Neural operators: Difference between revisions

Revision as of 18:01, 20 October 2023

Revision as of 18:28, 21 October 2023

Line 6:

'''Neural operators''' are a class of [[Deep learning|deep learning]] architectures designed to learn maps between infinite-dimensional [[Function space|function spaces]]. 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 journalbook |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> Standard PDE solvers can be time-consuming and computationally intensive, especially for complex systems. Neural operators have demonstrated improved performance in solving PDEs compared to existing machine learning methodologies while being significantly faster than numerical solvers.<ref name="FNO">{{cite journalarXiv |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 |journal=arXiv preprint arXiv:2010.08895 |date=2020 |urlclass=https://arxivcs.org/pdf/LG |eprint=2010.08895.pdf }}</ref>. The operator learning paradigm allows learning maps between function spaces, and is different from parallel ideas of learning maps from finite-dimensional spaces to function spaces <ref name="meshfreeflownet">{{cite journal | vauthors=((Esmaeilzadeh, S., Azizzadenesheli, K., Kashinath, K., Mustafa, M., Tchelepi, H. A., Marcus, P., Prabhat, M., Anandkumar, A., others)) | title=Meshfreeflownet: A physics-constrained deep continuous space-time super-resolution framework | pages=1--151–15 | publisher=IEEE | date=19 October 2020| arxiv=2005.01463 }}</ref><ref name="deeponet">{{cite journal | vauthors=((Lu, L., Jin, P., Pang, G., Zhang, Z., Karniadakis, G. E.)) | title=Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators | volume=3 | issue=3 | pages=218--229218–229 | publisher=Nature Publishing Group UK London | date=19 October 2021}}</ref>, and subsumes these settings when limited to fixed input resolution.

== Operator learning ==

Line 18:

== Definition and formulation ==

Architecturally, neural operators are similar to feed-forward neural networks in the sense that they are comprised 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|integral operators]] on function spaces and point-wise non-linearities.<ref name="NO journal" /> Using an analogous architecture to finite-dimensional neural networks, similar [[Universal approximation theorem|universal approximation theorems]] have been proven for neural operators. In particular, it has been shown that neural operators can approximate any continuous operator on a [[Compact space|compact]] set.<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>

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

Line 38:

<math>v_{t+1}(x) \approx \sigma\left(\sum_j^n \kappa_\phi(x, y_j, v_t(x), v_t(y_j))v_t(y_j)\Delta_{y_j} + W_t(v_t(y_j)) + b_t(x)\right).</math>

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 journalarXiv |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 |journal=arXiv preprint arXiv:2003.03485 |date=2020 |urlclass=https://arxivcs.org/pdf/LG |eprint=2003.03485.pdf }}</ref>.

There have been various parameterizations of neural operators for different applications<ref name="FNO" /><ref name="Graph NO">{{cite journalarXiv |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 |journal=arXiv preprint arXiv:2003.03485 |date=2020 |urlclass=https://arxivcs.org/pdf/LG |eprint=2003.03485.pdf }}</ref>. These typically differ in their parameterization of <math>\kappa</math>. The most popular instantiation is the Fourier neural operator (FNO). FNO takes <math>\kappa_\phi(x, y, v_t(x), v_t(y)) := \kappa_\phi(x-y)</math> and by applying the [[Convolution theorem|convolution theorem]], arrives at the following parameterization of the kernel integral operator:

<math>(\mathcal{K}_\phi v_t)(x) = \mathcal{F}^{-1} (R_\phi \cdot (\mathcal{F}v_t))(x), </math>

Line 53:

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 journalarXiv |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 |urlclass=https://arxivcs.org/abs/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 ==