Neural operators: Difference between revisions

Revision as of 23:22, 21 October 2023

Revision as of 23:22, 21 October 2023

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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|backpropagation]] and [[Gradient descent|gradient descent]]-based methods.

Another training paradigm is associated with physics-informed machine learning. In particular, [[Physics-informed neural networks|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 |journal=arXiv preprint arXiv:2111.03794 |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>.

== References ==