Recurrent Neural Networks (RNNs) have shown remarkable performances in system identification, particularly in nonlinear dynamical systems such as thermal processes. However, stability remains a critical challenge in practical applications: although the underlying process may be intrinsically stable, there may be no guarantee that the resulting RNN model captures this behavior. This paper addresses the stability issue by deriving a sufficient condition for Input-to-State Stability based on the infinity-norm (ISS$_{\infty}$) for Long Short-Term Memory (LSTM) networks. The obtained condition depends on fewer network parameters compared to prior works. A ISS$_{\infty}$-promoted training strategy is developed, incorporating a penalty term in the loss function that encourages stability and an ad hoc early stopping approach. The quality of LSTM models trained via the proposed approach is validated on a thermal system case study, where the ISS$_{\infty}$-promoted LSTM outperforms both a physics-based model and an ISS$_{\infty}$-promoted Gated Recurrent Unit (GRU) network while also surpassing non-ISS$_{\infty}$-promoted LSTM and GRU RNNs.

翻译：循环神经网络在系统辨识中表现出卓越的性能，尤其是在热过程等非线性动态系统中。然而，稳定性在实际应用中仍是一个关键挑战：尽管底层过程可能本质稳定，但无法保证所得到的RNN模型能够捕捉这种行为。本文通过推导长短期记忆网络基于无穷范数的输入-状态稳定性充分条件来解决该稳定性问题。相较于已有研究，所得条件依赖更少的网络参数。我们开发了一种ISS$_{\infty}$促进训练策略，通过在损失函数中加入促进稳定性的惩罚项，并采用特定的早停方法。通过热系统案例研究验证了采用所提方法训练的LSTM模型质量，其中ISS$_{\infty}$促进LSTM在性能上超越了基于物理的模型和ISS$_{\infty}$促进门控循环单元网络，同时也优于非ISS$_{\infty}$促进的LSTM与GRU循环神经网络。