A Tutorial on Neural Network-Based Solvers for Hyperbolic Conservation Laws: Supervised vs. Unsupervised Learning, and Applications to Traffic Modeling

Alexi Canesse, Zhe Fu, Nathan Lichtlé, Hossein Nick Zinat Matin, Zihe Liu, Maria Laura Delle Monache, Alexandre M. Bayen
Proceedings of the International Conference on Neuro-symbolic Systems, PMLR 288:708-720, 2025.

Abstract

Neural networks (NNs) are powerful tools for solving complex partial differential equations (PDEs) with high accuracy. However, many NN-based solvers are designed as general-purpose models or lack theoretical grounding, limiting their ability to capture essential solution properties such as regularity, conservation, and entropy conditions. This issue is especially critical for hyperbolic conservation laws, which govern wave propagation and shock formation, and are among the most challenging PDEs to solve accurately. This tutorial examines both supervised and unsupervised NN-based solvers from computational and theoretical perspectives, with a focus on NN-based finite volume methods (FVMs) tailored to conservation laws. In the supervised setting, NN solvers learn from available solution data, such as Riemann problems, to capture characteristic solution structures, while the unsupervised approach employs a weak formulation loss to enforce the correct weak solution behavior. In practice, both the supervised and unsupervised variants tend to learn the entropic solution, effectively handling discontinuities and shocks, and outperforming comparable numerical schemes in accuracy. This tutorial aims to provide a deeper understanding of NN-based solvers for PDEs and to present structure-preserving neural methods for scientific computing.

Cite this Paper

BibTeX
@InProceedings{pmlr-v288-canesse25a, title = {A Tutorial on Neural Network-Based Solvers for Hyperbolic Conservation Laws: Supervised vs. Unsupervised Learning, and Applications to Traffic Modeling}, author = {Canesse, Alexi and Fu, Zhe and Lichtl\'{e}, Nathan and Zinat Matin, Hossein Nick and Liu, Zihe and Delle Monache, Maria Laura and Bayen, Alexandre M.}, booktitle = {Proceedings of the International Conference on Neuro-symbolic Systems}, pages = {708--720}, year = {2025}, editor = {Pappas, George and Ravikumar, Pradeep and Seshia, Sanjit A.}, volume = {288}, series = {Proceedings of Machine Learning Research}, month = {28--30 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v288/main/assets/canesse25a/canesse25a.pdf}, url = {https://proceedings.mlr.press/v288/canesse25a.html}, abstract = {Neural networks (NNs) are powerful tools for solving complex partial differential equations (PDEs) with high accuracy. However, many NN-based solvers are designed as general-purpose models or lack theoretical grounding, limiting their ability to capture essential solution properties such as regularity, conservation, and entropy conditions. This issue is especially critical for hyperbolic conservation laws, which govern wave propagation and shock formation, and are among the most challenging PDEs to solve accurately. This tutorial examines both supervised and unsupervised NN-based solvers from computational and theoretical perspectives, with a focus on NN-based finite volume methods (FVMs) tailored to conservation laws. In the supervised setting, NN solvers learn from available solution data, such as Riemann problems, to capture characteristic solution structures, while the unsupervised approach employs a weak formulation loss to enforce the correct weak solution behavior. In practice, both the supervised and unsupervised variants tend to learn the entropic solution, effectively handling discontinuities and shocks, and outperforming comparable numerical schemes in accuracy. This tutorial aims to provide a deeper understanding of NN-based solvers for PDEs and to present structure-preserving neural methods for scientific computing.} }
Endnote
%0 Conference Paper %T A Tutorial on Neural Network-Based Solvers for Hyperbolic Conservation Laws: Supervised vs. Unsupervised Learning, and Applications to Traffic Modeling %A Alexi Canesse %A Zhe Fu %A Nathan Lichtlé %A Hossein Nick Zinat Matin %A Zihe Liu %A Maria Laura Delle Monache %A Alexandre M. Bayen %B Proceedings of the International Conference on Neuro-symbolic Systems %C Proceedings of Machine Learning Research %D 2025 %E George Pappas %E Pradeep Ravikumar %E Sanjit A. Seshia %F pmlr-v288-canesse25a %I PMLR %P 708--720 %U https://proceedings.mlr.press/v288/canesse25a.html %V 288 %X Neural networks (NNs) are powerful tools for solving complex partial differential equations (PDEs) with high accuracy. However, many NN-based solvers are designed as general-purpose models or lack theoretical grounding, limiting their ability to capture essential solution properties such as regularity, conservation, and entropy conditions. This issue is especially critical for hyperbolic conservation laws, which govern wave propagation and shock formation, and are among the most challenging PDEs to solve accurately. This tutorial examines both supervised and unsupervised NN-based solvers from computational and theoretical perspectives, with a focus on NN-based finite volume methods (FVMs) tailored to conservation laws. In the supervised setting, NN solvers learn from available solution data, such as Riemann problems, to capture characteristic solution structures, while the unsupervised approach employs a weak formulation loss to enforce the correct weak solution behavior. In practice, both the supervised and unsupervised variants tend to learn the entropic solution, effectively handling discontinuities and shocks, and outperforming comparable numerical schemes in accuracy. This tutorial aims to provide a deeper understanding of NN-based solvers for PDEs and to present structure-preserving neural methods for scientific computing.
APA
Canesse, A., Fu, Z., Lichtlé, N., Zinat Matin, H.N., Liu, Z., Delle Monache, M.L. & Bayen, A.M.. (2025). A Tutorial on Neural Network-Based Solvers for Hyperbolic Conservation Laws: Supervised vs. Unsupervised Learning, and Applications to Traffic Modeling. Proceedings of the International Conference on Neuro-symbolic Systems, in Proceedings of Machine Learning Research 288:708-720 Available from https://proceedings.mlr.press/v288/canesse25a.html.

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