{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T14:26:50Z","timestamp":1768746410148,"version":"3.49.0"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T00:00:00Z","timestamp":1758844800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T00:00:00Z","timestamp":1758844800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Swiss Federal Institute of Technology Zurich"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2025,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We prove deep neural network (DNN for short) expressivity rate bounds for solution sets of a model class of singularly perturbed, elliptic two-point boundary value problems, in Sobolev norms, on the bounded interval <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$(-1,1)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. We assume that the given source term and reaction coefficient are analytic in <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$[-1,1]$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>[<\/mml:mo>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>]<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. The expression rate bounds in Sobolev norms in terms of the NN size are robust, i.e. uniform with respect to the singular perturbation parameter <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\varepsilon \\in (0,1]$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03b5<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>]<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> for several classes of DNN architectures. In particular, ReLU NNs, spiking NNs, and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\tanh $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mo>tanh<\/mml:mo>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>- and sigmoid-activated NNs. The latter activations can represent \u201cexponential boundary layer solution features\u201d explicitly, in the last hidden layer of the DNN, i.e. in a shallow subnetwork, and afford improved robust expression rate bounds in terms of the NN size. All DNN architectures allow <jats:italic>robust exponential solution expression<\/jats:italic> in so-called \u2018energy\u2019 as well as in \u2018balanced\u2019 Sobolev norms, for analytic input data.<\/jats:p>","DOI":"10.1007\/s00211-025-01491-6","type":"journal-article","created":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T07:40:56Z","timestamp":1758872456000},"page":"1897-1936","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Neural networks for singular perturbations"],"prefix":"10.1007","volume":"157","author":[{"given":"J. A. A.","family":"Opschoor","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ch.","family":"Schwab","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C.","family":"Xenophontos","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,26]]},"reference":[{"key":"1491_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115169","volume":"402","author":"M Ainsworth","year":"2022","unstructured":"Ainsworth, M., Dong, J.: Galerkin neural network approximation of singularly-perturbed elliptic systems. Comput. Methods Appl. Mech. Eng. 402, 115169 (2022). https:\/\/doi.org\/10.1016\/j.cma.2022.115169","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1491_CR2","doi-asserted-by":"crossref","unstructured":"Gie, G-.M., Hong, Y., Jung, C-.Y., Lee, D.: Singular layer physics-informed neural network method for convection dominated boundary layer problems in two dimensions. J. Comput. Appl. Math. 474, 116918 (2026)","DOI":"10.1016\/j.cam.2025.116918"},{"key":"1491_CR3","volume-title":"Interpolation and approximation. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography","author":"PJ Davis","year":"1975","unstructured":"Davis, P.J.: Interpolation and approximation. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography. Dover Publications Inc, New York (1975)"},{"key":"1491_CR4","doi-asserted-by":"publisher","first-page":"732","DOI":"10.1016\/j.neunet.2021.08.015","volume":"143","author":"T De Ryck","year":"2021","unstructured":"De Ryck, T., Lanthaler, S., Mishra, S.: On the approximation of functions by tanh neural networks. Neural Netw. 143, 732\u2013750 (2021). https:\/\/doi.org\/10.1016\/j.neunet.2021.08.015","journal-title":"Neural Netw."