Learning neural operators on Riemannian manifolds

Open Access

Issue		Natl Sci Open Volume 3, Number 6, 2024


Article Number		20240001
Number of page(s)		20
Section		Information Sciences
DOI		https://doi.org/10.1360/nso/20240001
Published online		12 April 2024

Wang H, Fu T, Du Y, et al. Scientific discovery in the age of artificial intelligence. Nature 2023; 620: 47-60. [Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
Zhang R, Meng Q, Ma ZM. Deciphering and integrating invariants for neural operator learning with various physical mechanisms. Natl Sci Rev 2024; 11: nwad336. [Article] [CrossRef] [Google Scholar]
Li Z, Kovachki N, Azizzadenesheli K, et al. Neural operator: Graph kernel network for partial differential equations. arXiv: 2003.03485. [Google Scholar]
Li Z, Huang DZ, Liu B, et al. Fourier neural operator with learned deformations for pdes on general geometries. arXiv: 2207.05209. [Google Scholar]
Wang S, Wang H, Perdikaris P. Learning the solution operator of parametric partial differential equations with physics-informed deeponets. Sci Adv 2021; 7: eabi8605. [Article] [CrossRef] [Google Scholar]
Chen RT, Rubanova Y, Bettencourt J, et al. Neural ordinary differential equations. In: Proceedings of the 32nd International Conference on Neural Information Processing Systems. Red Hook: Curran Associates Inc., 2018. [Google Scholar]
Brunton SL, Kutz JN. Machine learning for partial differential equations. arXiv: 2303.17078. [Google Scholar]
Gopakumar V, Pamela S, Zanisi L, et al. Fourier neural operator for plasma modelling. arXiv: 2302.06542. [Google Scholar]
Corti M, Zingaro A, Dede’ L, et al. Impact of atrial fibrillation on left atrium haemodynamics: A computational fluid dynamics study. Comput Biol Med 2022; 150: 106143. [Article] [CrossRef] [PubMed] [Google Scholar]
Sabater C, Stürmer P, Bekemeyer P. Fast predictions of aircraft aerodynamics using deep-learning techniques. AIAA J 2022; 60: 5249-5261. [Article] [NASA ADS] [CrossRef] [Google Scholar]
Taverniers S, Korneev S, Pietrzyk KM, et al. Accelerating part-scale simulation in liquid metal jet additive manufacturing via operator learning. arXiv: 2202.03665. [Google Scholar]
Azzizadenesheli K, Kovachki N, Li Z, et al. Neural operators for accelerating scientific simulations and design. arXiv: 2309.15325. [Google Scholar]
Ramezankhani M, Crawford B, Narayan A, et al. Making costly manufacturing smart with transfer learning under limited data: A case study on composites autoclave processing. J Manuf Syst 2021; 59: 345-354. [Article] [CrossRef] [Google Scholar]
Yuan Y, Liu J, Jin D, et al. DeceFL: A principled fully decentralized federated learning framework. Natl Sci Open 2023; 2: 20220043. [Article] [CrossRef] [Google Scholar]
Rao C, Ren P, Wang Q, et al. Encoding physics to learn reaction-diffusion processes. Nat Mach Intell 2023; 5: 765-779. [Article] [CrossRef] [Google Scholar]
Chen J, Viquerat J, Hachem E. U-net architectures for fast prediction of incompressible laminar flows. arXiv: 1910.13532. [Google Scholar]
Wu H, Hu T, Luo H, et al. Solving high-dimensional PDEs with latent spectral models. arXiv: 2301.12664. [Google Scholar]
Velickovic P, Cucurull G, Casanova A, et al. Graph attention networks. arXiv: 1710.10903. [Google Scholar]
Chen J, Hachem E, Viquerat J. Graph neural networks for laminar flow prediction around random two-dimensional shapes. Phys Fluids 2021; 33: 123607. [Article] [CrossRef] [Google Scholar]
Lu L, Meng X, Cai S, et al. A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data. Comput Methods Appl Mech Eng 2022; 393: 114778. [Article] [CrossRef] [Google Scholar]
You H, Yu Y, D’Elia M, et al. Nonlocal kernel network (NKN): A stable and resolution-independent deep neural network. J Comput Phys 2022; 469: 111536. [Article] [CrossRef] [MathSciNet] [Google Scholar]
Lu L, Jin P, Pang G, et al. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat Mach Intell 2021; 3: 218-229. [Article] [CrossRef] [Google Scholar]
Li Z, Kovachki N, Azizzadenesheli K, et al. Fourier neural operator for parametric partial differential equations. arXiv: 2010.08895. [Google Scholar]
Kovachki N, Li Z, Liu B, et al. Neural operator: Learning maps between function spaces. arXiv: 2108.08481. [Google Scholar]
Lehmann F, Gatti F, Bertin M, et al. 3D elastic wave propagation with a Factorized Fourier Neural Operator (F-FNO). Comput Methods Appl Mech Eng 2024; 420: 116718. [Article] [Google Scholar]
Rahman MA, Ross ZE, Azizzadenesheli K. U-no: U-shaped neural operators. arXiv: 2204.11127. [Google Scholar]
Tripura T, Chakraborty S. Wavelet neural operator for solving parametric partial differential equations in computational mechanics problems. Comput Methods Appl Mech Eng 2023; 404: 115783. [Article] [CrossRef] [MathSciNet] [Google Scholar]
Li Z, Huang DZ, Liu B, et al. Fourier neural operator with learned deformations for pdes on general geometries. arXiv: 2207.05209. [Google Scholar]
