# Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problem

arXiv: https://arxiv.org/abs/2607.27433 · HTML: https://mlsanity.com/paper/2607.27433/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 57 references that are in the map.
- First posted on arXiv 29 Jul 2026.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Overview frequency principle/spectral bias in deep learning](https://mlsanity.com/paper/2201.07395/) (2022)
- [Physics-informed neural networks for the shallow-water equations on the sphere](https://mlsanity.com/paper/2104.00615/) (2021)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws](https://mlsanity.com/paper/2207.08483/) (2022)
- [Parametrization of subgrid scales in long-term simulations of the shallow-water equations using machine learning and convex limiting](https://mlsanity.com/paper/2602.00378/) (2026)

## Abstract (arXiv)

We develop two parametric data-driven reduced models: a physics-informed neural network (PINN) and a non-intrusive tensorial reduced-order model (TROM), and apply both approaches to the parametrized one-dimensional shallow-water dam-break problem. Neither reduced model requires time integration: both learn a direct parameter-to-solution map from space, time, and dam-break parameters to the physical state, with the PINN providing predictions at arbitrary times and the TROM reconstructing solutions at the stored snapshot times. In addition, we demonstrate that it is essential to introduce shock-aware collocation to improve the robustness of the PINN model.