Learning and enforcing physical structure with physics-informed dynamic mode decomposition (piDMD)*

Data-driven models that respect physical laws are robust to noise, require few training samples, and are highly generalisable. Although the dynamic mode decomposition (DMD) is a principal tool of data-driven fluid dynamics, it is rare for learned DMD models to obey physical laws such as symmetries, invariances, causalities, spatial locality and conservation laws. Thus, we present physics-informed dynamic mode decomposition (piDMD), a suite of tools that incorporate physical structures into linear system identification. Specifically, we develop efficient and accurate algorithms that produce DMD models that obey the matrix-analogues of user-specified physical constraints. Through a range of examples from fluid dynamics, we demonstrate the improved diagnostic, predictive and interpretative abilities of piDMD. We consider examples from stability analysis, data-driven resolvent analysis, reduced-order modelling, control, and the low-data and high-noise regimes. Conversely, if the physical structures are unknown then, through cross-validation, piDMD can be used to discover the physical structures present in the observed system. Finally, we show that the time and memory requirements of piDMD are competitive with standard DMD approaches.