## 1 Introduction  
Federated learning (FL) [1, 2, 3] enables collaborative model training while preserving data privacy [4] by keeping the raw data decentralised. In each collaborative round, clients update the global model locally using private training data and then transmit gradients to the central parameter server [5] for aggregation (typically by averaging) and global update, thereby addressing privacy concerns by exposing the raw data [6]. After the aggregation of these updates by the central server, the new model parameters are sent back to the clients and the global training round is completed [7]. This method is particularly relevant for deep neural networks (DNNs), which are increasingly used in various applications due to their ability to model complex patterns [8]. However, training DNNs in an FL environment poses some challenges, particularly with respect to gradient sharing for model updates [9, 10, 11], which highlights the possibility of reconstructing the original input from the shared gradients (especially in the absence of protection methods) resulting from the transfer of model gradients and parameters. However, recent studies [12, 13] have shown that shared gradients in FL can be exploited by gradient inversion attacks (GIAs), which could allow attackers to reconstruct private training data. While much of the research has focused on the reconstruction of image and text data, in this paper, we explore the application of GIAs using a schematic approach of privacy auditing to time-series data [14; 15] in the context of with and without differential privacy (DP). The project involves data from lithium-ion batteries used in electric vehicles (EVs), where protecting user-sensitive information--such as mileage, usage patterns, and battery health--is a key concern. At the same time, it is essential to collaboratively train a common Artificial Intelligence (AI) model for battery behaviour prediction using federated learning (FL). Analysing time-series data, such as the ageing of lithium-ion batteries, plays a crucial role in understanding patterns in data over time. It enables the prediction and analysis of performance degradation, the optimisation of maintenance schedules, the improvement in reliability and useful life extension, not only in battery ageing but also in similar critical systems through data-driven decisions. Accurate predictions of battery ageing (e.g., capacity degradation or remaining useful life) can improve the safe and reliable operation of battery electric vehicles, facilitate eco-transportation systems in smart cities and recommend appropriate points of interest [16]. There is, therefore, great interest in extending or at least predicting the remaining lifetime of batteries [17]. Recent studies on the ageing of lithium-ion batteries (LIBs) (e.g., persistent degradation during their lifetime) include data-based modelling techniques such as empirical models, equivalent circuit models, machine learning-based and deep learning modelling approaches [18]. Machine learning models provide predictive capabilities for the ageing of LIBs, while deep learning models use algorithms such as deep neural networks. To adapt GIAs for time-series data in FL, sequence modelling approaches such as transformer-based models or recurrent neural networks are used to capture temporal dependencies in the reconstructed data. Transformers in particular have a great capability in learning time-series and can be scaled for distributed, collaborative or FL scenarios [19]. We address the case where an adversary--such as a compromised or untrusted server--has access to the model updates (i.e., gradient updates, \(\Delta W\)) from a single client prior to aggregation. In this setting, the adversary may attempt to exploit the exposed updates to reconstruct the client's original input signal. Importantly, the number of federated learning (FL) clients is not a limiting factor in this attack model, as the adversary operates before any averaging or aggregation occurs. **Motivation for the research.