Physicsinformed dynamic mode decomposition (piDMD)
Abstract
In this work, we demonstrate how physical principles  such as symmetries, invariances, and conservation laws  can be integrated into the dynamic mode decomposition (DMD). DMD is a widelyused data analysis technique that extracts lowrank modal structures and dynamics from highdimensional measurements. However, DMD frequently produces models that are sensitive to noise, fail to generalize outside the training data, and violate basic physical laws. Our physicsinformed DMD (piDMD) optimization, which may be formulated as a Procrustes problem, restricts the family of admissible models to a matrix manifold that respects the physical structure of the system. We focus on five fundamental physical principles  conservation, selfadjointness, localization, causality, and shiftinvariance  and derive several closedform solutions and efficient algorithms for the corresponding piDMD optimizations. With fewer degrees of freedom, piDMD models are less prone to overfitting, require less training data, and are often less computationally expensive to build than standard DMD models. We demonstrate piDMD on a range of challenging problems in the physical sciences, including energypreserving fluid flow, travellingwave systems, the Schrödinger equation, solute advectiondiffusion, a system with causal dynamics, and threedimensional transitional channel flow. In each case, piDMD significantly outperforms standard DMD in metrics such as spectral identification, state prediction, and estimation of optimal forcings and responses.
 Publication:

arXiv eprints
 Pub Date:
 December 2021
 arXiv:
 arXiv:2112.04307
 Bibcode:
 2021arXiv211204307B
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Numerical Analysis;
 Mathematics  Optimization and Control;
 Physics  Data Analysis;
 Statistics and Probability;
 Physics  Fluid Dynamics
 EPrint:
 36 pages, 9 figures