Open Access
Issue
ESAIM: M2AN
Volume 60, Number 2, March-April 2026
Page(s) 541 - 603
DOI https://doi.org/10.1051/m2an/2026007
Published online 06 April 2026
  1. J. Aghili, E. Franck, R. Hild, V. Michel-Dansac and V. Vigon, Accelerating the convergence of Newton’s method for nonlinear elliptic PDEs using Fourier neural operators. Commun. Nonlinear Sci. 140 (2025) 108434. [Google Scholar]
  2. R.C. Almeida and R.S. Silva, A stable Petrov-Galerkin method for convection-dominated problems. Comput. Methods Appl. Mech. Eng. 140 (1997) 291–304. [Google Scholar]
  3. M.S. Alnæs, A. Logg, K.B. Ølgaard, M.E. Rognes and G.N. Wells, Unified form language: a domain-specific language for weak formulations of partial differential equations. ACM Trans. Math. Softw. 40 (2014) 1–37. [CrossRef] [Google Scholar]
  4. M. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M.E. Rognes and G.N. Wells, The FEniCS Project Version 1.5. Arch. Numer. Softw. 3 (2015). [Google Scholar]
  5. I. Babuška and J.M. Melenk, The partition of unity method. Int. J. Numer. Meth. Eng. 40 (1997) 727–758. [Google Scholar]
  6. I. Babuška, U. Banerjee and J.E. Osborn, Generalized finite element methods main ideas, results and perspective. Int. J. Comput. Methods 1 (2004) 67–103. [CrossRef] [Google Scholar]
  7. S. Badia, W. Li and A.F. Martín, Finite element interpolated neural networks for solving forward and inverse problems. Comput. Methods Appl. Mech. Eng. 418 (2024) 116505. [Google Scholar]
  8. I.A. Baratta, J.P. Dean, J.S. Dokken, M. Habera, J. Hale, C.N. Richardson, M.E. Rognes, M.W. Scroggs, N. Sime and G.N. Wells, DOLFINx: The next generation FEniCS problem solving environment (2023). [Google Scholar]
  9. Y. Bazilevs, L. Beirão Da Veiga, J.A. Cottrell, T.J.R. Hughes and G. Sangalli, Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci. 16 (2006) 1031–1090. [Google Scholar]
  10. T. Belytschko, R. Gracie and G. Ventura, A review of extended/generalized finite element methods for material modeling. Modell. Simul. Mater. Sci. Eng. 17 (2009) 043001. [Google Scholar]
  11. G. Berkooz, P. Holmes and J.L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25 (1993) 539–575. [Google Scholar]
  12. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer New York (2008). [Google Scholar]
  13. J.-N. Brunet, A. Mendizabal, A. Petit, N. Golse, É. Vibert and S. Cotin, Physics-based deep neural network for augmented reality during liver surgery, in Medical Image Computing and Computer Assisted Intervention MICCAI 2019. Springer International Publishing (2019) 137–145. [Google Scholar]
  14. R.E. Caflisch, Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 7 (1998) 1–49. [CrossRef] [Google Scholar]
  15. D. Canales, A. Leygue, D. Chinesta, D. González, E. Cueto, E. Feulvarch, J.-M. Bergheau and A. Huerta, Vademecum-based GFEM (V-GFEM): optimal enrichment for transient problems. Int. J. Numer. Methods Eng. 108 (2016) 971–989. [Google Scholar]
  16. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics (2002). [Google Scholar]
  17. B. Cockburn, B. Dong and J. Guzmán, A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77 (2008) 1887–1916. [CrossRef] [Google Scholar]
  18. S. Cotin, M. Duprez, V. Lleras, A. Lozinski and K. Vuillemot, φ-FEM: an efficient simulation tool using simple meshes for problems in structure mechanics and heat transfer, in Partition of Unity Methods. Wiley Online Library (2023) 191–216. [Google Scholar]
  19. S. Cuomo, V.S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi and F. Piccialli, Scientific machine learning through physics-informed neural networks: Where we are and What’s Next. J. Sci. Comput. 92 (2022) 88. [Google Scholar]
  20. T. De Ryck, S. Mishra and R. Molinaro, wPINNs: weak physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws. SIAM J. Numer. Anal. 62 (2024) 811–841. [Google Scholar]
  21. L.F. Demkowicz, Mathematical Theory of Finite Elements. Society for Industrial and Applied Mathematics (2023). [Google Scholar]
  22. V. Dolean, A. Heinlein, S. Mishra and B. Moseley, Multilevel domain decomposition-based architectures for physics-informed neural networks. Comput. Methods Appl. Mech. Eng. 429 (2024) 117116. [Google Scholar]
  23. M. Duprez and A. Lozinski, φ-FEM: a finite element method on domains defined by level-sets. SIAM J. Numer. Anal. 58 (2020) 1008–1028. [CrossRef] [MathSciNet] [Google Scholar]
  24. M. Duprez, V. Lleras and A. Lozinski, A new φ-FEM approach for problems with natural boundary conditions. Numer. Methods Part. Differ. Equ. 39 (2022) 281–303. [Google Scholar]
  25. M. Duprez, V. Lleras and A. Lozinski, φ-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations. ESAIM: Math. Modell. Numer. Anal. 57 (2023) 1111–1142. [Google Scholar]
