3 citations · 3 across the 6 of their papers we have counts for
8 papers
Variationally correct loss functions for nonlinear parametric problems and their neural approximations
Pablo Cortés Castillo, Wolfgang Dahmen, Jay Gopalakrishnan
We construct computable a posteriori error estimators for parametric families of nonlinear elliptic problems by measuring the residual of a variational formulation in the dual norm…
Global error estimators for parametric monotone nonlinearities and neural approximations
Pablo Cortés Castillo, Wolfgang Dahmen, Jay Gopalakrishnan
We construct computable error estimators, which double as loss functions for neural networks, for a class of parametric nonlinear partial differential equations with a monotonicity…
Sampling and reconstruction of convex functions
Andrea Bonito, Albert Cohen, Wolfgang Dahmen +3
We discuss optimal recovery for classes of multivariate convex functions from given point samples, as well as the sampling numbers of these classes, corresponding to optimal sample…
Preconditioning and Numerical Stability in Neural Network Training for Parametric PDEs
Markus Bachmayr, Wolfgang Dahmen, Chenguang Duan +1
In the context of training neural network-based approximations of solutions of parameter-dependent PDEs, we investigate the effect of preconditioning via well-conditioned frame rep…
Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation
Yuan Qiu, Wolfgang Dahmen, Peng Chen
Minimizing PDE-residual losses is a common strategy to promote physical consistency in neural operators. However, standard formulations often lack variational correctness, meaning…
Expansive Natural Neural Gradient Flows for Energy Minimization
Wolfgang Dahmen, Wuchen Li, Yuankai Teng +1
This paper develops expansive gradient dynamics in deep neural network-induced mapping spaces. Specifically, we generate tools and concepts for minimizing a class of energy functio…
DPG loss functions for learning parameter-to-solution maps by neural networks
Pablo Cortés Castillo, Wolfgang Dahmen, Jay Gopalakrishnan
We develop, analyze, and experimentally explore residual-based loss functions for machine learning of parameter-to-solution maps in the context of parameter-dependent families of p…
Variationally Correct Neural Residual Regression for Parametric PDEs: On the Viability of Controlled Accuracy
Markus Bachmayr, Wolfgang Dahmen, Mathias Oster
This paper is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all…