Physics-Informed Deep Operator Learning for Computational Hydraulics Modeling

TL;DR

This paper develops a physics-informed DeepONet framework to learn the solution operator of the 2D shallow water equations, enabling fast surrogates for computational hydraulics. By contrasting a purely data-driven SWE-DeepONet with a physics-informed PI-SWE-DeepONet, the authors demonstrate that incorporating $\text{SWEs}$ physics enhances out-of-distribution generalization at the cost of slight reductions in in-distribution accuracy. The Sacramento River case validates the approach on a real-world, irregular domain, showing that PI-SWE-DeepONet achieves better extrapolation robustness (e.g., larger breakdown distances) while maintaining reasonable predictions for $h$, $u$, and $v$. The work also discusses the trade-offs of physics constraints, the importance of loss weighting, and provides an open-source HydroNet workflow to facilitate practical adoption in engineering problems.

Abstract

Traditional 2D hydraulic models face significant computational challenges that limit their applications that are time-sensitive or require many model evaluations. This study presents a physics-informed Deep Operator Network (DeepONet) framework for computational hydraulics modeling that learns the solution operator of the 2D shallow water equations (SWEs) to create fast surrogate models. The framework can operate in two modes: a purely data-driven SWE-DeepONet that learns from numerical solver such as SRH-2D, and a physics-informed PI-SWE-DeepONet that additionally incorporates the continuous SWEs as constraints during training. Based on a real-world case, steady flows in a reach of the Sacramento River in California, it is demonstrated that PI-SWE-DeepONet possesses much enhanced prediction capability than SWE-DeepONet when applied to out-of-distribution scenarios. The physics-informed model is shown to exhibit slower error growth and larger breakdown distances in comparison with SWE-DeepONet. The gain of the physics-informed training, however, comes with costs, chief among which are the simulated results have slightly higher errors for in-distribution cases. It reflects the existence of a tension between the two competing training objectives: fitting the results from the traditional hydraulic model and satisfying the continuous governing equations. In this study, guidelines are developed for selecting the appropriate approach based on a real-world case: PI-SWE-DeepONet is preferred for out-of-distribution predictions, uncertain training data, or when physical consistency is a priority, while SWE-DeepONet is recommended if the modeling objective is to replicate faithfully the traditional hydraulic model results within the training distribution. Other challenges are also discussed, such as the loss weighting approach.

Figures (12)

• Figure 1: The physics-informed DeepONet framework for open channel flows governed by the 2D shallow water equations.
• Figure 2: The Sacramento River model domain: (a) bathymetry with the inlet and outlet locations marked, (b) Manning's $n$ values within the domain.
• Figure 3: Mesh and collocation points for the Sacramento River case: (a) SRH-2D mesh, (b) Collocation points in the confluence area for the PI-SWE-DeepONet model.
• Figure 4: Sampled inlet discharge values at selected inlet pairs: (a) $Q_1$ vs $Q_2$, (b) $Q_1$ vs $Q_3$, (c) $Q_2$ vs $Q_3$. The training/validation/test and out-of-distribution parameters are plotted in solid and open circles, respectively. Three out-of-distribution parameter sets with minimum, average, and maximum Wasserstein distances are circled in red, magenta, and blue, respectively.
• Figure 5: Evaluation metrics for the out-of-distribution cases: (a) Wasserstein distance, (b) error sensitivity, (c) error ratio, (d) model breakdown distance.