A Priori Analysis of Stable Neural Network Solutions to Numerical PDEs

Understand

Methods for solving PDEs using neural networks have recently become a very important topic.

• We provide an a priori error analysis for such methods which is based on the $\mathcal{K}_1(\mathbb{D})$-norm of the solution.
• We show that the resulting constrained optimization problem can be efficiently solved using a greedy algorithm, which replaces stochastic gradient descent.
• Following this, we show that the error arising from discretizing the energy integrals is bounded both in the deterministic case, i.e.