State estimation in minimal turbulent channel flow: A comparative study of 4DVar and PINN

The state of turbulent, minimal-channel flow is estimated from spatio-temporal sparse observations the velocity, using both a physics-informed neural network (PINN) and adjoint-variational data assimilation (4DVar). performance PINN assessed against benchmark results 4DVar. efficient to implement, takes advantage automatic differentiation evaluate governing equations, does not require development an adjoint model. In addition, evolution expressed in terms parameters which have far smaller dimension than predicted trajectory space or even just initial condition flow. Provided adequate observations, architecture training, can yield satisfactory estimates field, for missing velocity entirely unobserved pressure field. However, accuracy depends on architecture, dependence known priori. comparison 4DVar estimation becomes progressively more accurate over observation horizon, predictions are generally less maintain same level errors throughout time window. Another notable distinction capacity accurately forecast evolution: while prediction depart true gradually according Lyapunov exponent, inaccurate immediately beyond training horizon unless re-trained. Most importantly, satisfies discrete form equations point-wise machine precision, only satisfied L2 sense.