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<p>We establish error estimates for the approximation of parametric <i>p</i>-Dirichlet problems deploying the Deep Ritz Method. Parametric dependencies include, <i>e.g.</i>, varying geometries and exponents <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$p\in (1,\infty )$</EquationSource> </InlineEquation>. 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 demonstrates the method’s suitability for parametric problems. Finally, we present numerical experiments that illustrate potential applications.</p>

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The Deep Ritz Method for parametric p-Dirichlet problems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月16日

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  • Alex Kaltenbach,
  • Marius Zeinhofer

摘要

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 , ∞ ) $p\in (1,\infty )$ . 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 demonstrates the method’s suitability for parametric problems. Finally, we present numerical experiments that illustrate potential applications.