Chen, Shi; Ding, Zhiyan; Li, Qin; Wright, Stephen J. A reduced order Schwarz method for nonlinear multiscale elliptic equations based on two-layer neural networks. (English) Zbl 07806684 J. Comput. Math. 42, No. 2, 570-596 (2024). MSC: 65N55 35J66 41A46 68T07 PDFBibTeX XMLCite \textit{S. Chen} et al., J. Comput. Math. 42, No. 2, 570--596 (2024; Zbl 07806684) Full Text: DOI arXiv
Ying, Lexing Solving inverse problems with deep learning. (English) Zbl 07822589 Beliaev, Dmitry (ed.) et al., International congress of mathematicians 2022, ICM 2022, Helsinki, Finland, virtual, July 6–14, 2022. Volume 7. Sections 15–20. Berlin: European Mathematical Society (EMS). 5154-5175 (2023). MSC: 65N21 65N99 68T07 65R32 78A46 74J25 86A22 65F55 35S05 PDFBibTeX XMLCite \textit{L. Ying}, in: International congress of mathematicians 2022, ICM 2022, Helsinki, Finland, virtual, July 6--14, 2022. Volume 7. Sections 15--20. Berlin: European Mathematical Society (EMS). 5154--5175 (2023; Zbl 07822589) Full Text: DOI OA License
Simonnet, Eric Computing non-equilibrium trajectories by a deep learning approach. (English) Zbl 07771266 J. Comput. Phys. 491, Article ID 112349, 33 p. (2023). MSC: 60Hxx 65Cxx 65Mxx PDFBibTeX XMLCite \textit{E. Simonnet}, J. Comput. Phys. 491, Article ID 112349, 33 p. (2023; Zbl 07771266) Full Text: DOI arXiv
Shang, Yong; Wang, Fei; Sun, Jingbo Randomized neural network with Petrov-Galerkin methods for solving linear and nonlinear partial differential equations. (English) Zbl 1525.65128 Commun. Nonlinear Sci. Numer. Simul. 127, Article ID 107518, 20 p. (2023). MSC: 65N30 68T07 PDFBibTeX XMLCite \textit{Y. Shang} et al., Commun. Nonlinear Sci. Numer. Simul. 127, Article ID 107518, 20 p. (2023; Zbl 1525.65128) Full Text: DOI
Bittracher, Andreas; Mollenhauer, Mattes; Koltai, Péter; Schütte, Christof Optimal reaction coordinates: variational characterization and sparse computation. (English) Zbl 07705311 Multiscale Model. Simul. 21, No. 2, 449-488 (2023). Reviewer: Mikhail P. Moklyachuk (Kyïv) MSC: 60G25 60J25 65K10 62D99 PDFBibTeX XMLCite \textit{A. Bittracher} et al., Multiscale Model. Simul. 21, No. 2, 449--488 (2023; Zbl 07705311) Full Text: DOI arXiv
Strahan, John; Finkel, Justin; Dinner, Aaron R.; Weare, Jonathan Predicting rare events using neural networks and short-trajectory data. (English) Zbl 07696951 J. Comput. Phys. 488, Article ID 112152, 16 p. (2023). MSC: 68Txx 60Hxx 65Cxx PDFBibTeX XMLCite \textit{J. Strahan} et al., J. Comput. Phys. 488, Article ID 112152, 16 p. (2023; Zbl 07696951) Full Text: DOI arXiv
Fan, Yuwei; Ying, Lexing Solving traveltime tomography with deep learning. (English) Zbl 1515.65142 Commun. Math. Stat. 11, No. 1, 3-19 (2023). MSC: 65J22 68T07 74J25 PDFBibTeX XMLCite \textit{Y. Fan} and \textit{L. Ying}, Commun. Math. Stat. 11, No. 1, 3--19 (2023; Zbl 1515.65142) Full Text: DOI arXiv
Beck, Christian; Hutzenthaler, Martin; Jentzen, Arnulf; Kuckuck, Benno An overview on deep learning-based approximation methods for partial differential equations. (English) Zbl 07675828 Discrete Contin. Dyn. Syst., Ser. B 28, No. 6, 3697-3746 (2023). MSC: 65-02 68T07 35-02 65M99 PDFBibTeX XMLCite \textit{C. Beck} et al., Discrete Contin. Dyn. Syst., Ser. B 28, No. 6, 3697--3746 (2023; Zbl 07675828) Full Text: DOI arXiv
