Close Search Advanced
Journals
Resources
About Us
Open Access

Machine Learning and Computational Mathematics

Machine Learning and Computational Mathematics
Preview
Add to basket

Year:    2020

Author:    Weinan E

Communications in Computational Physics, Vol. 28 (2020), Iss. 5 : pp. 1639–1670

Abstract

Neural network-based machine learning is capable of approximating functions in very high dimension with unprecedented efficiency and accuracy. This has opened up many exciting new possibilities, not just in traditional areas of artificial intelligence, but also in scientific computing and computational science. At the same time, machine learning has also acquired the reputation of being a set of "black box" type of tricks, without fundamental principles. This has been a real obstacle for making further progress in machine learning.
In this article, we try to address the following two very important questions: (1) How machine learning has already impacted and will further impact computational mathematics, scientific computing and computational science? (2) How computational mathematics, particularly numerical analysis, can impact machine learning? We describe some of the most important progress that has been made on these issues. Our hope is to put things into a perspective that will help to integrate machine learning with computational mathematics.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2020-0185

Communications in Computational Physics, Vol. 28 (2020), Iss. 5 : pp. 1639–1670

Published online:    2020-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    32

Keywords:    Neural network-based machine learning machine learning-based algorithm.

Author Details

Weinan E

  1. Advances of machine learning in materials science: Ideas and techniques

    Chong, Sue Sin | Ng, Yi Sheng | Wang, Hui-Qiong | Zheng, Jin-Cheng

    Frontiers of Physics, Vol. 19 (2024), Iss. 1

    https://doi.org/10.1007/s11467-023-1325-z [Citations: 12]

  2. Differentiable physics-enabled closure modeling for Burgers’ turbulence

    Shankar, Varun | Puri, Vedant | Balakrishnan, Ramesh | Maulik, Romit | Viswanathan, Venkatasubramanian

    Machine Learning: Science and Technology, Vol. 4 (2023), Iss. 1 P.015017

    https://doi.org/10.1088/2632-2153/acb19c [Citations: 5]

  3. AI-driven research in pure mathematics and theoretical physics

    He, Yang-Hui

    Nature Reviews Physics, Vol. 6 (2024), Iss. 9 P.546

    https://doi.org/10.1038/s42254-024-00740-1 [Citations: 0]

  4. A MCMC method based on surrogate model and Gaussian process parameterization for infinite Bayesian PDE inversion

    Hu, Zheng | Wang, Hongqiao | Zhou, Qingping

    Journal of Computational Physics, Vol. 507 (2024), Iss. P.112970

    https://doi.org/10.1016/j.jcp.2024.112970 [Citations: 2]

  5. Computing multi-eigenpairs of high-dimensional eigenvalue problems using tensor neural networks

    Wang, Yifan | Xie, Hehu

    Journal of Computational Physics, Vol. 506 (2024), Iss. P.112928

    https://doi.org/10.1016/j.jcp.2024.112928 [Citations: 1]

  6. Subspace decomposition based DNN algorithm for elliptic type multi-scale PDEs

    Li, Xi-An | Xu, Zhi-Qin John | Zhang, Lei

    Journal of Computational Physics, Vol. 488 (2023), Iss. P.112242

    https://doi.org/10.1016/j.jcp.2023.112242 [Citations: 6]

  7. Thermodynamics of Learning Physical Phenomena

    Cueto, Elias | Chinesta, Francisco

    Archives of Computational Methods in Engineering, Vol. 30 (2023), Iss. 8 P.4653

    https://doi.org/10.1007/s11831-023-09954-5 [Citations: 11]

  8. An unsupervised K-means machine learning algorithm via overlapping to improve the nodes selection for solving elliptic problems

    Soleymani, Fazlollah | Zhu, Shengfeng | Hu, Xindi

    Engineering Analysis with Boundary Elements, Vol. 168 (2024), Iss. P.105919

    https://doi.org/10.1016/j.enganabound.2024.105919 [Citations: 0]

