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 representations of operators and demonstrate a significant improvement on the performance of standard training methods. We also observe that standard representations of preconditioned matrices are insufficient for obtaining numerical stability and propose a generally applicable form of stable representations that enables computations with single- and half-precision floating point numbers without loss of precision.
@article{arxiv.2601.23185,
title = {Preconditioning and Numerical Stability in Neural Network Training for Parametric PDEs},
author = {Markus Bachmayr and Wolfgang Dahmen and Chenguang Duan and Mathias Oster},
journal= {arXiv preprint arXiv:2601.23185},
year = {2026}
}