NN-Poly: Approximating common neural networks with Taylor polynomials to imbue dynamical system constraints

Recent advances in deep learning have bolstered our ability to forecast the evolution of dynamical systems, but common neural networks do not adhere to physical laws, critical information that could lead to sounder state predictions. This contribution addresses this concern by proposing a neural network to polynomial (NN-Poly) approximation, a method that furnishes algorithmic guarantees of adhering to physics while retaining state prediction accuracy. To achieve these goals, this article shows how to represent a trained fully connected perceptron, convolution, and recurrent neural networks of various activation functions as Taylor polynomials of arbitrary order. This solution is not only analytic in nature but also least squares optimal. The NN-Poly system identification or state prediction method is evaluated against a single-layer neural network and a polynomial trained on data generated by dynamic systems. Across our test cases, the proposed method maintains minimal root mean-squared state error, requires few parameters to form, and enables model structure for verification and safety. Future work will incorporate safety constraints into state predictions, with this new model structure and test high-dimensional dynamical system data.