4 papers
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…