  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.

Computational Physics