arXiv subjects

Pablo Cortés Castillo

Publications and source records attributed to Pablo Cortés Castillo.

3 recordsLinked to original sources

arXiv2026

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 squared error in the natural trial norm, for every trial function, not merely for those near the exact solution; this global property rests on monotonicity. The construction rests on splitting the nonlinear operator into a linear part and a strongly monotone closure, and on measuring the linear residual in a discrete dual norm. Since the closure contributes a dual norm that admits no closed form when the trial norm is stronger than an $L_2$ norm, the estimator is built around a computable surrogate for it, required only to satisfy a pairing bound and a Lipschitz bound. The main theorem then yields two-sided bounds with explicit constants and covers two instances. A first-order system least-squares estimator on conforming trial spaces is the first instance studied: its two-sided bound holds on the whole trial space and therefore applies to arbitrary approximations, including those not in a discrete finite element space. The second instance is a discontinuous Petrov-Galerkin estimator on trial spaces of finite element functions for each parameter value, built with broken test spaces, for which no conformity is required and the dual norm is computed by independent element-local problems. All assumptions are verified for a model class of nonlinear fluxes, with the constants tracked explicitly in terms of the parameter range. Being computable for an arbitrary input, both estimators serve as variationally correct loss functions for neural network approximations of parameter-to-solution maps.

math.NA↗

arXiv2026

DPG loss functions for learning parameter-to-solution maps by neural networks

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 partial differential equations (PDEs). Our primary concern is on rigorous accuracy certification to enhance the prediction capability of the resulting deep neural network reduced models. This is achieved by the use of variationally correct loss functions. Through one specific example of an elliptic PDE, details for establishing the variational correctness of a loss function from an ultraweak Discontinuous Petrov Galerkin (DPG) discretization are worked out. Despite the focus on the example, the proposed concepts apply to a much wider scope of problems, namely problems for which stable DPG formulations are available. The issue of high-contrast diffusion fields and ensuing difficulties with degrading ellipticity are discussed. Both numerical results and theoretical arguments illustrate that for high-contrast diffusion parameters the proposed DPG loss functions deliver much more robust performance than simpler least-squares losses.

math.NA↗

arXiv2026

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 $β$, 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 on each element from facet and interior bubbles and later augmented. The Fortin operator is shown to be bounded, uniformly over shape-regular mesh families, in the natural $β$-weighted broken test graph norm built on $L_q$ for every $1 < q < \infty$, where $q$ is the exponent conjugate to the trial exponent $p$. A non-characteristic facet condition is assumed when $k \ge 2$, while the lowest-order case requires no such condition and admits characteristic facets. As applications we prove that the practical fully discrete residual minimization method is quasioptimal in the DPG energy norm for every $1 < p < \infty$, with a quasioptimality constant governed solely by the Fortin operator, that its computable residual is a globally reliable and efficient a posteriori error estimator, and that augmenting the test space and changing the test norm improves the results in the lowest-order case.

math.NA↗