  26. M. Duprez, V. Lleras, A. Lozinski and K. Vuillemot, φ-FEM for the heat equation: optimal convergence on unfitted meshes in space. C. R. Math. 361 (2023) 1699–1710. [Google Scholar]
  27. W. E and B. Yu, The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6 (2018) 1–12. [Google Scholar]
  28. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Springer New York (2004). [Google Scholar]
  29. A. Ern and M. Steins, Convergence analysis for the wave equation discretized with hybrid methods in space (HHO, HDG and WG) and the leapfrog scheme in time. J. Sci. Comput. 101 (2024) 7. [Google Scholar]
  30. L.C. Evans, Partial Differential Equations. Number 19 in Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence, Rhode Island (2022). [Google Scholar]
  31. X. Feng, H. Shangguan, T. Tang, X. Wan and T. Zhou, A hybrid FEM-PINN method for time-dependent partial differential equations. Preprint arXiv: 2409.02810 [math] (2024). [Google Scholar]
  32. S. Frambati, H. Barucq, H. Calandra and J. Diaz, Practical unstructured splines: algorithms, multi-patch spline spaces, and some applications to numerical analysis. J. Comput. Phys. 471 (2022) 111625. [Google Scholar]
  33. E. Franck, V. Michel-Dansac and L. Navoret, Approximately well-balanced discontinuous galerkin methods using bases enriched with physics-informed neural networks. J. Comput. Phys. 512 (2024) 113144. [Google Scholar]
  34. T.-P. Fries and T. Belytschko, The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84 (2010) 253–304. [Google Scholar]
  35. T.G. Grossmann, U.J. Komorowska, J. Latz and C.-B. Schönlieb, Can physics-informed neural networks beat the finite element method? IMA J. Appl. Math. 89 (2024) 143–174. [Google Scholar]
  36. R. Hiptmair, A. Moiola and I. Perugia, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. 82 (2012) 247–268. [CrossRef] [Google Scholar]
  37. T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194 (2005) 4135–4195. [CrossRef] [Google Scholar]
  38. A. Hungria, D. Prada and F.-J. Sayas, HDG methods for elastodynamics. Comput. Math. Appl. 74 (2017) 2671–2690. [Google Scholar]
  39. L.-M. Imbert-Gérard, A. Moiola and P. Stocker, A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients. Math. Comput. 92 (2022) 1211–1249. [Google Scholar]
  40. A.D. Jagtap, E. Kharazmi and G.E. Karniadakis, Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems. Comput. Methods Appl. Mech. Eng. 365 (2020) 113028. [Google Scholar]
  41. V. John, P. Knobloch and J. Novo, Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story? Comput. Vis. Sci. 19 (2018) 47–63. [Google Scholar]
  42. W. Johnson, The curious history of Faà di Bruno’s formula. Am. Math. Monthly 109 (2002) 217–234. [Google Scholar]
  43. E. Kharazmi, Z. Zhang and G.E.M. Karniadakis, hp-VPINNs: variational physics-informed neural networks with domain decomposition. Comput. Methods Appl. Mech. Eng. 374 (2021) 113547. [Google Scholar]
  44. D. Kingma and J. Ba, Adam: a method for stochastic optimization, in International Conference on Learning Representations (ICLR), San Diego, CA, USA (2015). [Google Scholar]
  45. I.E. Lagaris, A. Likas and D.I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Netw. 9 (1998) 987–1000. [Google Scholar]
  46. H. Lee and I.S. Kang, Neural algorithm for solving differential equations. J. Comput. Phys. 91 (1990) 110–131. [Google Scholar]
  47. Z. Li, N.B. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart and A. Anandkumar, Fourier neural operator for parametric partial differential equations, in International Conference on Learning Representations (2021). [Google Scholar]
  48. N. Margenberg, R. Jendersie, C. Lessig and T. Richter, DNN-MG: a hybrid neural network/finite element method with applications to 3D simulations of the Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 420 (2024) 116692. [Google Scholar]
  49. A.J. Meade and A.A. Fernandez, The numerical solution of linear ordinary differential equations by feedforward neural networks. Math. Comput. Model. 19 (1994) 1–25. [Google Scholar]
  50. A. Moiola and I. Perugia, A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation. Numer. Math. 138 (2017) 389–435. [Google Scholar]
  51. J.J. Park, P. Florence, J. Straub, R. Newcombe and S. Lovegrove, DeepSDF: learning continuous signed distance functions for shape representation, in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). EEE (2019) 165-174. [Google Scholar]
  52. A. Paszke, F. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Tilleen, Z. Lin, N. Gimelshein, L. Antiga and A. Desmaison, PyTorch: An Imperative Style, High-Performance Deep Learning Library. Curran Associates Inc. Red Hook, NY, USA (2019) 8026-8037. [Google Scholar]
