This challenge becomes apparent when the model on batteries, particularly within the realm of sustainable deviates from the expert's path and continues to make errors mobility driven by electric vehicles (EVs) [1]. This transition that lead it into unfamiliar states, thus exacerbating the underscores the vital role of batteries in promoting ecofriendly initial mistake [11]. Dataset Aggregation (DAGGER) was transportation. However, it also highlights the pressing introduced by [12] as a method to address the challenge of need to enhance battery efficiency, long-lasting battery distributional shift. This iterative algorithm aims to minimize performance, and safety, particularly during the charging the compounding of errors resulting from the shift by iteratively phase. To address these challenges, advanced battery management integrating the decisions made by both the learning systems, often employing Model Predictive Control model and an expert policy. This integration prevents the (MPC), have gained prominence [2], [3].

Combining machine learning with physics is a trending approach for discovering unknown dynamics, and one of the most intensively studied frameworks is the physics-informed neural network (PINN). However, PINN often fails to optimize the network due to its difficulty in concurrently minimizing multiple losses originating from the system's governing equations. This problem can be more serious when the system's states are unmeasurable, like lithium-ion batteries (LiBs). In this work, we introduce a robust method for training PINN that uses fewer loss terms and thus constructs a less complex landscape for optimization. In particular, instead of having loss terms from each differential equation, this method embeds the dynamics into a loss function that quantifies the error between observed and predicted system outputs. This is accomplished by numerically integrating the predicted states from the neural network(NN) using known dynamics and transforming them to obtain a sequence of predicted outputs. Minimizing such a loss optimizes the NN to predict states consistent with observations given the physics. Further, the system's parameters can be added to the optimization targets. To demonstrate the ability of this method to perform various modeling and control tasks, we apply it to a battery model to concurrently estimate its states and parameters.