{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T20:37:48Z","timestamp":1770064668016,"version":"3.49.0"},"reference-count":31,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,23]],"date-time":"2025-03-23T00:00:00Z","timestamp":1742688000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation (NSF) of USA","doi-asserted-by":"publisher","award":["DMS-2110914"],"award-info":[{"award-number":["DMS-2110914"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation (NSF) of USA","doi-asserted-by":"publisher","award":["DMS-2306991"],"award-info":[{"award-number":["DMS-2306991"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>A new high-order hybrid method integrating neural networks and corrected finite differences is developed for solving elliptic equations with irregular interfaces and discontinuous solutions. Standard fourth-order finite difference discretization becomes invalid near such interfaces due to the discontinuities and requires corrections based on Cartesian derivative jumps. In traditional numerical methods, such as the augmented matched interface and boundary (AMIB) method, these derivative jumps can be reconstructed via additional approximations and are solved together with the unknown solution in an iterative procedure. Nontrivial developments have been carried out in the AMIB method in treating sharply curved interfaces, which, however, may not work for interfaces with geometric singularities. In this work, machine learning techniques are utilized to directly predict these Cartesian derivative jumps without involving the unknown solution. To this end, physics-informed neural networks (PINNs) are trained to satisfy the jump conditions for both closed and open interfaces with possible geometric singularities. The predicted Cartesian derivative jumps can then be integrated in the corrected finite differences. The resulting discrete Laplacian can be efficiently solved by fast Poisson solvers, such as fast Fourier transform (FFT) and geometric multigrid methods, over a rectangular domain with Dirichlet boundary conditions. This hybrid method is both easy to implement and efficient. Numerical experiments in two and three dimensions demonstrate that the method achieves fourth-order accuracy for the solution and its derivatives.<\/jats:p>","DOI":"10.3390\/computation13040083","type":"journal-article","created":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T10:53:54Z","timestamp":1742900034000},"page":"83","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A High-Order Hybrid Approach Integrating Neural Networks and Fast Poisson Solvers for Elliptic Interface Problems"],"prefix":"10.3390","volume":"13","author":[{"given":"Yiming","family":"Ren","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3023-2107","authenticated-orcid":false,"given":"Shan","family":"Zhao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2138","DOI":"10.1016\/j.jcp.2007.03.012","article-title":"A coupling interface method for elliptic interface problems","volume":"225","author":"Chern","year":"2007","journal-title":"J. Comput. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"7503","DOI":"10.1016\/j.jcp.2008.04.027","article-title":"Piecewise-polynomial discretization and Krylov-accelerated multigrid for elliptic interface problems","volume":"227","author":"Chen","year":"2008","journal-title":"J. Comput. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1016\/j.jcp.2013.06.019","article-title":"A kernel-free boundary integral method for implicitly defined surfaces","volume":"252","author":"Ying","year":"2013","journal-title":"J. Comput. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"109269","DOI":"10.1016\/j.jcp.2020.109269","article-title":"Solving elliptic interface problems with jump conditions on Cartesian grids","volume":"407","author":"Bochkov","year":"2020","journal-title":"J. Comput. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1137\/060666482","article-title":"Immersed-interface finite-element Methods for elliptic interface problems with non-homogeneous jump conditions","volume":"46","author":"Gong","year":"2008","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_6","first-page":"284","article-title":"Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions","volume":"8","author":"He","year":"2011","journal-title":"Int. J. Numer. Anal. Model."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1137\/0731054","article-title":"The immersed interface method for elliptic equations with discontinuous coefficients and singular sources","volume":"31","author":"LeVeque","year":"1994","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1137\/S0036142995291329","article-title":"A fast iterative algorithm for elliptic interface problem","volume":"35","author":"Li","year":"1998","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1137\/15M1040244","article-title":"Accurate solution and gradient computation for elliptic interface problems with variable coefficients","volume":"55","author":"Li","year":"2017","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1006\/jcph.1999.6236","article-title":"A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method)","volume":"152","author":"Fedkiw","year":"1999","journal-title":"J. Comput. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1016\/j.jcp.2007.08.003","article-title":"Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities","volume":"227","author":"Yu","year":"2007","journal-title":"J. Comput. