed neural networks|physics-informed neural networks]] utilize complete physics laws to fit neural networks to solutions of PDEs. The general extension to operator learning is physics informed neural operator paradigm (PINO),<ref name="PINO">{{cite journal |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 |journal=https://arxiv.org/pdf/2111.03794.pdf |date=2021 |url=https://arxiv.org/abs/2111.03794}}</ref>, where the supervision can also be channeled through physics equations and can process learning through partially available physics. PINO is mainly a supervised learning setting that is suitable for cases where partial data or partial physics in available. In short, in PINO, in addition to the data loss mentioned above, physics loss <math>\mathcal{L}_PDE((a, \mathcal{G}_\theta (a))</math>, is used for further training. 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>.

== References ==