A robust sparse identification of nonlinear dynamics approach by combining neural networks and an integral form
Ali Forootani, Pawan Goyal, Peter Benner
Abstract
One widely used methodology for uncovering governing equations from data is sparse regression for nonlinear dynamics , commonly known as Sparse Identification of Nonlinear Dynamics ( SINDy ). However, noisy and limited data remain a significant challenge for the success of the SINDy approach. In this work, we propose a robust strategy to discover nonlinear governing equations from both noisy and scarce data. Specifically, we employ neural networks to learn an implicit representation from measurement data, thereby ensuring that the network output remains close to the measurements while also admitting a dynamical system interpretation for its time evolution. Moreover, we identify this dynamical system in the spirit of the SINDy framework. By leveraging the neural network’s implicit representation, we employ automatic differentiation to obtain the derivative information required by SINDy . To further enhance the robustness of our approach, we incorporate an integral constraint on the output of the implicit networks. In addition, we extend our method to handle data acquired from multiple initial conditions. Through several examples, we demonstrate the proposed method’s effectiveness in discovering governing equations under noisy, data-scarce conditions and compare its performance against existing methods.