PCDMD: Physics-constrained dynamic mode decomposition for accurate and robust forecasting of dynamical systems with imperfect data and physics

The dynamic mode decomposition (DMD) method has attracted widespread attention as a representative modal-decomposition method and can build a predictive model. However, DMD may give predicted results that deviate from physical reality in some scenarios, such as dealing with translation problems or noisy data. Here, we propose a physics-constrained DMD (PCDMD) method to address this issue. The proposed PCDMD method first employs a data-driven model using DMD, then calculates the residual of the physical equations, and finally corrects the predicted results using Kalman filter and gain coefficients. In this way, the PCDMD method can integrate the physics-informed equations with the data-driven model generated by DMD. Numerical experiments are conducted using PCDMD, including the Allen–Cahn, advection-diffusion, Burgers' equations and lid-driven cavity flow. The results demonstrate that the proposed PCDMD method can reduce the reconstruction and prediction errors by 1%-10% by incorporating physical constraints. Regarding noisy datasets and imperfect physical constraints, PCDMD can still ensure that the predicted results satisfy the physical constraints, thereby reducing errors. Program title: PCDMD. Dataset link: https://github.com/YinYuhuiTJU/PCDMD. Licensing provisions: MIT. Programming language: Python