Modeling active fluids via physically constrained machine learning

We investigate an experimental fluid flow driven by microtubules confined to an oil/water interface. Deriving a mathematical model of this active fluid from first principles is difficult, as not all the relevant physical processes are well understood. Instead, we use sparse physics-informed discovery of empirical relations (SPIDER) to learn the governing equations directly from experimental data. General physical constraints such as locality, causality, and symmetry are used to construct libraries of candidate relations between the flow field and the director field describing the orientation of microtubules. Sparse regression is then used to identify a parsimonious two-dimensional model of this system. Three PDEs are identified from data: an incompressibility condition and momentum balance describing the fluid flow and a separate equation for the director field. The latter two governing equations are distinct from those appearing in the literature. In particular, neither the advection terms nor the time derivative of the flow velocity appears in the momentum equation, consistent with the low Reynold's number of the flow. We also find that elastic effects cannot be described by weakly nonlinear terms in the evolution equation for the director field.