Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

no code implementations • 15 Dec 2021 • Carlo Marcati, Christoph Schwab

We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial differential equations.

Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities

no code implementations • 23 Oct 2020 • Carlo Marcati, Joost A. A. Opschoor, Philipp C. Petersen, Christoph Schwab

We prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in $H^1(\Omega)$ for weighted analytic function classes in certain polytopal domains $\Omega$, in space dimension $d=2, 3$.