  9. TCAS-PINN: Physics-informed neural networks with a novel temporal causality-based adaptive sampling method

    Guo 郭, Jia 嘉 | Wang 王, Haifeng 海峰 | Gu 古, Shilin 仕林 | Hou 侯, Chenping 臣平

    Chinese Physics B, Vol. 33 (2024), Iss. 5 P.050701

    https://doi.org/10.1088/1674-1056/ad21f3 [Citations: 0]

  10. Programming for the near future: Concepts and pragmatic considerations

    Ilyin, V. P.

    Вестник Российской академии наук, Vol. 93 (2023), Iss. 2 P.150

    https://doi.org/10.31857/S086958732302007X [Citations: 0]

  11. Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations

    Guo, Ling | Wu, Hao | Yu, Xiaochen | Zhou, Tao

    Computer Methods in Applied Mechanics and Engineering, Vol. 400 (2022), Iss. P.115523

    https://doi.org/10.1016/j.cma.2022.115523 [Citations: 31]

  12. An adaptive energy-based sequential method for training PINNs to solve gradient flow equations

    Guo, Jia | Wang, Haifeng | Hou, Chenping

    Applied Mathematics and Computation, Vol. 479 (2024), Iss. P.128890

    https://doi.org/10.1016/j.amc.2024.128890 [Citations: 0]

  13. Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systems

    Hernández, Quercus | Badías, Alberto | Chinesta, Francisco | Cueto, Elías

    Computational Mechanics, Vol. 72 (2023), Iss. 3 P.553

    https://doi.org/10.1007/s00466-023-02296-w [Citations: 5]

  14. A Hybrid Iterative Method for Elliptic Variational Inequalities of the Second Kind

    Cao, Yujian | Huang, Jianguo | Wang, Haoqin

    Communications on Applied Mathematics and Computation, Vol. (2024), Iss.

    https://doi.org/10.1007/s42967-024-00423-9 [Citations: 0]

  15. Piano Teaching Improvement Based on Machine Learning and Artificial Intelligence

    Zhang, Nan | Cui, Tao

    Security and Communication Networks, Vol. 2022 (2022), Iss. P.1

    https://doi.org/10.1155/2022/9672068 [Citations: 3]

  16. Modeling forest above-ground biomass using freely available satellite and multisource datasets

    Hojo, Ai | Avtar, Ram | Nakaji, Tatsuro | Tadono, Takeo | Takagi, Kentaro

    Ecological Informatics, Vol. 74 (2023), Iss. P.101973

    https://doi.org/10.1016/j.ecoinf.2023.101973 [Citations: 9]

  17. Data quantity governance for machine learning in materials science

    Liu, Yue | Yang, Zhengwei | Zou, Xinxin | Ma, Shuchang | Liu, Dahui | Avdeev, Maxim | Shi, Siqi

    National Science Review, Vol. 10 (2023), Iss. 7

    https://doi.org/10.1093/nsr/nwad125 [Citations: 57]

  18. Failure-Informed Adaptive Sampling for PINNs, Part II: Combining with Re-sampling and Subset Simulation

    Gao, Zhiwei | Tang, Tao | Yan, Liang | Zhou, Tao

    Communications on Applied Mathematics and Computation, Vol. 6 (2024), Iss. 3 P.1720

    https://doi.org/10.1007/s42967-023-00312-7 [Citations: 4]

  19. Failure-Informed Adaptive Sampling for PINNs

    Gao, Zhiwei | Yan, Liang | Zhou, Tao

    SIAM Journal on Scientific Computing, Vol. 45 (2023), Iss. 4 P.A1971

    https://doi.org/10.1137/22M1527763 [Citations: 35]

  20. [Retracted] Considering Computational Mathematics IGHG3 as Malignant Melanoma Is Associated with Immune Infiltration of Malignant Melanoma

    Si, Mengqing | Cao, Xianwei | Parameshachari, B. D.