Gao H, Sun L, Wang JX. PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain. J Comput Phys 2021; 428: 110079. [Article] [CrossRef] [Google Scholar]
Seiler J, Jonscher M, Schoberl M, et al. Resampling images to a regular grid from a non-regular subset of pixel positions using frequency selective reconstruction. IEEE Trans Image Process 2015; 24: 4540-4555. [Article] [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
Aflalo Y, Brezis H, Kimmel R. On the optimality of shape and data representation in the spectral domain. SIAM J Imag Sci 2015; 8: 1141-1160. [Article] [CrossRef] [MathSciNet] [Google Scholar]
Vaswani A, Shazeer N, Parmar N, et al. Attention is all you need. In: Proceedings of the 31st International Conference on Neural Information Processing Systems. Red Hook: Curran Associates Inc., 2017. [Google Scholar]
Tschannen M, Bachem O, Lucic M. Recent advances in autoencoder-based representation learning. arXiv: 1812.05069. [Google Scholar]
Bhattacharya K, Hosseini B, Kovachki NB, et al. Model reduction and neural networks for parametric PDEs. SMAI J Comput Math 2021; 7: 121-157. [Article] [CrossRef] [MathSciNet] [Google Scholar]
Seidman JH, Kissas G, Perdikaris P, et al. Nomad: Nonlinear manifold decoders for operator learning. arXiv: 2206.03551. [Google Scholar]
Rai N, Mondal S. Spectral methods to solve nonlinear problems: A review. Partial Differ Equ Appl Math 2021; 4: 100043. [Article] [CrossRef] [Google Scholar]
Patanè G. Laplacian spectral basis functions. Comput Aided Geometric Des 2018; 65: 31-47. [Article] [CrossRef] [MathSciNet] [Google Scholar]
Aflalo Y, Kimmel R. Spectral multidimensional scaling. Proc Natl Acad Sci USA 2013; 110: 18052-18057. [Article] [NASA ADS] [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
Terence Tao. Fourier transform. 2016. https://www.math.ucla.edu/tao/preprints/fourier.pdf [Google Scholar]
Reuter M, Wolter FE, Peinecke N. Laplace-Beltrami spectra as “Shape-DNA” of surfaces and solids. Comput-Aided Des 2006; 38: 342-366. [Article] [CrossRef] [Google Scholar]
Alexa M, Herholz P, Kohlbrenner M, et al. Properties of laplace operators for tetrahedral meshes. Comput Graphics Forum 2020; 39: 55-68. [Article] [CrossRef] [Google Scholar]
Lanthaler S, Mishra S, Karniadakis GE. Error estimates for DeepONets: A deep learning framework in infinite dimensions. Trans Math Its Appl 2022; 6: tnac001. [Article] [MathSciNet] [Google Scholar]
Hamilton W, Ying Z, Leskovec J. Inductive representation learning on large graphs. In: Proceedings of the 31st International Conference on Neural Information Processing Systems. Red Hook: Curran Associates Inc., 2017. [Google Scholar]
Yuan Y, Li X, Li L, et al. Machine discovery of partial differential equations from spatiotemporal data: A sparse Bayesian learning framework. Chaos-An Interdiscip J Nonlinear Sci 2023; 33: 113122. [Article] [CrossRef] [Google Scholar]
Rouse H. Modern conceptions of the mechanics of fluid turbulence. Trans Am Soc Civ Eng 1937; 102: 463-505. [Article] [CrossRef] [Google Scholar]
Li Y, Li W, Han T, et al. Transforming heat transfer with thermal metamaterials and devices. Nat Rev Mater 2021; 6: 488-507. [Article] [CrossRef] [Google Scholar]
Shen Y, Lu Y, Liu S, et al. Self-resistance electric heating of shaped CFRP laminates: Temperature distribution optimization and validation. Int J Adv Manuf Technol 2022; 121: 1755-1768. [Article] [CrossRef] [Google Scholar]
Struzziero G, Teuwen JJE, Skordos AA. Numerical optimisation of thermoset composites manufacturing processes: A review. Compos Part A-Appl Sci Manuf 2019; 124: 105499. [Article] [CrossRef] [Google Scholar]
Secomb TW. Hemodynamics. Compr Physiol 2016; 6: 975-1003. [CrossRef] [PubMed] [Google Scholar]
Maul N, Zinn K, Wagner F, et al. Transient hemodynamics prediction using an efficient octree-based deep learning model. In: Information Processing in Medical Imaging. Cham: Springer, 2023; 13939: 183-194. [CrossRef] [Google Scholar]
Wen CY, Yang AS, Tseng LY, et al. Investigation of pulsatile flowfield in healthy thoracic aorta models. Ann Biomed Eng 2010; 38: 391-402. [Article] [CrossRef] [PubMed] [Google Scholar]
Masci J, Boscaini D, Bronstein M, et al. Geodesic convolutional neural networks on riemannian manifolds. In: Proceedings of the 2015 IEEE International Conference on Computer Vision Workshop (ICCVW). Santiago: IEEE, 2015. [Google Scholar]
Herzog R, Loayza-Romero E. A manifold of planar triangular meshes with complete riemannian metric. arXiv: 2012.05624. [Google Scholar]
Yan Q, Jiang SW, Harlim J. Spectral methods for solving elliptic PDEs on unknown manifolds. J Comput Phys 2023; 486: 112132. [Article] [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
Sharp N, Crane K. A laplacian for nonmanifold triangle meshes. Comput Graphics Forum 2020; 39: 69-80. [Article] [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.