Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 54 min ago
PreprintCC BY 4.0arXiv

Neural Network-Assisted Refinement of Traditional Schemes for One-Dimensional Scalar Conservation Laws

  • Imre Fekete
  • Ferenc Izsák
  • Vendel P. Kupás
Open at sourcePDFDOICite

Abstract

A clear link is established between conventional numerical methods and neural network approximations for solving one-dimensional scalar conservation laws. The focus is on the construction of an appropriate flux term in the case of convex flux functions for improving the classical schemes. The first neural network developed here is able to rediscover Godunov's method, while the second one emulates the behavior of a second-order slope-limiter function. In this way, by merging them, second-order reconstruction-based schemes can be developed. The networks presented here employ a minimal number of parameters, significantly reducing the complexity compared to previous approaches. These networks can also be linked consecutively to get a deep one corresponding to multiple time steps. Training them with an appropriate loss leads to stable schemes, improving even the classical methods without increasing their complexity.

Versions

Related

  1. Hybrid Joint-Selective Optimization: Reduced-Space Levenberg-Marquardt Refinement of Low-Dimensional Parameters of Interest
    Muhammad Luthfi Shahab, Gabriella Alfa Indahsari, Imam Mukhlash +1
    arXivPreprintCC BY 4.0Full textNumerical analysis & scientific computingOptimisation & controlarXiv
  2. Substochastic Path Operators for Refinement-Consistent Network Risks
    Luca Grilli, Domenico Santoro, Giacinto Angelo Sgarro
    AxiomsPeer-reviewedCC BY 4.0General mathematicsCrossref
  3. Optimal Refinement in Dynamic Discretisation Discovery for Continuous-Time Service Network Design
    Alexander Helber, Oliver Bachtler, Tom Wüllner
    arXivPreprintCC BY 4.0Full textOptimisation & controlarXiv
  4. Local conservation laws and symmetries of three-dimensional incompressible Euler flow
    Alexey Shevyakov
    arXivPreprintarXiv licenceFull textAnalysis & PDEsMathematical physicsarXiv
  5. High-dimensional covariate-assisted ranking
    Yujia Liu, Yuanyuan Lin, Ruijian Han +1
    arXivPreprintarXiv licenceFull textStatisticsarXiv

Cite

BibTeXRIS
BibTeX
@misc{fekete2026neural,
  title = {{Neural Network-Assisted Refinement of Traditional Schemes for One-Dimensional Scalar Conservation Laws}},
  author = {Imre Fekete and Ferenc Izsák and Vendel P. Kupás},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32887},
  eprint = {2609.32887},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32887},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Neural Network-Assisted Refinement of Traditional Schemes for One-Dimensional Scalar Conservation Laws
AU  - Imre Fekete
AU  - Ferenc Izsák
AU  - Vendel P. Kupás
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32887
UR  - https://arxiv.org/abs/2609.32887
AB  - A clear link is established between conventional numerical methods and neural network approximations for solving one-dimensional scalar conservation laws. The focus is on the construction of an appropriate flux term in the case of convex flux functions for improving the classical schemes. The first neural network developed here is able to rediscover Godunov's method, while the second one emulates the behavior of a second-order slope-limiter function. In this way, by merging them, second-order reconstruction-based schemes can be developed. The networks presented here employ a minimal number of parameters, significantly reducing the complexity compared to previous approaches. These networks can also be linked consecutively to get a deep one corresponding to multiple time steps. Training them with an appropriate loss leads to stable schemes, improving even the classical methods without increasing their complexity.
ER  -