  53. A.T. Patera and E.M. Rønquist, Reduced basis approximation and a posteriori error estimation for a Boltzmann model. Comput. Methods Appl. Mech. Eng. 196 (2007) 2925-2942. [Google Scholar]
  54. H. Pham, F. Faucher and H. Barucq, Numerical investigation of stabilization in the Hybridizable Discontinuous Galerkin method for linear anisotropic elastic equation. Comput. Methods Appl. Mech. Eng. 428 (2024) 117080. [Google Scholar]
  55. C. Prud’homme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera and G. Turinici, Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods. J. Fluids Eng. 124 (2001) 70-80. [Google Scholar]
  56. M. Raissi, P. Perdikaris and G.E. Karniadakis, Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378 (2019) 686-707. [Google Scholar]
  57. S.S. Ravindran, A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Int. J. Numer. Methods Fluids 34 (2000) 425-448. [CrossRef] [Google Scholar]
  58. J.N. Reddy, Introduction to the Finite Element Method, 4th edition. McGraw-Hill Education, New York, NY (2019). [Google Scholar]
  59. M.W. Scroggs, I.A. Baratta, C.N. Richardson and G.N. Wells, Basix: a runtime finite element basis evaluation library. J. Open Source Softw. 7 (2022) 3982. [Google Scholar]
  60. M.W. Scroggs, J.S. Dokken, C.N. Richardson and G.N. Wells, Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes. ACM Trans. Math. Softw. 48 (2022) 1-23. [CrossRef] [Google Scholar]
  61. M. Sikora, P. Krukowski, A. Paszyńska and M. Paszyński, Comparison of physics informed neural networks and finite element method solvers for advection-dominated diffusion problems. J. Comput. Sci. 81 (2024) 102340. [Google Scholar]
  62. V. Sitzmann, J. Martel, A. Bergman, D. Lindell and G. Wetzstein, Implicit neural representations with periodic activation functions, in Advances in Neural Information Processing Systems, edited by H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan and H. Lin. Vol. 33. Curran Associates, Inc. (2020) 7462-7473. [Google Scholar]
  63. K. Škardová, A. Daby-Seesaram and M. Genet, Finite element neural network interpolation: Part I interpretable and adaptive discretization for solving PDEs. Comput. Mech. (2025) 1-21. [Google Scholar]
  64. H. Son, J.W. Jang, W.J. Han and H.J. Hwang, Sobolev training for physics informed neural networks. Preprint arXiv: 2101.08932 (2021). [Google Scholar]
  65. T. Strouboulis, K. Copps and I. Babuška, The generalized finite element method. Comput. Methods Appl. Mech. Eng. 190 (2001) 4081-4193. [Google Scholar]
  66. W. Sun and Y.-X. Yuan, Quasi-Newton methods, in Numerical Optimization. Springer, New York, NY (2006) 7462-7473. [Google Scholar]
  67. N. Sukumar and A. Arivastava, Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Comput. Methods Appl. Mech. Eng. 389 (2022) 114333. [Google Scholar]
  68. N. Sukumar, N. Moës, B. Moran and T. Belytschko, Extended finite element method for three-dimensional crack modelling. Int. J. Numer. Methods Eng. 48 (2000) 1549-1670. [Google Scholar]
  69. M. Tancik, P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Raghavan, U. Singhal, R. Ramamoorthi, J. Barron and R. Ng, Fourier features let networks learn high frequency functions in low dimensional domains, in Advances in Neural Information Processing Systems. Vol. 33. Curran Associates, Inc. (2020) 7537-7547. [Google Scholar]
  70. J.M. Taylor, D. Pardo and J. Muñoz-Matute, Regularity-conforming neural networks (ReCoNNs) for solving partial differential equations. J. Comput. Phys. 532 (2025) 113954. [Google Scholar]
  71. L.B. Wahlbin, Superconvergence in Galerkin Finite Element Methods. Springer Berlin Heidelberg (1995). [Google Scholar]
  72. D. Wang, H. Li and Q. Zhang, General enrichments of stable GFEM for interface problems: theory and extreme learning machine construction. Appl. Numer. Math. 214 (2025) 143-159. [Google Scholar]
  73. Z. Xiang, W. Peng, W. Yao and W. Zhou, Hybrid finite-difference physics-informed neural networks partial differential equation solver for complex geometries. J. Thermophys. Heat Trans. 39 (2025) 686-703. [Google Scholar]
  74. W. Xiong, X. Long, S.P.A. Bordas and C. Jiang, The deep finite element method: a deep learning framework integrating the physics-informed neural networks with the finite element method. Comput. Methods Appl. Mech. Eng. 436 (2025) 117681. [Google Scholar]
  75. W. Zhang and M. Al Kobaisi, On the monotonicity and positivity of physics-informed neural networks for highly anisotropic diffusion equations. Energies 15 (2022) 6823. [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.

Recommended for you