Yves Atchade

Indrajit Kalita, Lucia Vilallonga, Yves Atchade

With a warming planet, tropical regions are expected to experience the brunt of climate change, with more intense and more volatile rainfall events. Currently, state-of-the-art numerical weather prediction (NWP) models are known to struggle to produce skillful rainfall forecasts in tropical regions of Africa. There is thus a pressing need for improved rainfall forecasting in these regions. Over the last decade or so, the increased availability of large-scale meteorological datasets and the development of powerful machine learning models have opened up new opportunities for data-driven weather forecasting. Focusing on Ghana in this study, we use these tools to develop two U-Net convolutional neural network (CNN) models, to predict 24h rainfall at 12h and 30h lead-time. The models were trained using data from the ERA5 reanalysis dataset, and the GPM-IMERG dataset. A special attention was paid to interpretability. We developed a novel statistical methodology that allowed us to probe the relative importance of the meteorological variables input in our model, offering useful insights into the factors that drive precipitation in the Ghana region. Empirically, we found that our 12h lead-time model has performances that match, and in some accounts are better than the 18h lead-time forecasts produced by the ECMWF (as available in the TIGGE dataset). We also found that combining our data-driven model with classical NWP further improves forecast accuracy.

Yi Sun, Debarghya Mukherjee, Yves Atchade

Solving partial differential equations (PDEs) and their inverse problems using Physics-informed neural networks (PINNs) is a rapidly growing approach in the physics and machine learning community. Although several architectures exist for PINNs that work remarkably in practice, our theoretical understanding of their performances is somewhat limited. In this work, we study the behavior of a Bayesian PINN estimator of the solution of a PDE from $n$ independent noisy measurement of the solution. We focus on a class of equations that are linear in their parameters (with unknown coefficients $θ_\star$). We show that when the partial differential equation admits a classical solution (say $u_\star$), differentiable to order $β$, the mean square error of the Bayesian posterior mean is at least of order $n^{-2β/(2β+ d)}$. Furthermore, we establish a convergence rate of the linear coefficients of $θ_\star$ depending on the order of the underlying differential operator. Last but not least, our theoretical results are validated through extensive simulations.