Echo State Networks (ESNs) are recurrent neural networks usually employed for modeling nonlinear dynamic systems with relatively ease of training. By incorporating physical laws into the training of ESNs, Physics-Informed ESNs (PI-ESNs) were proposed initially to model chaotic dynamic systems without external inputs. They require less data for training since Ordinary Differential Equations (ODEs) of the considered system help to regularize the ESN. In this work, the PI-ESN is extended with external inputs to model controllable nonlinear dynamic systems. Additionally, an existing self-adaptive balancing loss method is employed to balance the contributions of the residual regression term and the physics-informed loss term in the total loss function. The experiments with two nonlinear systems modeled by ODEs, the Van der Pol oscillator and the four-tank system, and with one differential-algebraic (DAE) system, an electric submersible pump, revealed that the proposed PI-ESN outperforms the conventional ESN, especially in scenarios with limited data availability, showing that PI-ESNs can regularize an ESN model with external inputs previously trained on just a few datapoints, reducing its overfitting and improving its generalization error (up to 92% relative reduction in the test error). Further experiments demonstrated that the proposed PI-ESN is robust to parametric uncertainties in the ODE equations and that model predictive control using PI-ESN outperforms the one using plain ESN, particularly when training data is scarce.

翻译：回声状态网络（ESN）是一种循环神经网络，通常用于相对易于训练的非线性动态系统建模。通过将物理定律融入ESN的训练过程，物理信息ESN（PI-ESN）最初被提出用于建模无外部输入的混沌动态系统。由于所考虑系统的常微分方程（ODE）有助于正则化ESN，该方法需要更少的训练数据。本研究将PI-ESN扩展至包含外部输入，以建模可控非线性动态系统。此外，采用现有的自适应平衡损失方法，以平衡总损失函数中残差回归项与物理信息损失项的贡献。通过对范德波尔振荡器和四水箱系统两个ODE建模的非线性系统，以及电潜泵这一微分代数方程（DAE）系统进行实验，结果表明所提出的PI-ESN优于传统ESN，特别是在数据有限的情况下。实验证明PI-ESN能够正则化仅用少量数据点预训练的外部输入ESN模型，减少其过拟合并改善泛化误差（测试误差相对降低高达92%）。进一步实验表明，所提出的PI-ESN对ODE方程中的参数不确定性具有鲁棒性，且使用PI-ESN的模型预测控制优于使用普通ESN的方法，在训练数据稀缺时尤为显著。