[1] Metzler R and Klafter J, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 2000, 339(1): 1-77.
[2] Shlesinger M F, West B J, and Klafter J, Lévy dynamics of enhanced diffusion: Application to turbulence, Physical Review Letters, 1987, 58(11): 1100-1103.
[3] de Pablo A, Quirós F, Rodrıguez A, et al., A fractional porous medium equation, Advances in Mathematics, 2011, 226(2): 1378-1409.
[4] Laskin N, Fractional quantum mechanics and Lévy path integrals, Physics Letters A, 2000, 268(4- 6): 298-305.
[5] Laskin N, Fractional Schrödinger equation, Physical Review E, 2002, 66(5): 056108.
[6] Longhi S, Fractional Schrödinger equation in optics, Optics Letters, 2015, 40(6): 1117-1120.
[7] Zhang Y, Zhong H, Belić M R, et al., PT symmetry in a fractional Schrödinger equation, Laser & Photonics Reviews, 2016, 10(3): 526-531.
[8] Ainsworth M and Glusa C, Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver, Computer Methods in Applied Mechanics and Engineering, 2017, 327: 4-35.
[9] Ainsworth M and Glusa C, Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains, Contemporary Computational Mathematics — A Celebration of the 80th Birthday of Ian Sloan, Eds. by Dick J, Kuo F Y, and Wózniakowski H, Springer, Cham, 2018.
[10] Del Teso F, Endal J, and Jakobsen E R, Robust numerical methods for nonlocal (and local) equations of porous medium type, Part II: Schemes and experiments, SIAM Journal on Numerical Analysis, 2018, 56(6): 3611-3647.
[11] Mao Z, Chen S, and Shen J, Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations, Applied Numerical Mathematics, 2016, 106: 165-181.
[12] Kyprianou A E, Osojnik A, and Shardlow T, Unbiased ‘walk-on-spheres’ Monte Carlo methods for the fractional Laplacian, IMA Journal of Numerical Analysis, 2018, 38(3): 1550-1578.
[13] Shardlow T, A walk outside spheres for the fractional Laplacian: Fields and first eigenvalue, Mathematics of Computation, 2019, 88(320): 2767-2792.
[14] D’Elia M, Du Q, Glusa C, et al., Numerical methods for nonlocal and fractional models, Acta Numerica, 2020, 29: 1-124.
[15] Lischke A, Pang G, Gulian M, et al., What is the fractional Laplacian? A comparative review with new results, Journal of Computational Physics, 2020, 404: 109009.
[16] Bonito A, Borthagaray J P, Nochetto R H, et al., Numerical methods for fractional diffusion, Computing and Visualization in Science, 2018, 19: 19-46.
[17] Dyda B, Kuznetsov A, and Kwásnicki M, Eigenvalues of the fractional Laplace operator in the unit ball, Journal of the London Mathematical Society, 2017, 95(2): 500-518.
[18] Dyda B, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fractional Calculus and Applied Analysis, 2012, 15(4): 536-555.
[19] Bao W, Chen L, Jiang X, et al., A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator, Journal of Computational Physics, 2020, 421: 109733.
[20] Borthagaray J P, Del Pezzo L M, and Martınez S, Finite element approximation for the fractional eigenvalue problem, Journal of Scientific Computing, 2018, 77(1): 308-329.
[21] Samardzija N and Waterland R L, A neural network for computing eigenvectors and eigenvalues, Biological Cybernetics, 1991, 65(4): 211-214.
[22] Cichocki A and Unbehauen R, Neural networks for computing eigenvalues and eigenvectors, Biological Cybernetics, 1992, 68: 155-164.
[23] Carleo G and Troyer M, Solving the quantum many-body problem with artificial neural networks, Science, 2017, 355(6325): 602-606.
[24] Choo K, Mezzacapo A, and Carleo G, Fermionic neural-network states for ab-initio electronic structure, Nature Communications, 2020, 11(1): 2368.
[25] Han J, Zhang L, and E W, Solving many-electron Schrödinger equation using deep neural networks, Journal of Computational Physics, 2019, 399: 108929.
[26] Han J, Lu J, and Zhou M, Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach, Journal of Computational Physics, 2020, 423: 109792.
[27] Li H and Ying L, A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks, Journal of Computational Physics, 2022, 453: 110939.
[28] Zhang W, Li T, and Schütte C, Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks, Journal of Computational Physics, 2022, 465: 111377.
[29] Elhamod M, Bu J, Singh C, et al., CoPhy-PGNN: Learning physics-guided neural networks with competing loss functions for solving eigenvalue problems, ACM Transactions on Intelligent Systems and Technology, 2022, 13(6): 1-23.
[30] Finol D, Lu Y, Mahadevan V, et al., Deep convolutional neural networks for eigenvalue problems in mechanics, International Journal for Numerical Methods in Engineering, 2019, 118(5): 258-275.
[31] Lu J and Lu Y, A priori generalization error analysis of two-layer neural networks for solving high dimensional Schrödinger eigenvalue problems, Communications of the American Mathematical Society, 2022, 2(1): 1-21.
[32] Kwásnicki M, Ten equivalent definitions of the fractional Laplace operator, Fractional Calculus and Applied Analysis, 2017, 20(1): 7-51.
[33] Valdinoci E, From the long jump random walk to the fractional Laplacian, Bulletin of the Spanish Society of Applied Mathematics, 2009, 49: 33-44.
[34] Frank R L, Eigenvalue bounds for the fractional Laplacian: A review, Recent Developments in Nonlocal Theory, Eds. by Palatucci G and Kuusi T, Walter de Gruyter, Berlin, 2017, 210-235.
[35] Chen Z Q and Song R, Two-sided eigenvalue estimates for subordinate processes in domains, Journal of Functional Analysis, 2005, 226(1): 90-113.
[36] Bao W, Ruan X, Shen J, et al., Fundamental gaps of the fractional Schrödinger operator, Communications in Mathematical Sciences, 2019, 17(2): 447-471.
[37] Grubb G, Fractional Laplacians on domains, a development of Hörmander’s theory of μ- transmission pseudodifferential operators, Advances in Mathematics, 2015, 268: 478-528.
[38] Khoo Y, Lu J, and Ying L. Solving for high-dimensional committor functions using artificial neural networks, Research in the Mathematical Sciences, 2019, 6: 1-13.
[39] Liu X and Oishi S, Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM Journal on Numerical Analysis, 2013, 51(3): 1634-1654.
[40] Yuan Q and He Z, Bounds to eigenvalues of the Laplacian on L-shaped domain by variational methods, Journal of Computational and Applied Mathematics, 2009, 233(4): 1083-1090.
[41] Kac M, Can one hear the shape of a drum? The American Mathematical Monthly, 1966, 73(492): 1-33.
[42] Gordon C, Webb D L, and Wolpert S, One cannot hear the shape of a drum, Bulletin of the American Mathematical Society, 1992, 27(1): 134-138.
[43] Gordon C, Webb D L, and Wolpert S, Isospectral plane domains and surfaces via Riemannian orbifolds, Inventiones Mathematicae, 1992, 110(1): 1-22.
[44] Driscoll T A, Eigenmodes of isospectral drums, SIAM Review, 1997, 39(1): 1-17.
[45] Borzı A and Borzı G, Algebraic multigrid methods for solving generalized eigenvalue problems, International Journal for Numerical Methods in Engineering, 2006, 65(8): 1186-1196.
[46] Li H and Zhang Z, Efficient spectral and spectral element methods for eigenvalue problems of Schrödinger equations with an inverse square potential, SIAM Journal on Scientific Computing, 2017, 39(1): A114-A140.