Evans, Luke; Cameron, Maria K.; Tiwary, Pratyush Computing committors in collective variables via Mahalanobis diffusion maps. (English) Zbl 1528.65007 Appl. Comput. Harmon. Anal. 64, 62-101 (2023). MSC: 65C99 60H10 PDFBibTeX XMLCite \textit{L. Evans} et al., Appl. Comput. Harmon. Anal. 64, 62--101 (2023; Zbl 1528.65007) Full Text: DOI arXiv
Darbon, Jérôme; Dower, Peter M.; Meng, Tingwei Neural network architectures using min-plus algebra for solving certain high-dimensional optimal control problems and Hamilton-Jacobi PDEs. (English) Zbl 1507.49019 Math. Control Signals Syst. 35, No. 1, 1-44 (2023). MSC: 49K20 35F21 93B70 PDFBibTeX XMLCite \textit{J. Darbon} et al., Math. Control Signals Syst. 35, No. 1, 1--44 (2023; Zbl 1507.49019) Full Text: DOI arXiv
Gao, Wenhan; Wang, Chunmei Active learning based sampling for high-dimensional nonlinear partial differential equations. (English) Zbl 07649269 J. Comput. Phys. 475, Article ID 111848, 20 p. (2023). MSC: 68Txx 65Mxx 65Nxx PDFBibTeX XMLCite \textit{W. Gao} and \textit{C. Wang}, J. Comput. Phys. 475, Article ID 111848, 20 p. (2023; Zbl 07649269) Full Text: DOI arXiv
Chen, Xiaoli; Duan, Jinqiao; Hu, Jianyu; Li, Dongfang Data-driven method to learn the most probable transition pathway and stochastic differential equation. (English) Zbl 1504.60107 Physica D 443, Article ID 133559, 15 p. (2023). MSC: 60H35 60H10 65C30 68T07 70H03 PDFBibTeX XMLCite \textit{X. Chen} et al., Physica D 443, Article ID 133559, 15 p. (2023; Zbl 1504.60107) Full Text: DOI arXiv
Chen, Yian; Hoskins, Jeremy; Khoo, Yuehaw; Lindsey, Michael Committor functions via tensor networks. (English) Zbl 07620352 J. Comput. Phys. 472, Article ID 111646, 22 p. (2023). MSC: 60Hxx 65Fxx 65Nxx PDFBibTeX XMLCite \textit{Y. Chen} et al., J. Comput. Phys. 472, Article ID 111646, 22 p. (2023; Zbl 07620352) Full Text: DOI arXiv
Lu, Jianfeng; Lu, Yulong A priori generalization error analysis of two-layer neural networks for solving high dimensional Schrödinger eigenvalue problems. (English) Zbl 07750988 Commun. Am. Math. Soc. 2, 1-21 (2022). MSC: 65-XX 35P15 65N25 68T07 35Q40 PDFBibTeX XMLCite \textit{J. Lu} and \textit{Y. Lu}, Commun. Am. Math. Soc. 2, 1--21 (2022; Zbl 07750988) Full Text: DOI arXiv
Lucente, Dario; Rolland, Joran; Herbert, Corentin; Bouchet, Freddy Coupling rare event algorithms with data-based learned committor functions using the analogue Markov chain. (English) Zbl 07581190 J. Stat. Mech. Theory Exp. 2022, No. 8, Article ID 083201, 40 p. (2022). MSC: 82-XX PDFBibTeX XMLCite \textit{D. Lucente} et al., J. Stat. Mech. Theory Exp. 2022, No. 8, Article ID 083201, 40 p. (2022; Zbl 07581190) Full Text: DOI arXiv
Liu, Shu; Li, Wuchen; Zha, Hongyuan; Zhou, Haomin Neural parametric Fokker-Planck equation. (English) Zbl 1493.65132 SIAM J. Numer. Anal. 60, No. 3, 1385-1449 (2022). Reviewer: Bülent Karasözen (Ankara) MSC: 65M06 65L12 49Q22 68T07 65M15 65M12 60H30 35A24 35R02 35R60 35Q84 PDFBibTeX XMLCite \textit{S. Liu} et al., SIAM J. Numer. Anal. 60, No. 3, 1385--1449 (2022; Zbl 1493.65132) Full Text: DOI arXiv
