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We establish error estimates for the approximation of parametric$p$-Dirichlet problems deploying the Deep Ritz Method. Parametric dependenciesinclude, e.g., varying geometries and exponents $p\in (1,\infty)$. Combiningthe derived error estimates with quantitative approximation theorems yieldserror decay rates and establishes that the Deep Ritz Method retains thefavorable approximation capabilities of neural networks in the approximation ofhigh dimensional functions which makes the method attractive for parametricproblems. Finally, we present numerical examples to illustrate potentialapplications.

The Deep Ritz Method for Parametric $p$-Dirichlet Problems