    BioMed Research International, Vol. 2022 (2022), Iss. 1

    https://doi.org/10.1155/2022/4168937 [Citations: 1]

  21. Improved freezing rain forecast using machine learning

    Wen, Qiuzi Han | Wan, Dingyu | Dong, Quan | Yan, Yan | Zhang, Pingwen

    Weather and Climate Extremes, Vol. 44 (2024), Iss. P.100690

    https://doi.org/10.1016/j.wace.2024.100690 [Citations: 0]

  22. Supply chain risk management with machine learning technology: A literature review and future research directions

    Yang, Mei | Lim, Ming K. | Qu, Yingchi | Ni, Du | Xiao, Zhi

    Computers & Industrial Engineering, Vol. 175 (2023), Iss. P.108859

    https://doi.org/10.1016/j.cie.2022.108859 [Citations: 44]

  23. Subspace Decomposition Based Dnn Algorithm for Elliptic-Type Multi-Scale Pdes

    Li, Xi-An | Xu, Zhi-Qin John | Zhang, Lei

    SSRN Electronic Journal, Vol. (2022), Iss.

    https://doi.org/10.2139/ssrn.4020731 [Citations: 3]

  24. Classifying a Lending Portfolio of Loans with Dynamic Updates via a Machine Learning Technique

    Soleymani, Fazlollah | Masnavi, Houman | Shateyi, Stanford

    Mathematics, Vol. 9 (2020), Iss. 1 P.17

    https://doi.org/10.3390/math9010017 [Citations: 8]

  25. Computing ground states of Bose-Einstein condensation by normalized deep neural network

    Bao, Weizhu | Chang, Zhipeng | Zhao, Xiaofei

    Journal of Computational Physics, Vol. 520 (2025), Iss. P.113486

    https://doi.org/10.1016/j.jcp.2024.113486 [Citations: 0]

  26. Neural control of discrete weak formulations: Galerkin, least squares & minimal-residual methods with quasi-optimal weights

    Brevis, Ignacio | Muga, Ignacio | van der Zee, Kristoffer G.

    Computer Methods in Applied Mechanics and Engineering, Vol. 402 (2022), Iss. P.115716

    https://doi.org/10.1016/j.cma.2022.115716 [Citations: 6]
  27. Parallel Computational Technologies

    Parallel Intelligent Computing in Algebraic Problems

    Il’in, Valery

    2021

    https://doi.org/10.1007/978-3-030-81691-9_8 [Citations: 1]

  28. Seismic deghosting using convolutional neural networks

    Almuteri, Khalid | Sava, Paul

    GEOPHYSICS, Vol. 88 (2023), Iss. 3 P.V113

    https://doi.org/10.1190/geo2021-0521.1 [Citations: 2]

  29. Programming for the Near Future: Concepts and Pragmatic Considerations

    Ilyin, V. P.

    Herald of the Russian Academy of Sciences, Vol. 93 (2023), Iss. 2 P.92

    https://doi.org/10.1134/S1019331623010112 [Citations: 0]

  30. Investigation on aortic hemodynamics based on physics-informed neural network

    Du, Meiyuan | Zhang, Chi | Xie, Sheng | Pu, Fang | Zhang, Da | Li, Deyu

    Mathematical Biosciences and Engineering, Vol. 20 (2023), Iss. 7 P.11545

    https://doi.org/10.3934/mbe.2023512 [Citations: 9]

  31. Bayesian chemical reaction neural network for autonomous kinetic uncertainty quantification

    Li, Qiaofeng | Chen, Huaibo | Koenig, Benjamin C. | Deng, Sili

    Physical Chemistry Chemical Physics, Vol. 25 (2023), Iss. 5 P.3707

    https://doi.org/10.1039/D2CP05083H [Citations: 13]