Shape-morphing modes for reduced-order modeling of advection-dominated flows with shallow neural networks

Reduced-order modeling of fluid flows that are dominated by advection is notoriously difficult. We introduce a new method which tackles this issue by incorporating time-dependent shifts in the modes to which the flow is reduced. The evolution of the shift parameter is determined automatically using the method of reduced-order nonlinear solutions (RONS). We show that these shifts are equivalent to a rotation of the reduced linear subspace and can be interpreted as a shallow neural network with time-dependent biases. In addition, any number of conserved quantities of the flow can be readily enforced in the reduced model. We demonstrate the application of our method on a number of examples, including vortex dynamics and surface waves.