Stable time-stepping via residual minimization: finite-element and neural-network approximations for transient parabolic problems
Abstract
We propose a time-stepping minimum-residual (MinRes) framework for transient coercive variational problems, applicable to finite-element and neural-network trial approximations. For the Backward Euler scheme, we measure the residual at each time level in the dual norm induced by the steady test-space norm, scaled by the time step. This choice yields stability estimates controlling a discrete parabolic energy quantity, up to the time-discretization defect. We develop conforming and broken-test formulations and specialize the analysis to diffusion-advection-reaction problems. For finite-dimensional test spaces, we derive fully discrete reliability estimates by supplementing the computable discrete residual with a complementary contribution that accounts for residual components the test space does not resolve. We then extend the framework to neural-network trial classes and obtain computable residual decompositions for conforming and broken polynomial test spaces. Numerical experiments confirm the expected finite-element convergence rates, examine the effect of test-space refinement for a global space-time neural approximation, and demonstrate residual-driven spatial refinement for a transient problem with a moving localized feature.
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BibTeX
@misc{munozmatute2026stable,
title = {{Stable time-stepping via residual minimization: finite-element and neural-network approximations for transient parabolic problems}},
author = {Judit Muñoz-Matute and Sergio Rojas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27548},
eprint = {2609.27548},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27548},
}
RIS
TY - UNPB
TI - Stable time-stepping via residual minimization: finite-element and neural-network approximations for transient parabolic problems
AU - Judit Muñoz-Matute
AU - Sergio Rojas
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27548
UR - https://arxiv.org/abs/2609.27548
AB - We propose a time-stepping minimum-residual (MinRes) framework for transient coercive variational problems, applicable to finite-element and neural-network trial approximations. For the Backward Euler scheme, we measure the residual at each time level in the dual norm induced by the steady test-space norm, scaled by the time step. This choice yields stability estimates controlling a discrete parabolic energy quantity, up to the time-discretization defect. We develop conforming and broken-test formulations and specialize the analysis to diffusion-advection-reaction problems. For finite-dimensional test spaces, we derive fully discrete reliability estimates by supplementing the computable discrete residual with a complementary contribution that accounts for residual components the test space does not resolve. We then extend the framework to neural-network trial classes and obtain computable residual decompositions for conforming and broken polynomial test spaces. Numerical experiments confirm the expected finite-element convergence rates, examine the effect of test-space refinement for a global space-time neural approximation, and demonstrate residual-driven spatial refinement for a transient problem with a moving localized feature.
ER -