arXiv subjects

Paulina Sepúlveda

Publications and source records attributed to Paulina Sepúlveda.

3 recordsLinked to original sources

arXiv2026

Robust Deep FOSLS for Transmission Problems

This work presents a robust, energy-based deep learning framework for solving transmission problems in heterogeneous media, including cases with discontinuous material scenarios. We introduce a weighted First-Order System Least-Squares (FOSLS) formulation involving an energy-norm Poincaré constant and prove its equivalence to a natural energy norm of the underlying equations, with constants independent of material parameters. As a result, the optimization landscape remains aligned with a meaningful error approximation even under high material contrast, where standard neural network losses often deteriorate. We further prove that the FOSLS formulation, together with its integral-loss representation, exhibits a passive variance reduction property, whereby the gradient variance progressively decreases as the loss diminishes, in contrast to methods such as VPINNs and Deep Ritz. From a numerical standpoint, we adopt a reduced-order perspective by constructing a low-dimensional space described by a neural network. The optimal coefficients are computed via a least-squares solver, and the space is subsequently improved through gradient-based updates. By selecting the activation function ReQU, the method mitigates the spurious overshoots typically observed in smooth networks when approximating discontinuities. Numerical experiments in 1D and 2D interface settings corroborate these findings.

math.NA↗

arXiv2025

Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations

The appearance of singularities in the function of interest constitutes a fundamental challenge in scientific computing. It can significantly undermine the effectiveness of numerical schemes for function approximation, numerical integration, and the solution of partial differential equations (PDEs), etc. The problem becomes more sophisticated if the location of the singularity is unknown, which is often encountered in solving PDEs. Detecting the singularity is therefore critical for developing efficient adaptive methods to reduce computational costs in various applications. In this paper, we consider singularity detection in a purely data-driven setting. Namely, the input only contains given data, such as the vertex set from a mesh. To overcome the limitation of the raw unlabeled data, we propose a self-supervised learning (SSL) framework for estimating the location of the singularity. A key component is a filtering procedure as the pretext task in SSL, where two filtering methods are presented, based on $k$ nearest neighbors and kernel density estimation, respectively. We provide numerical examples to illustrate the potential pathological or inaccurate results due to the use of raw data without filtering. Various experiments are presented to demonstrate the ability of the proposed approach to deal with input perturbation, label corruption, and different kinds of singularities such interior circle, boundary layer, concentric semicircles, etc.

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arXiv2023

Adaptive stabilized finite elements via residual minimization onto bubble enrichments

The Adaptive Stabilized Finite Element method (AS-FEM) developed in Calo et. al. combines the idea of the residual minimization method with the inf-sup stability offered by the discontinuous Galerkin (dG) frameworks. As a result, the discretizations deliver stabilized approximations and residual representatives in the dG space that can drive automatic adaptivity. We generalize AS FEM by considering continuous test spaces; thus, we propose a residual minimization method on a stable Continuous Interior Penalty (CIP) formulation that considers a C0-conforming trial FEM space and a test space based on the enrichment of the trial space by bubble functions. In our numerical experiments, the test space choice results in a significant reduction of the total degrees of freedom compared to the dG test spaces of Calo et. al. that converge at the same rate. Moreover, as trial and test spaces are C0-conforming, implementing a full dG data structure is unnecessary, simplifying the method's implementation considerably and making it appealing for industrial applications, see Labanda et. al.

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