},{"issue":"1","key":"1491_CR5","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s00365-021-09541-6","volume":"55","author":"D Elbr\u00e4chter","year":"2022","unstructured":"Elbr\u00e4chter, D., Grohs, P., Jentzen, A., Schwab, C.: DNN expression rate analysis of high-dimensional PDEs: application to option pricing. Constr. Approx. 55(1), 3\u201371 (2022). https:\/\/doi.org\/10.1007\/s00365-021-09541-6","journal-title":"Constr. Approx."},{"key":"1491_CR6","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198506720.001.0001","volume-title":"Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation","author":"W Gautschi","year":"2004","unstructured":"Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2004)"},{"key":"1491_CR7","series-title":"Applied Mathematical Sciences","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-00638-9","volume-title":"Singular Perturbations and Boundary Layers","author":"GM Gie","year":"2018","unstructured":"Gie, G.M., Hamouda, M., Jung, C.Y., Temam, R.M.: Singular Perturbations and Boundary Layers. Applied Mathematical Sciences, vol. 200. Springer, Cham (2018)"},{"issue":"2","key":"1491_CR8","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1007\/s10915-021-01718-2","volume":"90","author":"L Herrmann","year":"2022","unstructured":"Herrmann, L., Opschoor, J.A.A., Schwab, C.: Constructive deep ReLU neural network approximation. J. Sci. Comput. 90(2), 75 (2022). https:\/\/doi.org\/10.1007\/s10915-021-01718-2","journal-title":"J. Sci. Comput."},{"key":"1491_CR9","doi-asserted-by":"publisher","DOI":"10.3389\/fnins.2024.1346805","author":"Y Kim","year":"2024","unstructured":"Kim, Y., Kahana, A., Yin, R., Li, Y., Stinis, P., Karniadakis, G.E., Panda, P.: Rethinking skip connections in spiking neural networks with time-to-first-spike coding. Front. Neurosci. (2024). https:\/\/doi.org\/10.3389\/fnins.2024.1346805","journal-title":"Front. Neurosci."},{"issue":"2","key":"1491_CR10","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1162\/neco.1997.9.2.279","volume":"9","author":"W Maass","year":"1997","unstructured":"Maass, W.: Fast sigmoidal networks via spiking neurons. Neural Comput. 9(2), 279\u2013304 (1997). https:\/\/doi.org\/10.1162\/neco.1997.9.2.279","journal-title":"Neural Comput."},{"issue":"9","key":"1491_CR11","doi-asserted-by":"publisher","first-page":"1659","DOI":"10.1016\/S0893-6080(97)00011-7","volume":"10","author":"W Maass","year":"1997","unstructured":"Maass, W.: Networks of spiking neurons: the third generation of neural network models. Neural Netw. 10(9), 1659\u20131671 (1997). https:\/\/doi.org\/10.1016\/S0893-6080(97)00011-7","journal-title":"Neural Netw."},{"issue":"3","key":"1491_CR12","doi-asserted-by":"publisher","first-page":"1043","DOI":"10.1007\/s10208-022-09565-9","volume":"23","author":"C Marcati","year":"2023","unstructured":"Marcati, C., Opschoor, J.A.A., Petersen, P.C., Schwab, C.: Exponential ReLU neural network approximation rates for point and edge singularities. J. Found. Comput. Math. 23(3), 1043\u20131127 (2023). https:\/\/doi.org\/10.1007\/s10208-022-09565-9","journal-title":"J. Found. Comput. Math."},{"issue":"3","key":"1491_CR13","doi-asserted-by":"publisher","first-page":"1513","DOI":"10.1137\/21M1465718","volume":"61","author":"C Marcati","year":"2023","unstructured":"Marcati, C., Schwab, C.: Exponential convergence of deep operator networks for elliptic partial differential equations. SIAM J. Numer. Anal. 61(3), 1513\u20131545 (2023). https:\/\/doi.org\/10.1137\/21M1465718","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"1491_CR14","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1093\/imanum\/17.4.577","volume":"17","author":"JM Melenk","year":"1997","unstructured":"Melenk, J.M.: On the robust exponential convergence of $$hp$$ finite element method for problems with boundary layers. IMA J. Numer. Anal. 17(4), 577\u2013601 (1997). https:\/\/doi.org\/10.1093\/imanum\/17.4.577","journal-title":"IMA J. Numer. Anal."