Physically-Enhanced Training of Neural Networks for Hydraulic Modelling of Rivers and Flood Events

Machine Learning is gaining increasing attention from the scientific community in hydrological and hydraulic research. However, this field faces a consistent challenge in applying data-driven approaches due to the evident poor generalization capabilities, which are partly a result of inherent data scarcity.

We propose incorporating expert knowledge into data-driven models for river hydraulics and flood mapping by integrating physically-based information without relying on the underlying mathematical formulation (e.g., the calculation of the residuals of differential equations). This approach appears to be particularly valuable for flood simulations, where hydraulically relevant distributed parameters such as roughness, lithology, topography etc. pose significant uncertainties. The method is versatile and applicable to physical systems and scenarios in which the underlying mathematical formulation is not fully known, but expert knowledge enables the introduction of meaningful, physically-inspired constraints.

Specifically, the physical information is integrated into the model by including an additional term, weighted by the hyperparameter in the guise of a regularization term in the loss function :

 

Here, represents the data-driven error metric, while the physical loss term is an error metric that depends not only on the true and predicted outputs ( ), but also potentially on the inputs . Indeed, this term employs physical principles, laws, and quantities, which are not explicitly formulated in the original dataset. In this sense, we can note its similarity to data augmentation, a widely used technique in machine learning that extracts additional insights by offering alternative interpretations of the same dataset.

We clarify that this approach does not aim to replace numerical solvers or serve as an alternative numerical model, as Physics-Informed Neural Networks do: indeed, their similarity is limited to the formulation of the modified loss function.

We assessed the methodology and empirically quantified the effectiveness of the method in a simplified, well-controlled problem, evaluating the gain in generalisation capability of Neural Networks (NNs) in the reconstruction of the steady state, one-dimensional, water surface profile in a rectangular channel. We found improved predictive capabilities, even when extrapolating beyond the boundaries of the training dataset and in data-scarce scenarios. This kind of assessment is of great relevance to the application of NNs to flood mapping, where cases featuring values of the observed quantities falling out of the range of the recorded series need to be predicted.

New experiments have been also conducted on two-dimensional domains. The data-driven model was trained on a single catchment and tested on its ability to determine flooded areas in unseen catchments. Preliminary results show that an encoder-decoder model with convolutional layers exhibits improved generalization when a physical training strategy is employed. Future applications could include flood mapping for ungauged basins, leveraging similarities with other basins.