The Deep Ritz Method for Parametric $p$-Dirichlet Problems
This work addresses parametric problems in computational mathematics, such as varying geometries and exponents, but is incremental as it extends existing Deep Ritz Method analysis to parametric settings.
The paper tackles the approximation of parametric p-Dirichlet problems using the Deep Ritz Method, establishing error estimates and decay rates that demonstrate the method retains neural networks' high-dimensional approximation capabilities, with numerical examples provided.
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 $p\in (1,\infty)$. 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 makes the method attractive for parametric problems. Finally, we present numerical examples to illustrate potential applications.