Convergence and error analysis of PINNs

Physics-informed neural networks (PINNs) are a promising approach that combines the power of neural networks with the interpretability of physical modeling. PINNs have shown good practical performance in solving partial differential equations (PDEs) and in hybrid modeling scenarios, where physical models enhance data-driven approaches. However, it is essential to establish their theoretical properties in order to fully understand their capabilities and limitations. In this study, we highlight that classical training of PINNs can suffer from systematic overfitting. This problem can be addressed by adding a ridge regularization to the empirical risk, which ensures that the resulting estimator is risk-consistent for both linear and nonlinear PDE systems. However, the strong convergence of PINNs to a solution satisfying the physical constraints requires a more involved analysis using tools from functional analysis and calculus of variations. In particular, for linear PDE systems, an implementable Sobolev-type regularization allows to reconstruct a solution that not only achieves statistical accuracy but also maintains consistency with the underlying physics

Human spatial dynamics for electricity demand forecasting: the case of France during the 2022 energy crisis

Accurate electricity demand forecasting is crucial to meet energy security and efficiency, especially when relying on intermittent renewable energy sources. Recently, massive savings have been observed in Europe, following an unprecedented global energy crisis. However, assessing the impact of such crisis and of government incentives on electricity consumption behaviour is challenging. Moreover, standard statistical models based on meteorological and calendar data have difficulty adapting to such brutal changes. Here, we show that mobility indices based on mobile network data significantly improve the performance of the state-of-the-art models in electricity demand forecasting during the sobriety period. We start by documenting the drop in the French electricity consumption during the winter of 2022-2023. We then show how our mobile network data captures work dynamics and how adding these mobility indices outperforms the state-of-the-art during this atypical period. Our results characterise the effect of work behaviours on the electricity demand

Human spatial dynamics for electricity demand forecasting: the case of France during the 2022 energy crisis

<p>This is a dataset for electricity load forecasting in France between 2013 and 2023. <i>dataset_national.csv </i>is at the national level and <i>dataset_regional.csv</i> at the regional one. It contains seasonal data from the French open platform, electricity load data from RTE, and weather data from MétéoFrance. The code for the creation of the dataset and the benchmark of electricity load forecasting related to it is available at https://github.com/NathanDoumeche/Mobility_data_assimilation.</p&gt

Forecasting Electric Vehicle Charging Station Occupancy: Smarter Mobility Data Challenge

The transport sector is a major contributor to greenhouse gas emissions in Europe. Shifting to electric vehicles (EVs) powered by a low-carbon energy mix would reduce carbon emissions. However, to support the development of electric mobility, a better understanding of EV charging behaviours and more accurate forecasting models are needed. To fill that gap, the Smarter Mobility Data Challenge has focused on the development of forecasting models to predict EV charging station occupancy. This challenge involved analysing a dataset of 91 charging stations across four geographical areas over seven months in 2020-2021. The forecasts were evaluated at three levels of aggregation (individual stations, areas and global) to capture the inherent hierarchical structure of the data. The results highlight the potential of hierarchical forecasting approaches to accurately predict EV charging station occupancy, providing valuable insights for energy providers and EV users alike. This open dataset addresses many real-world challenges associated with time series, such as missing values, non-stationarity and spatio-temporal correlations. Access to the dataset, code and benchmarks are available at https://gitlab.com/smarter-mobility-data-challenge/tutorials to foster future research