Can physical information aid the generalization ability of Neural Networks for hydraulic modeling?

TL;DR

The paper addresses the challenge of generalizing neural models in river hydraulics under data scarcity. It proposes using soft physical information to train neural operators, instead of solving hard PDE residuals as in traditional PINNs, enabling broader physical guidance while avoiding strict solver constraints. Through a synthetic 1D steady-flow test case with a weir and hydraulic jumps, the authors show that physics-informed training improves generalization across multiple architectures, especially in limited-data scenarios, though some constraints can hinder performance for certain configurations. The findings suggest a practical path to more robust flood-mapping tools for ungauged basins by enabling reliable extrapolation beyond observed data ranges.

Abstract

Application of Neural Networks to river hydraulics is fledgling, despite the field suffering from data scarcity, a challenge for machine learning techniques. Consequently, many purely data-driven Neural Networks proved to lack predictive capabilities. In this work, we propose to mitigate such problem by introducing physical information into the training phase. The idea is borrowed from Physics-Informed Neural Networks which have been recently proposed in other contexts. Physics-Informed Neural Networks embed physical information in the form of the residual of the Partial Differential Equations (PDEs) governing the phenomenon and, as such, are conceived as neural solvers, i.e. an alternative to traditional numerical solvers. Such approach is seldom suitable for environmental hydraulics, where epistemic uncertainties are large, and computing residuals of PDEs exhibits difficulties similar to those faced by classical numerical methods. Instead, we envisaged the employment of Neural Networks as neural operators, featuring physical constraints formulated without resorting to PDEs. The proposed novel methodology shares similarities with data augmentation and regularization. We show that incorporating such soft physical information can improve predictive capabilities.

Figures (11)

• Figure 1: Sketch of the analyzed physical problem, i.e. the steady state water profile in a rectangular prismatic channel. Upper panel describes the subcritical case; lower panel depicts the mixed regime case. For the meaning of the symbols, the reader is referred to the text.
• Figure 2: Profile reconstruction using the three different approaches; input and output are depicted as blue and orange circles, respectively; Single-Point (upper panel) outputs the flow depth at a specific stationing; Integrator (middle panel) requires the neighboring downstream value, regardless of the stationing; Vector-To-Sequence outputs the entire vector of flow depth (i.e. the whole water profile) at once.
• Figure 3: The employed FFNN architectures. Left to right: Single-Point, Integrator, Vector-To-Sequence.
• Figure 4: Typical outcome of the comparison between the employed models, namely DD, EN, FR, and the reference solution, FD: only the case of the SP architecture is shown for the sake of clarity. The depicted profile, determined by the quantities in the box, contains a hydraulic jump.
• Figure 5: Cumulative frequency distributions of NMAE and NNSE values over the test set. Markers represent the positioning of the results for the example case shown in Figure \ref{['SPex']}