Performance comparison of a variable modification-based finite difference neural network and physics-informed neural networks in open channel modelling

ABSTRACT The physics of many natural systems can be represented by partial differential equations. Physics-informed neural networks (PINNs) solve these equations using artificial intelligence (AI) and automatic differentiation (AD) to approximate derivatives. However, AD is not appropriate for all kinds of partial differential equations. In this article, we present VANFID (variable modification finite-difference neural network), a modification of the classical PINN in which the finite-difference method replaces AD and a prior variable modification is applied. The PINN used in this study also incorporates VANFID's variable modification. Both VANFID and the PINN are applied to reference and real-world cases (cascade and step pool). For the real-world cases, two strategies are tested to handle varying cross-sections: (i) an equivalent cross-sectional representation and (ii) an additional neural network. Moreover, a methodology for handling transcritical flow is evaluated in both PINN and VANFID. Results show that the tanh activation function performs better across reference and real scenarios; VANFID outperforms PINN in reference cases, while PINN performs better in most real cases. Adding a neural network to represent cross-sectional properties in real cases has a limited impact on cascade high flow.