Physics-Constrained Forecasting of Fluid Dynamics with Reservoir Computing

We present a new approach for incorporating physical constraints into Reservoir Computers (RCs). The long-term aim of this work is to increase the interpretability of RCs while mitigating the computational costs associated with training RCs for high dimensional problems, such as spatiotemporal fluid flows. A Reservoir Computer is, itself, a dynamical system, commonly implemented as a single-layer recurrent neural network in which only the linear output layer is trained and all other parameters are randomly initialized and fixed. Therefore, training an RC only involves solving a least squares problem which—while interpretable and efficient for small problems—scales poorly for problems of higher dimension. Due in part to this limitation, RCs have seldom been applied to spatial fluid flows despite significant interest in doing so. We show that physical constraints such as conservation laws and boundary conditions can be imposed in the training procedure and can be guaranteed to hold for forecasting. To enforce the constraints, we modify the RC training procedure with a linear homogeneous constraint represented by a differentiation matrix. We demonstrate the efficacy of this method by imposing zero-divergence and periodic boundary condition constraints for 2D incompressible fluid flow.