Lin, Bo; Li, Qianxiao; Ren, Weiqing Computing the invariant distribution of randomly perturbed dynamical systems using deep learning. (English) Zbl 1490.35502 J. Sci. Comput. 91, No. 3, Paper No. 77, 17 p. (2022). MSC: 35Q84 37M05 65Z05 PDFBibTeX XMLCite \textit{B. Lin} et al., J. Sci. Comput. 91, No. 3, Paper No. 77, 17 p. (2022; Zbl 1490.35502) Full Text: DOI arXiv
Zhu, Wei; Khademi, Wesley; Charalampidis, Efstathios G.; Kevrekidis, Panayotis G. Neural networks enforcing physical symmetries in nonlinear dynamical lattices: the case example of the Ablowitz-Ladik model. (English) Zbl 1512.35512 Physica D 434, Article ID 133264, 14 p. (2022). MSC: 35Q51 35Q55 35Q35 35C08 65M99 65K10 68T07 37K10 37K15 37K60 35R60 PDFBibTeX XMLCite \textit{W. Zhu} et al., Physica D 434, Article ID 133264, 14 p. (2022; Zbl 1512.35512) Full Text: DOI arXiv
Huang, Jianguo; Wang, Chunmei; Wang, Haoqin A deep learning method for elliptic hemivariational inequalities. (English) Zbl 1485.65070 East Asian J. Appl. Math. 12, No. 3, 487-502 (2022). MSC: 65K15 68T07 68U99 PDFBibTeX XMLCite \textit{J. Huang} et al., East Asian J. Appl. Math. 12, No. 3, 487--502 (2022; Zbl 1485.65070) Full Text: DOI
Lyu, Liyao; Zhang, Zhen; Chen, Minxin; Chen, Jingrun MIM: a deep mixed residual method for solving high-order partial differential equations. (English) Zbl 07517751 J. Comput. Phys. 452, Article ID 110930, 16 p. (2022). MSC: 65Nxx 65Mxx 68Txx PDFBibTeX XMLCite \textit{L. Lyu} et al., J. Comput. Phys. 452, Article ID 110930, 16 p. (2022; Zbl 07517751) Full Text: DOI arXiv
Li, Haoya; Ying, Lexing A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks. (English) Zbl 07517713 J. Comput. Phys. 453, Article ID 110939, 15 p. (2022). MSC: 35Qxx 60Hxx 68Txx PDFBibTeX XMLCite \textit{H. Li} and \textit{L. Ying}, J. Comput. Phys. 453, Article ID 110939, 15 p. (2022; Zbl 07517713) Full Text: DOI arXiv
E, Weinan; Han, Jiequn; Jentzen, Arnulf Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning. (English) Zbl 1490.60202 Nonlinearity 35, No. 1, 278-310 (2022). MSC: 60H30 60H35 65C05 65C30 65M75 PDFBibTeX XMLCite \textit{W. E} et al., Nonlinearity 35, No. 1, 278--310 (2022; Zbl 1490.60202) Full Text: DOI arXiv
Gu, Yiqi; Yang, Haizhao; Zhou, Chao SelectNet: self-paced learning for high-dimensional partial differential equations. (English) Zbl 07513823 J. Comput. Phys. 441, Article ID 110444, 18 p. (2021). MSC: 68Txx 65Mxx 65Lxx PDFBibTeX XMLCite \textit{Y. Gu} et al., J. Comput. Phys. 441, Article ID 110444, 18 p. (2021; Zbl 07513823) Full Text: DOI arXiv
Darbon, Jérôme; Meng, Tingwei On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton-Jacobi partial differential equations. (English) Zbl 07508503 J. Comput. Phys. 425, Article ID 109907, 16 p. (2021). MSC: 68-XX 65-XX PDFBibTeX XMLCite \textit{J. Darbon} and \textit{T. Meng}, J. Comput. Phys. 425, Article ID 109907, 16 p. (2021; Zbl 07508503) Full Text: DOI arXiv
Chen, Zhen; Xiu, Dongbin On generalized residual network for deep learning of unknown dynamical systems. (English) Zbl 07505957 J. Comput. Phys. 438, Article ID 110362, 16 p. (2021). MSC: 68Txx 35Qxx 37Mxx PDFBibTeX XMLCite \textit{Z. Chen} and \textit{D. Xiu}, J. Comput. Phys. 438, Article ID 110362, 16 p. (2021; Zbl 07505957) Full Text: DOI arXiv
