Neural operators: Difference between revisions

Revision as of 19:43, 10 December 2023

Revision as of 00:29, 15 December 2023

Line 1:

{{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]] .<ref name="patel1">{{cite arXiv |last1=Patel |first1=Ravi G. |last2=Desjardins |first2=Olivier |title=Nonlinear integro-differential operator regression with neural networks |date=2018 |class=cs.LG |eprint=1810.08552}}</ref>. 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> <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&nbsp;accenture&nbsp;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 name="patel2">{{cite journal |last1=Patel |first1=Ravi G. |last2=Trask |first2=Nathaniel A. |last3=Wood |first3=Mitchell A. |last4=Cyr |first4=Eric C. |title=A physics-informed operator regression framework for extracting data-driven continuum models |journal=Computer Methods in Applied Mechanics and Engineering |date=January 2021 |volume=373 |pages=113500 |doi=10.1016/j.cma.2020.113500|arxiv=2009.11992 }}</ref><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 |date=2020 |class=cs.LG |eprint=2010.08895 }}</ref><ref>Hao, K. (2021, October 20). Ai has cracked a key mathematical puzzle for understanding our world. MIT Technology Review. https://www.technologyreview.com/2020/10/30/1011435/ai-fourier-neural-network-cracks-navier-stokes-and-partial-differential-equations/ </ref><ref> Ananthaswamy, A., &amp; Quanta Magazine moderates comments to&nbsp;facilitate an informed, substantive. (2021, September 10). Latest neural nets solve world’s hardest equations faster than ever before. Quanta Magazine. https://www.quantamagazine.org/latest-neural-nets-solve-worlds-hardest-equations-faster-than-ever-before-20210419/ </ref> Neural operators have also been applied to various scientific and engineering disciplines such as turbulent flow modeling, computational mechanics, graph-structured data,<ref>Sharma, A., Singh, S. & Ratna, S. Graph Neural Network Operators: a Review. Multimed Tools Appl (2023). https://doi.org/10.1007/s11042-023-16440-4

Line 57:

<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 .<ref name="patel1" />.

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>.