The Deep Ritz Method for Parametric p-Dirichlet Problems

2022/07/05 by Alex Kaltenbach, Kaltenbach, Alex, Marius Zeinhofer +1 · 2 citations
Computer Science · Engineering · Physics and Astronomy · #35A35 #65N15 #68T07 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Analysis Techniques #Analysis of PDEs (math.AP) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Neural and Evolutionary Computing (cs.NE) #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2207.01894

openalex publication_date 2022/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish error estimates for the approximation of parametric p-Dirichlet problems deploying the Deep Ritz Method. Parametric dependencies include, e.g., varying geometries and exponents p∈ (1,∞). Combining the derived error estimates with quantitative approximation theorems yields error decay rates and establishes that the Deep Ritz Method retains the favorable approximation capabilities of neural networks in the approximation of high dimensional functions which makes the method attractive for parametric problems. Finally, we present numerical examples to illustrate potential applications.

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