Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks
This work addresses the gap between theory and practice in machine learning for numerical analysis, though it is incremental as it validates existing theoretical predictions.
The study investigated how approximation theory affects practical learning in solving parametric partial differential equations, finding that performance depends mildly on parameter dimension and is influenced by the solution manifold's intrinsic dimension, with strong support for theoretical effects in numerical analysis.
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.