Author

P. Castillo

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Preprint Oct 2026

Variationally correct loss functions for nonlinear parametric problems and their neural approximations

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 of a closed subspace of the test space. A single abstract theorem shows that the resulting estimator is bounded above and below by the erro...

P. Castillo, W. Dahmen, J. Gopalakrishnan · 0 citations

Preprint Aug 2026

Fortin operators for DPG advection discretizations

We construct Fortin operators for discontinuous Petrov-Galerkin (DPG) discretizations of the advection equation with a piecewise constant divergence-free advection vector $\beta$, on simplicial meshes of any spatial dimension $N$ and for any polynomial degree $k \ge 1$ of the trial space. A minimal test space is built...

P. Castillo, Jay Gopalakrishnan · 0 citations

Preprint Oct 2026

Global error estimators for parametric monotone nonlinearities and neural approximations

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 property, and prove that they are globally reliable and efficient. The value of such a loss function is bounded above and below by the squa...

P. Castillo, W. Dahmen, Jay Gopalakrishnan · 1 citation