Approximations with deep neural networks in Sobolev time-space

2020/12/23 by Ahmed Abdeljawad, Philipp Grohs, Abdeljawad, Ahmed +1 · 4 citations
Computer Science · Engineering · Physics and Astronomy · #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.2101.06115

openalex publication_date 2020/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Solutions of evolution equation generally lies in certain Bochner-Sobolev spaces, in which the solution may has regularity and integrability properties for the time variable that can be different for the space variables. Therefore, in this paper, we develop a framework shows that deep neural networks can approximate Sobolev-regular functions with respect to Bochner-Sobolev spaces. In our work we use the so-called Rectified Cubic Unit (ReCU) as an activation function in our networks, which allows us to deduce approximation results of the neural networks while avoiding issues caused by the non regularity of the most commonly used Rectivied Linear Unit (ReLU) activation function.

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