Investigation into labelled-data-free physics-informed neural networks to solve the shallow water equations

Abstract

This PhD thesis investigates the application of machine learning (ML) to solve free-surface flow problems governed by the shallow water equations (SWEs). As analytical solutions to the SWEs are only available under very specific, idealised cases, the simulation of real-world problems has so far been based on the hydraulic models developed from numerical approximation methods (e.g., finite difference, finite volume) to solve the SWEs. However, the accuracy and computational performance of numerical methods available (in particular, explicit numerical schemes) are very sensitive to the size of computational cells used, which often translates into simulations that take infeasibly long.

The use of ML techniques for finding approximate solutions to partial differential equations (PDEs) has emerged as a consequence of the ability of ML models to capture any nonlinear relationship between inputs and outputs. Particular attention is placed in this thesis towards Physics-Informed Neural Networks (PINNs), which are essentially ML algorithms which use the information contained in the physical laws to train the model. The type of PINNs proposed here can eliminate the need for labelled data –in particular, known model outputs within the (excluding the boundaries of) domain which correspond to given inputs– for training the network. This is achieved by requiring the model to conform with the governing PDEs (along with the given boundary and initial conditions). The network solutions are penalised by residuals of the PDEs instead of the conventional approach which penalises residuals with regard to known outputs (labelled data, typically obtained through other numerical solution methods). As a result, this new approach, named ‘Labelled-Data-Free PINN’ in this thesis, can offer a completely independent numerical method to solve problems governed by the SWEs. Partial derivatives in the governing PDEs are determined by taking advantage of automatic differentiation (AD), which is computationally efficient and provides the exact values of the derivatives (although they are derivatives of approximate solutions).

Two types of Artificial Neural Networks (ANNs) are employed in the research: Convolutional Neural Networks (C-PINNs) and Fully Connected Neural Networks (F-PINNs). The initial investigation focuses on testing the ability of PINNs to accurately approximate solutions to idealised 1D flow problems, demonstrating good agreement with benchmark results in each test case. Subsequently, comparative studies, covering accuracy and training speed, are conducted among C-PINNs, F-PINNs, and a state-of-the-art finite volume solver. These models address two idealised 2D flow problems and a real-world flood event. Results show that PINNs may deliver remarkably accurate results, and may also offer, under some scenarios, competitive computational efficiency compared to the finite volume solver. The C-PINN model demonstrates a superior trade-off between computational speed and accuracy than F-PINN. Overall, the results in this thesis suggest that PINNs may become competitive alternatives to conventional numerical solvers. The thesis suggests that this technique needs further research and development, including the exploration of more advanced automatic training methods and the practical implementation of building a real-time flood forecasting system.

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Identifiers

ORCID for Xin Qi: orcid.org/0000-0002-3635-7453

ORCID for Gustavo De Almeida: orcid.org/0000-0002-3291-3985

ORCID for Sergio Maldonado: orcid.org/0000-0001-6072-122X

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