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jcp.2005.07.022","article-title":"High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular source","volume":"213","author":"Zhou","year":"2006","journal-title":"J. Comput. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"109677","DOI":"10.1016\/j.jcp.2020.109677","article-title":"A fourth order finite difference method for solving elliptic interface problems with the FFT acceleration","volume":"419","author":"Feng","year":"2020","journal-title":"J. Comput. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"111924","DOI":"10.1016\/j.jcp.2023.111924","article-title":"A FFT accelerated fourth order finite difference method for solving three-dimensional elliptic interface problems","volume":"477","author":"Ren","year":"2023","journal-title":"J. Comput. Phys."},{"key":"ref_15","unstructured":"Ren, Y., and Zhao, S. (Int. J. Numer. Anal. Model., 2025). A multigrid-based fourth order finite difference method for elliptic interface problems with variable coefficients, Int. J. Numer. Anal. Model., submitted."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Li, C., Zhao, S., Pentecost, B., Ren, Y., and Guan, Z. (J. Sci. Comput., 2025). A spatially fourth-order Cartesian grid method for fast solution of elliptic and parabolic problems on irregular domains with sharply curved boundaries, J. Sci. Comput., submitted.","DOI":"10.2139\/ssrn.4812718"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40304-018-0127-z","article-title":"The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems","volume":"6","author":"E","year":"2018","journal-title":"Commun. Math. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1162","DOI":"10.4208\/cicp.OA-2021-0201","article-title":"Deep unfitted Nitsche method for elliptic interface problems","volume":"31","author":"Guo","year":"2022","journal-title":"Commun. Comput. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"114358","DOI":"10.1016\/j.cam.2022.114358","article-title":"A mesh-free method using piecewise deep neural network for elliptic interface problems","volume":"412","author":"He","year":"2022","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"109175","DOI":"10.1016\/j.aml.2024.109175","article-title":"Generalization of PINNs for elliptic interface problems","volume":"157","author":"Jiang","year":"2024","journal-title":"Appl. Math. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"112359","DOI":"10.1016\/j.jcp.2023.112359","article-title":"A cusp-capturing PINN for elliptic interface problems","volume":"491","author":"Tseng","year":"2023","journal-title":"J. Comput. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"111588","DOI":"10.1016\/j.jcp.2022.111588","article-title":"INN: Interfaced neural networks as an accessible meshless approach for solving interface PDE problems","volume":"470","author":"Wu","year":"2022","journal-title":"J. Comput. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"C633","DOI":"10.1137\/24M1632309","article-title":"An Accurate and Efficient Continuity-Preserved Method Based on Randomized Neural Networks for Elliptic Interface Problems","volume":"46","author":"Ying","year":"2024","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"111576","DOI":"10.1016\/j.jcp.2022.111576","article-title":"A discontinuity capturing shallow neural network for elliptic interface problems","volume":"469","author":"Hu","year":"2022","journal-title":"J. Comput. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1090","DOI":"10.4208\/cicp.OA-2022-0284","article-title":"An efficient neural-network and finite-difference hybrid method for elliptic interface problems with applications","volume":"33","author":"Hu","year":"2023","journal-title":"Commun. Comput. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"110762","DOI":"10.1016\/j.jcp.2021.110762","article-title":"A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains","volume":"448","author":"Ren","year":"2022","journal-title":"J. Comput. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1137\/0111030","article-title":"An algorithm for least-squares estimation of nonlinear parameters","volume":"11","author":"Marquardt","year":"1963","journal-title":"SIAM J. Appl. Math."},{"key":"ref_29","first-page":"1","article-title":"Automatic differentiation in machine learning: A survey","volume":"18","author":"Baydin","year":"2018","journal-title":"J. Mach. Learn. Res."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Li, Z., and Ito, K. (2006). The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9780898717464"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s10915-021-01570-4","article-title":"A high order compact FD framework for elliptic BVPs involving singular sources, interfaces, and irregular domains","volume":"88","author":"Pan","year":"2021","journal-title":"J. Sci. Comput."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/4\/83\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:58:46Z","timestamp":1760029126000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/4\/83"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,23]]},"references-count":31,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["computation13040083"],"URL":"https:\/\/doi.org\/10.3390\/computation13040083","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,23]]}}}