Patterns in Ecosystems, Powered by Physics Informed Neural Networks

Abstract

Multi-scale branching tidal channel networks carry several important functions for coastal wetland ecosystems, as well as for their surroundings. Many species living in and around the ecosystem are supported and protected by the channel network, as the channels carry important nutrients and resources. Furthermore, the channels increase the resilience of the landscape against sea-level rise by transporting sediment. Accurate predictions and understanding about the formation of these channel networks is therefore crucial. The speed and accuracy of numerical simulations is limited by, for example, temporal and spatial scale-differences between the processes in the landscape and working with realistic boundary shapes is tedious. Physics-Informed Neural Networks (PINNs) can produce mesh-less and differentiable solutions, while providing fast inference and flexibility towards boundary shapes. These properties can be highly beneficial to coastal wetland simulations, as the continuous nature of their solutions can aid in bridging the spatial and temporal gap between the different landscape-processes. In this work, we embed a complex multi-scale hydro-morphodynamical system to capture the formation of tidal channels into a physics-informed framework. We show that a PINN can be effectively trained, without data, to solve the 2D Shallow Water Equations (SWEs) in a closed water basin with flat bottom topography. The network solves these three coupled equations with steps of one second per iteration and is capable of working in complexly shaped domains with non-trivial obstacles. Effective training relies on oscillators introducing surface-disturbances. We propose that a PINN can further be trained, without data, to solve the 2D SWEs in a partially open landscape, by employing a numerical solver for the bottom topography and vegetation. A simple Convolutional Neural Network (CNN) is designed to translate from the discrete numerical solutions to the mesh-less representations of the PINN. We show that the PINN trains more effectively on landscapes with a high degree of spatial complexity. The accuracy of the hydrodynamic predictions made by the PINN influences the spatial complexity of the landscapes in its training set, and thereby the training effectiveness of the PINN itself. Important aspects of the hydrodynamic solutions are qualitatively accurate, and we show that they are detailed enough to trigger a channelisation feedback in the bed topography. Our physics-informed hydrodynamics are not sufficiently accurate to support the full ensemble of scale-dependent feedbacks behind the multi-scale tidal channels. We propose an ensemble of three networks to solve the morphological equations, in addition to the hydrodynamical ones, by means of a PINN. Learning effectiveness of one network in the ensemble heavily depends on the quality and accuracy of the partial solutions of the neighbouring networks. Our multi-network setup facilitates multi-stage or asynchronous training, which could potentially lead to more effective training. By showing that a PINN can be trained without data to find fast, reasonably accurate, consistent and flexible solutions, we set the stage for an improved hydrodynamical solver that can be coupled to both a numerical and a physics-informed solver for the corresponding morphodynamics to simulate the complete hydro-morphodynamical system behind tidal channel formation. This could potentially also be used in other hydrodynamical applications.