Khoo, Yuehaw; Lu, Jianfeng; Ying, Lexing Solving parametric PDE problems with artificial neural networks. (English) Zbl 1501.65154 Eur. J. Appl. Math. 32, No. 3, 421-435 (2021). MSC: 65N99 68T07 35R60 PDFBibTeX XMLCite \textit{Y. Khoo} et al., Eur. J. Appl. Math. 32, No. 3, 421--435 (2021; Zbl 1501.65154) Full Text: DOI arXiv
Liao, Yulei; Ming, Pingbing Deep Nitsche method: deep Ritz method with essential boundary conditions. (English) Zbl 1473.65309 Commun. Comput. Phys. 29, No. 5, 1365-1384 (2021). MSC: 65N30 65M12 41A46 35J25 PDFBibTeX XMLCite \textit{Y. Liao} and \textit{P. Ming}, Commun. Comput. Phys. 29, No. 5, 1365--1384 (2021; Zbl 1473.65309) Full Text: DOI arXiv
Bringmann, Kathrin; Kaszian, Jonas; Milas, Antun; Nazaroglu, Caner Integral representations of rank two false theta functions and their modularity properties. (English) Zbl 1482.11069 Res. Math. Sci. 8, No. 4, Paper No. 54, 31 p. (2021). Reviewer: M. P. Chaudhary (New Delhi) MSC: 11F27 17B69 11F37 81T40 PDFBibTeX XMLCite \textit{K. Bringmann} et al., Res. Math. Sci. 8, No. 4, Paper No. 54, 31 p. (2021; Zbl 1482.11069) Full Text: DOI arXiv
Kharazmi, Ehsan; Zhang, Zhongqiang; Karniadakis, George E. M. hp-VPINNs: variational physics-informed neural networks with domain decomposition. (English) Zbl 1506.68105 Comput. Methods Appl. Mech. Eng. 374, Article ID 113547, 25 p. (2021). MSC: 68T07 92B20 65M55 PDFBibTeX XMLCite \textit{E. Kharazmi} et al., Comput. Methods Appl. Mech. Eng. 374, Article ID 113547, 25 p. (2021; Zbl 1506.68105) Full Text: DOI arXiv
Li, Zhihan; Fan, Yuwei; Ying, Lexing Multilevel fine-tuning: closing generalization gaps in approximation of solution maps under a limited budget for training data. (English) Zbl 1468.65220 Multiscale Model. Simul. 19, No. 1, 344-373 (2021). MSC: 65N55 65C20 65C05 62J07 62M45 68T05 PDFBibTeX XMLCite \textit{Z. Li} et al., Multiscale Model. Simul. 19, No. 1, 344--373 (2021; Zbl 1468.65220) Full Text: DOI arXiv
Huang, Jianguo; Wang, Haoqin; Yang, Haizhao Int-Deep: a deep learning initialized iterative method for nonlinear problems. (English) Zbl 07507236 J. Comput. Phys. 419, Article ID 109675, 24 p. (2020). MSC: 65-XX 68-XX PDFBibTeX XMLCite \textit{J. Huang} et al., J. Comput. Phys. 419, Article ID 109675, 24 p. (2020; Zbl 07507236) Full Text: DOI arXiv
Feliu-Fabà, Jordi; Fan, Yuwei; Ying, Lexing Meta-learning pseudo-differential operators with deep neural networks. (English) Zbl 07505630 J. Comput. Phys. 408, Article ID 109309, 18 p. (2020). MSC: 65-XX 35-XX PDFBibTeX XMLCite \textit{J. Feliu-Fabà} et al., J. Comput. Phys. 408, Article ID 109309, 18 p. (2020; Zbl 07505630) Full Text: DOI arXiv
Li, Yingzhou; Cheng, Xiuyuan; Lu, Jianfeng Butterfly-net: optimal function representation based on convolutional neural networks. (English) Zbl 1507.68271 Commun. Comput. Phys. 28, No. 5, 1838-1885 (2020). MSC: 68T07 15A23 41A25 65D05 65F10 PDFBibTeX XMLCite \textit{Y. Li} et al., Commun. Comput. Phys. 28, No. 5, 1838--1885 (2020; Zbl 1507.68271) Full Text: DOI arXiv
Trstanova, Z.; Leimkuhler, B.; Lelièvre, T. Local and global perspectives on diffusion maps in the analysis of molecular systems. (English) Zbl 1472.82029 Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 476, No. 2233, Article ID 20190036, 22 p. (2020). MSC: 82C31 PDFBibTeX XMLCite \textit{Z. Trstanova} et al., Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 476, No. 2233, Article ID 20190036, 22 p. (2020; Zbl 1472.82029) Full Text: DOI arXiv
