Gearboxes often operate under varying speed conditions, making them susceptible to tooth crack faults over time. Failure to identify these faults early can lead to progression and potentially severe accidents. This study focuses on enhancing the detection of gearbox faults under varying speed conditions. A speed-integrated long short-term memory (SI-LSTM) is developed, with hyperparameters optimized through a physics-informed hyperparameter selection strategy aimed at maximizing the discrepancy between healthy and physics-informed faulty states. Artificially induced tooth crack impulses are incorporated into the validation dataset to simulate faulty state data. A case study using a helical gearbox dataset demonstrates that the physics-informed SI-LSTM model, based on the maximal discrepancy, outperforms the traditional SI-LSTM model optimized for minimal mean square error in detecting tooth crack faults.