Because of physical assumptions and numerical approximations, low-order models are affected by uncertainties in the state and parameters, and by model biases. Model biases, also known as model errors or systematic errors, are difficult to infer because they are `unknown unknowns', i.e., we do not necessarily know their functional form a priori. With biased models, data assimilation methods may be ill-posed because either (i) they are 'bias-unaware' because the estimators are assumed unbiased, (ii) they rely on an a priori parametric model for the bias, or (iii) they can infer model biases that are not unique for the same model and data. First, we design a data assimilation framework to perform combined state, parameter, and bias estimation. Second, we propose a mathematical solution with a sequential method, i.e., the regularized bias-aware ensemble Kalman Filter (r-EnKF), which requires a model of the bias and its gradient (i.e., the Jacobian). Third, we propose an echo state network as the model bias estimator. We derive the Jacobian of the network, and design a robust training strategy with data augmentation to accurately infer the bias in different scenarios. Fourth, we apply the r-EnKF to nonlinearly coupled oscillators (with and without time-delay) affected by different forms of bias. The r-EnKF infers in real-time parameters and states, and a unique bias. The applications that we showcase are relevant to acoustics, thermoacoustics, and vibrations; however, the r-EnKF opens new opportunities for combined state, parameter and bias estimation for real-time and on-the-fly prediction in nonlinear systems.

翻译：由于物理假设和数值近似，低阶模型会受到状态与参数不确定性以及模型偏差的影响。模型偏差（亦称模型误差或系统误差）难以推断，因其属于“未知的未知”，即我们未必先验知晓其函数形式。使用有偏模型时，数据同化方法可能面临不适定性，原因在于：（i）它们“忽略偏差”——假设估计量无偏；（ii）依赖偏差的先验参数化模型；或（iii）对相同模型与数据推断出的模型偏差不唯一。首先，我们设计了一个数据同化框架，用于实现状态、参数与偏差的联合估计。其次，我们提出一种基于序贯方法的数学方案——正则化偏感知集合卡尔曼滤波（r-EnKF），该方法需要偏差模型及其梯度（即雅可比矩阵）。第三，我们采用回声状态网络作为模型偏差估计器，推导了该网络的雅可比矩阵，并通过数据增强设计出稳健的训练策略，以在不同场景下精确推断偏差。第四，将r-EnKF应用于受不同形式偏差影响的非线性耦合振子（含时滞与不含时滞）。r-EnKF能实时估计参数、状态及唯一偏差值。本文展示的应用与声学、热声学及振动相关；然而，r-EnKF为非线性系统中实时与动态预测的状态、参数与偏差联合估计开辟了新途径。