On the approximation of rough functions with deep neural networks

• Autores: Tim De Ryck, Siddhartha Mishra , Deep Ray
• Localización: SeMA Journal: Boletín de la Sociedad Española de Matemática Aplicada, ISSN-e 2254-3902, ISSN 2254-3902, Vol. 79, Nº. Extra 3, 2022, págs. 399-440
• Idioma: inglés
• DOI: 10.1007/s40324-022-00299-w
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• Resumen

  • The essentially non-oscillatory (ENO) procedure and its variant, the ENO-SR procedure, are very efficient algorithms for interpolating (reconstructing) rough functions. We prove that the ENO (and ENO-SR) procedure are equivalent to deep ReLU neural networks. This demonstrates the ability of deep ReLU neural networks to approximate rough functions to high-order of accuracy. Numerical tests for the resulting trained neural networks show excellent performance for interpolating functions, approximating solutions of nonlinear conservation laws and at data compression.