E, Weinan; Ma, Chao; Wu, Lei Machine learning from a continuous viewpoint. I. (English) Zbl 1472.68136 Sci. China, Math. 63, No. 11, 2233-2266 (2020). MSC: 68T05 49M20 49Q20 PDFBibTeX XMLCite \textit{W. E} et al., Sci. China, Math. 63, No. 11, 2233--2266 (2020; Zbl 1472.68136) Full Text: DOI arXiv
Helfmann, Luzie; Ribera Borrell, Enric; Schütte, Christof; Koltai, Péter Extending transition path theory: periodically driven and finite-time dynamics. (English) Zbl 1475.82013 J. Nonlinear Sci. 30, No. 6, 3321-3366 (2020). Reviewer: Piotr Garbaczewski (Opole) MSC: 82C05 82C26 60J22 60J10 PDFBibTeX XMLCite \textit{L. Helfmann} et al., J. Nonlinear Sci. 30, No. 6, 3321--3366 (2020; Zbl 1475.82013) Full Text: DOI arXiv
Wang, Yating; Cheung, Siu Wun; Chung, Eric T.; Efendiev, Yalchin; Wang, Min Deep multiscale model learning. (English) Zbl 1454.76008 J. Comput. Phys. 406, Article ID 109071, 17 p. (2020). MSC: 76-10 68T07 76S05 65M55 PDFBibTeX XMLCite \textit{Y. Wang} et al., J. Comput. Phys. 406, Article ID 109071, 17 p. (2020; Zbl 1454.76008) Full Text: DOI arXiv
Fan, Yuwei; Ying, Lexing Solving electrical impedance tomography with deep learning. (English) Zbl 1453.65041 J. Comput. Phys. 404, Article ID 109119, 19 p. (2020). MSC: 65D18 68T07 65N21 PDFBibTeX XMLCite \textit{Y. Fan} and \textit{L. Ying}, J. Comput. Phys. 404, Article ID 109119, 19 p. (2020; Zbl 1453.65041) Full Text: DOI arXiv
Geneva, Nicholas; Zabaras, Nicholas Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks. (English) Zbl 1454.65130 J. Comput. Phys. 403, Article ID 109056, 32 p. (2020). MSC: 65M99 35Q35 68T07 PDFBibTeX XMLCite \textit{N. Geneva} and \textit{N. Zabaras}, J. Comput. Phys. 403, Article ID 109056, 32 p. (2020; Zbl 1454.65130) Full Text: DOI arXiv
Darbon, Jérôme; Langlois, Gabriel P.; Meng, Tingwei Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures. (English) Zbl 1445.35119 Res. Math. Sci. 7, No. 3, Paper No. 20, 50 p. (2020). MSC: 35F21 35F25 92C20 35R30 PDFBibTeX XMLCite \textit{J. Darbon} et al., Res. Math. Sci. 7, No. 3, Paper No. 20, 50 p. (2020; Zbl 1445.35119) Full Text: DOI arXiv
Zang, Yaohua; Bao, Gang; Ye, Xiaojing; Zhou, Haomin Weak adversarial networks for high-dimensional partial differential equations. (English) Zbl 1436.65156 J. Comput. Phys. 411, Article ID 109409, 13 p. (2020). MSC: 65M99 65K10 68T07 35G50 35D30 PDFBibTeX XMLCite \textit{Y. Zang} et al., J. Comput. Phys. 411, Article ID 109409, 13 p. (2020; Zbl 1436.65156) Full Text: DOI arXiv
Li, Yingzhou; Lu, Jianfeng; Mao, Anqi Variational training of neural network approximations of solution maps for physical models. (English) Zbl 1435.68292 J. Comput. Phys. 409, Article ID 109338, 14 p. (2020). MSC: 68T07 35Q55 PDFBibTeX XMLCite \textit{Y. Li} et al., J. Comput. Phys. 409, Article ID 109338, 14 p. (2020; Zbl 1435.68292) Full Text: DOI arXiv
Khoo, Yuehaw; Ying, Lexing SwitchNet: a neural Network Model for forward and inverse scattering problems. (English) Zbl 1425.65208 SIAM J. Sci. Comput. 41, No. 5, A3182-A3201 (2019). MSC: 65R10 65T50 PDFBibTeX XMLCite \textit{Y. Khoo} and \textit{L. Ying}, SIAM J. Sci. Comput. 41, No. 5, A3182--A3201 (2019; Zbl 1425.65208) Full Text: DOI arXiv