Beam breakup (BBU) instability limits the performance of energy-recovery linacs (ERLs). This study presents a new approach to estimating the BBU threshold current using a physics-inspired neural network. The application of neural networks to the highly nonlinear beam dynamics of ERLs opens a new paradigm for the development of efficient future particle accelerators. It is found that figure-of-merit-based data can predict the threshold current with higher accuracy than directly feeding the proposed network with raw parameter-based data. This study investigates how embedding such a physically meaningful feature influences predictive performance and generalization. In this work, the effect of the variation in HOM (higher-order modes) parameters due to the geometrical tolerances in the RF cavity fabrication on the BBU threshold current is analyzed with the proposed multi-layered Bidirectional Long Short-Term Memory (BiLSTM) neural network.