Dynamic system models, based on partial differential equations (PDEs), are often unsuitable for direct use in control or state estimation purposes, due to the high computational cost arising from the necessity to apply sophisticated numerical methods for a solution, such as semi-discretization, also known as spatial discretization. Hence, there is often an inevitable trade-off between accuracy and computational efficiency during the model reduction step to ensure real-time applicability.

In this contribution, we propose a state–space model formulation, using so-called physics-informed neural networks. This modeling approach enables a highly efficient inclusion of complex physical system descriptions within the design of control or state estimation setups. The resulting state–space model does not require any numerical solution techniques during the state propagation, as each time step is based on the evaluation of a reasonably sized neural net that approximates the solution of the PDE. Thus, this approach is suitable for real-time applications of various complex dynamic systems that can be described by one or a set of PDEs.

Besides the modeling approach itself, the contribution also provides an illustrative example of the state–space modeling method in the context of model predictive control, as well as state estimation with an extended Kalman filter. These methods will be applied to a system based on a numerical solution of the Burgers equation.

基于偏微分方程（PDE）的动态系统模型通常不适合直接用于控制或状态估计，这是由于需要将复杂的数值方法用于解决方案（例如半离散化）而导致的高计算成本，也称为空间离散化。因此，在模型简化步骤中，为了确保实时适用性，常常在准确性和计算效率之间不可避免地要权衡取舍。

在这项贡献中，我们提出了一种状态空间模型的公式化，它使用了所谓的物理信息神经网络。这种建模方法可以在控制或状态估计设置的设计中高效地包含复杂的物理系统描述。最终的状态空间模型在状态传播过程中不需要任何数值解技术，因为每个时间步都是基于对合理大小的神经网络（近似于PDE的解）的评估。因此，该方法适用于可以由一个或一组PDE进行描述的各种复杂动态系统的实时应用。

除了建模方法本身之外，该贡献还提供了模型预测控制和扩展卡尔曼滤波器状态估计中的状态空间建模方法的示例。这些方法将应用于基于Burgers方程数值解的系统。