# Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks

arXiv: https://arxiv.org/abs/2004.12131 · HTML: https://mlsanity.com/paper/2004.12131/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Apr 2020.

## Built on by (most cited first, 27 of 27)

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## Abstract (arXiv)

We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysis. As the underlying model, we study the machine-learning-based solution of parametric partial differential equations. Here, approximation theory predicts that the performance of the model should depend only very mildly on the dimension of the parameter space and is determined by the intrinsic dimension of the solution manifold of the parametric partial differential equation. We use various methods to establish comparability between test-cases by minimizing the effect of the choice of test-cases on the optimization and sampling aspects of the learning problem. We find strong support for the hypothesis that approximation-theoretical effects heavily influence the practical behavior of learning problems in numerical analysis.