A well-balanced shock-capturing variational physics-informed neural method for shallow-water flow with topography
Abstract
We introduce a variational physics-informed neural discretization for the nonlinear shallow-water equations with topography, in which a single global neural network represents the space-time trial state while the governing equations are enforced weakly in space through local cell moments and strongly in time at selected time nodes. Written in the free-surface variables (η, q), the output map preserves the non-negative water depth by construction, inherits the initial condition without a penalty term, and embeds wet and wet-dry lakes at rest in the same trial manifold. For shock-forming calculations, the conservative variational residual is supplemented with a one-sided mechanicalentropy loss, which penalizes only non-admissible positive energy production, and with a localized viscosity that concentrates dissipation near steep transitions. We establish two scheme-specific analytical results: an exact identity for the well-balanced defect generated by the positivity map, yielding distributional preservation for the positive part while quantifying the loss floor induced by a numerical film or a softplus map, and a two-regime consistency estimate for the nonlinear viscosity, which is O(h 2 ) in smooth regions, reaches the O(h) scale near an O(h) transition layer, and vanishes in distributions along refined sequences with uniformly bounded spatial variation. Wet-and dry-bed dam breaks, fully wet and partially dry equilibria, and one-and two-dimensional moving-shoreline solutions then assess these mechanisms numerically.
Related
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BibTeX
@misc{cardonna2026wellbalanced,
title = {{A well-balanced shock-capturing variational physics-informed neural method for shallow-water flow with topography}},
author = {Sacha Cardonna and Ali Haidar},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05837658v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A well-balanced shock-capturing variational physics-informed neural method for shallow-water flow with topography
AU - Sacha Cardonna
AU - Ali Haidar
PY - 2026
DA - 2026/10/23
PB - HAL
UR - https://hal.science/hal-05837658v1
AB - We introduce a variational physics-informed neural discretization for the nonlinear shallow-water equations with topography, in which a single global neural network represents the space-time trial state while the governing equations are enforced weakly in space through local cell moments and strongly in time at selected time nodes. Written in the free-surface variables (η, q), the output map preserves the non-negative water depth by construction, inherits the initial condition without a penalty term, and embeds wet and wet-dry lakes at rest in the same trial manifold. For shock-forming calculations, the conservative variational residual is supplemented with a one-sided mechanicalentropy loss, which penalizes only non-admissible positive energy production, and with a localized viscosity that concentrates dissipation near steep transitions. We establish two scheme-specific analytical results: an exact identity for the well-balanced defect generated by the positivity map, yielding distributional preservation for the positive part while quantifying the loss floor induced by a numerical film or a softplus map, and a two-regime consistency estimate for the nonlinear viscosity, which is O(h 2 ) in smooth regions, reaches the O(h) scale near an O(h) transition layer, and vanishes in distributions along refined sequences with uniformly bounded spatial variation. Wet-and dry-bed dam breaks, fully wet and partially dry equilibria, and one-and two-dimensional moving-shoreline solutions then assess these mechanisms numerically.
ER -