A well-balanced meshless tsunami propagation and inundation model

We present a novel meshless tsunami propagation and inundation model. We discretize the nonlinear shallow-water equations using a well-balanced scheme relying on radial basis function based finite differences. The inundation model relies on radial basis function generated extrapolation from the wet points closest to the wet-dry interface into the dry region. Numerical results against standard one- and two-dimensional benchmarks are presented.Comment: 20 pages, 13 figure

M-ENIAC: A machine learning recreation of the first successful numerical weather forecasts

In 1950 the first successful numerical weather forecast was obtained by solving the barotropic vorticity equation using the Electronic Numerical Integrator and Computer (ENIAC), which marked the beginning of the age of numerical weather prediction. Here, we ask the question of how these numerical forecasts would have turned out, if machine learning based solvers had been used instead of standard numerical discretizations. Specifically, we recreate these numerical forecasts using physics-informed neural networks. We show that physics-informed neural networks provide an easier and more accurate methodology for solving meteorological equations on the sphere, as compared to the ENIAC solver.Comment: 10 pages, 1 figur

Casimir-dissipation stabilized stochastic rotating shallow water equations on the sphere

We introduce a structure preserving discretization of stochastic rotating shallow water equations, stabilized with an energy conserving Casimir (i.e. potential enstrophy) dissipation. A stabilization of a stochastic scheme is usually required as, by modeling subgrid effects via stochastic processes, small scale features are injected which often lead to noise on the grid scale and numerical instability. Such noise is usually dissipated with a standard diffusion via a Laplacian which necessarily also dissipates energy. In this contribution we study the effects of using an energy preserving selective Casimir dissipation method compared to diffusion via a Laplacian. For both, we analyze stability and accuracy of the stochastic scheme. The results for a test case of a barotropically unstable jet show that Casimir dissipation allows for stable simulations that preserve energy and exhibit more dynamics than comparable runs that use a Laplacian

Variational integrators for the rotating shallow water equations

The numerical simulation of the Earth’s atmosphere plays an important role in developing our understanding of climate change. The atmosphere and ocean can be seen as a shallow fluid on the globe; here, we use the shallow water equations as a first step to approximate these geophysical flows. Then, the numerical model can only be accurate if it has good conservation properties, e.g. without conserving mass the simulation can not be physical. Obtaining such a numerical model can be achieved using numerical variational integration. Here, we have derived a numerical variational integrator for the rotating shallow water equations on the sphere using the Euler–Poincaré framework. First, the continuous Lagrangian is discretized; then, the numerical scheme is obtained by computing the discrete variational principle. The conservational properties and accuracy of the model are verified with standard test cases. However, in order to obtain more realistic simulations, the shallow water equations need to include physical parametrizations. Thus, we introduce a new representation of the rotating shallow water equations based on a stochastic transport principle. Then, benchmarks are carried out to demonstrate that the spatial part of the stochastic scheme preserves the total energy. The proposed random model better captures the structure of a large-scale flow than a comparable deterministic model. Furthermore, to be able to carry out long term simulations we extend the discrete Euler–Poincaré framework with a selective decay. The selective decay dissipates an otherwise conserved quantity while conserving energy. We apply the new framework to the shallow water equations to dissipate the potential enstrophy. Then, we carry out standard benchmarks to demonstrate the conservation properties. We show that the selective decay resolves more small scales compared to a standard dissipation

Improving physics-informed DeepONets with hard constraints

Current physics-informed (standard or operator) neural networks still rely on accurately learning the initial conditions of the system they are solving. In contrast, standard numerical methods evolve such initial conditions without needing to learn these. In this study, we propose to improve current physics-informed deep learning strategies such that initial conditions do not need to be learned and are represented exactly in the predicted solution. Moreover, this method guarantees that when a DeepONet is applied multiple times to time step a solution, the resulting function is continuous.Comment: 15 pages, 5 figures, 4 tables; release versio

Rotating shallow water flow under location uncertainty: Part II: some numerical results

International audienc

M‐ENIAC: A Physics‐Informed Machine Learning Recreation of the First Successful Numerical Weather Forecasts

Abstract In 1950 the first successful numerical weather forecast was obtained by solving the barotropic vorticity equation using the Electronic Numerical Integrator and Computer (ENIAC), which marked the beginning of the age of numerical weather prediction. Here, we ask the question of how these numerical forecasts would have turned out, if machine learning based solvers had been used instead of standard numerical discretizations. Specifically, we recreate these numerical forecasts using physics‐informed neural networks. We show that physics‐informed neural networks provide an easier and more accurate methodology for solving meteorological equations on the sphere, as compared to the ENIAC solver

Computing the Ensemble Spread From Deterministic Weather Predictions Using Conditional Generative Adversarial Networks

Abstract Ensemble prediction systems are an invaluable tool for weather forecasting. Practically, ensemble predictions are obtained by running several perturbations of the deterministic control forecast. However, ensemble prediction is associated with a high computational cost and often involves statistical post‐processing steps to improve its quality. Here we propose to use deep‐learning‐based algorithms to learn the statistical properties of an ensemble prediction system, the ensemble spread, given only the deterministic control forecast. Thus, once trained, the costly ensemble prediction system will not be needed anymore to obtain future ensemble forecasts, and the statistical properties of the ensemble can be derived from a single deterministic forecast. We adapt the classical pix2pix architecture to a three‐dimensional model and train them against several years of operational (ensemble) weather forecasts for the 500 hPa geopotential height. The results demonstrate that the trained models indeed allow obtaining a highly accurate ensemble spread from the control forecast only

Rotating shallow water flow under location uncertainty: Part II: some numerical results

International audienc

Rotating shallow water flow under location uncertainty with a structure-preserving discretization

International audienceWe introduce a physically relevant stochastic representation of the rotating shallow waterequations. The derivation relies mainly on a stochastic transport principle and on a decomposition of thefluid flow into a large-scale component and a noise term that models the unresolved flow components. Asfor the classical (deterministic) system, this scheme, referred to as modeling under location uncertainty (LU),conserves the global energy of any realization and provides the possibility to generate an ensemble of physicallyrelevant random simulations with a good trade-off between the model error representation and the ensemble'sspread. To maintain numerically the energy conservation feature, we combine an energy (in space) preservingdiscretization of the underlying deterministic model with approximations of the stochastic terms that are basedon standard finite volume/difference operators. The LU derivation, built from the very same conservationprinciples as the usual geophysical models, together with the numerical scheme proposed can be directly usedin existing dynamical cores of global numerical weather prediction models. The capabilities of the proposedframework is demonstrated for an inviscid test case on the f-plane and for a barotropically unstable jet on thesphere