},{"issue":"1","key":"1491_CR15","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1007\/s10092-015-0139-y","volume":"53","author":"JM Melenk","year":"2016","unstructured":"Melenk, J.M., Xenophontos, C.: Robust exponential convergence of $$hp$$-FEM in balanced norms for singularly perturbed reaction-diffusion equations. Calcolo 53(1), 105\u2013132 (2016). https:\/\/doi.org\/10.1007\/s10092-015-0139-y","journal-title":"Calcolo"},{"key":"1491_CR16","unstructured":"Opschoor, J.A.A.: Constructive deep neural network approximations of weighted analytic solutions to partial differential equations in polygons. Ph.D. thesis, ETH Z\u00fcrich. Diss. ETH No. 29278 (2023)"},{"key":"1491_CR17","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1016\/j.camwa.2024.06.008","volume":"169","author":"JAA Opschoor","year":"2024","unstructured":"Opschoor, J.A.A.: Schwab, Christoph Deep ReLU networks and high-order finite element methods II: Cheby\u0161ev emulation. Comput. Math. Appl. 169, 142\u2013162 (2024)","journal-title":"Comput. Math. Appl."},{"key":"1491_CR18","unstructured":"Opschoor, J.A.A., Schwab, C.: Deep ReLU networks and high-order finite element methods II: Chebyshev emulation. Technical Report 2023-38, Seminar for Applied Mathematics, ETH Z\u00fcrich, Switzerland (2023)"},{"key":"1491_CR19","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1016\/j.neunet.2018.08.019","volume":"108","author":"P Petersen","year":"2018","unstructured":"Petersen, P., Voigtlaender, F.: Optimal approximation of piecewise smooth functions using deep ReLU neural networks. Neural Netw. 108, 296\u2013330 (2018). https:\/\/doi.org\/10.1016\/j.neunet.2018.08.019","journal-title":"Neural Netw."},{"issue":"304","key":"1491_CR20","doi-asserted-by":"publisher","first-page":"661","DOI":"10.1090\/mcom\/3113","volume":"86","author":"H Rauhut","year":"2017","unstructured":"Rauhut, H., Schwab, C.: Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations. Math. Comput. 86(304), 661\u2013700 (2017). https:\/\/doi.org\/10.1090\/mcom\/3113","journal-title":"Math. Comput."},{"key":"1491_CR21","volume-title":"The Chebyshev Polynomials. Pure and Applied Mathematics","author":"TJ Rivlin","year":"1974","unstructured":"Rivlin, T.J.: The Chebyshev Polynomials. Pure and Applied Mathematics. Wiley-Interscience (Wiley), New York-London-Sydney (1974)"},{"key":"1491_CR22","volume-title":"$$p$$- and $$hp$$-Finite Element Methods. Numerical Mathematics and Scientific Computation","author":"C Schwab","year":"1998","unstructured":"Schwab, C.: $$p$$- and $$hp$$-Finite Element Methods. Numerical Mathematics and Scientific Computation. The Clarendon Press, Oxford University Press, New York (1998)"},{"issue":"216","key":"1491_CR23","doi-asserted-by":"publisher","first-page":"1403","DOI":"10.1090\/S0025-5718-96-00781-8","volume":"65","author":"C Schwab","year":"1996","unstructured":"Schwab, C., Suri, M.: The $$p$$ and $$hp$$ versions of the finite element method for problems with boundary layers. Math. Comput. 65(216), 1403\u20131429 (1996). https:\/\/doi.org\/10.1090\/S0025-5718-96-00781-8","journal-title":"Math. Comput."},{"key":"1491_CR24","doi-asserted-by":"crossref","unstructured":"Stanojevic, A., Wo\u017aniak, S., Bellec, G., Cherubini, G., Pantazi, A., Gerstner, W.: An exact mapping from ReLU networks to spiking neural networks. ArXiv:2212.12522 (2022)","DOI":"10.1016\/j.neunet.2023.09.011"},{"key":"1491_CR25","unstructured":"Tang, S., Li, B., Yu, H.Y.: ChebNet: efficient and stable constructions of deep neural networks with rectified power units via Chebyshev approximations ArXiv. arXiv:1911.05467v3."},{"key":"1491_CR26","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611975949","volume-title":"Approximation Theory and Approximation Practice","author":"LN Trefethen","year":"2019","unstructured":"Trefethen, L.N.: Approximation Theory and Approximation Practice, Extended Society for Industrial and Applied Mathematics, Philadelphia (2019)","edition":"Extended"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01491-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-025-01491-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01491-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T04:03:28Z","timestamp":1760501008000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-025-01491-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,26]]},"references-count":26,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["1491"],"URL":"https:\/\/doi.org\/10.1007\/s00211-025-01491-6","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,26]]},"assertion":[{"value":"14 January 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 January 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 July 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 September 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}