For real-life nonlinear systems, the exact form of nonlinearity is often not known and the known governing equations are often based on certain assumptions and approximations. Such representation introduced model-form error into the system. In this paper, we propose a novel gray-box modeling approach that not only identifies the model-form error but also utilizes it to improve the predictive capability of the known but approximate governing equation. The primary idea is to treat the unknown model-form error as a residual force and estimate it using duel Bayesian filter based joint input-state estimation algorithms. For improving the predictive capability of the underlying physics, we first use machine learning algorithm to learn a mapping between the estimated state and the input (model-form error) and then introduce it into the governing equation as an additional term. This helps in improving the predictive capability of the governing physics and allows the model to generalize to unseen environment. Although in theory, any machine learning algorithm can be used within the proposed framework, we use Gaussian process in this work. To test the performance of proposed framework, case studies discussing four different dynamical systems are discussed; results for which indicate that the framework is applicable to a wide variety of systems and can produce reliable estimates of original system's states.
翻译:对于实际存在的非线性系统,非线性的确切形式往往不为人所知,已知的治理方程式往往以某些假设和近似值为基础。这种表示方式在系统中引入了模型形式错误。在本文中,我们提议一种新的灰箱模型方法,不仅识别模型形式错误,而且还利用它来提高已知但近似治理方程式的预测能力。主要想法是将未知模型形式错误作为残余力处理,并使用基于比亚西亚过滤器的联合投入状态估计算法来估计它。为了提高基础物理的预测能力,我们首先使用机器学习算法来学习对估计状态和输入的映射(模型形式错误),然后将其引入管理方程式,作为额外的术语。这有助于提高管理物理的预测能力,使模型能够概括到看不见的环境。虽然在理论上,任何机器学习算法都可以在拟议框架内使用,但我们在这项工作中使用了高斯语进程。测试拟议框架的性能,我们首先使用机器学习算法来学习四个不同的动态系统;讨论四个动态系统的案例研究,然后将结果作为管理方程的补充,从而显示最初的系统。