PF-DMD: Physics-fusion dynamic mode decomposition for accurate and robust forecasting of dynamical systems with imperfect data and physics

The DMD (Dynamic Mode Decomposition) method has attracted widespread attention as a representative modal-decomposition method and can build a predictive model. However, the DMD may give predicted results that deviate from physical reality in some scenarios, such as dealing with translation problems or noisy data. Therefore, this paper proposes a physics-fusion dynamic mode decomposition (PFDMD) method to address this issue. The proposed PFDMD method first obtains a data-driven model using DMD, then calculates the residual of the physical equations, and finally corrects the predicted results using Kalman filtering and gain coefficients. In this way, the PFDMD method can integrate the physics-informed equations with the data-driven model generated by DMD. Numerical experiments are conducted using the PFDMD, including the Allen-Cahn, advection-diffusion, and Burgers' equations. The results demonstrate that the proposed PFDMD method can significantly reduce the reconstruction and prediction errors by incorporating physics-informed equations, making it usable for translation and shock problems where the standard DMD method has failed.