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math.NA2026
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…
math.NA2025
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…
math.NA2025
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…