Symbolic Regression via Neural Networks

Machine learning - specifically deep learning - techniques have shown their capabilities in approximating dynamics from data, but a shortcoming of traditional deep learning is that there is little insight into the underlying mapping beyond its numerical output for a given input. This limits their utility in analysis beyond simple prediction. Simultaneously, a number of strategies exist which identify models based on a fixed dictionary of basis functions, but most either require some intuition or insight about the system, or are susceptible to overfitting or a lack of parsimony. Here we present a novel approach that combines the flexibility and accuracy of deep learning approaches with the utility of symbolic solutions: a deep neural network that generates a symbolic expression for the governing equations. We first describe the architecture for our model, then show the accuracy of our algorithm across a range of classical dynamical systems. The dynamics of quantities of interest are widely modeled A number of authors have approached system identificaas differential equations, often derived from first princi-tion by fitting coefficients of a linear combination of basis 3ples. However, this is not always possible, especially whenfunctions, dating at least back to Crutchfield and McNamara . The The set of basis functions typically includes nonlinear terms, identification of models from data has seen significant ad-for example terms which would arise in a Taylor series exvances with the advent of machine learning. While deeppansion about the origin of the system3-6 or a broader class neural networks have enabled sufficient accuracy in fore-of functions7. The coefficients of the basis functions are decasting dynamic data with unprecedented versatility, thetermined through comparison of the original data points with models they represent lack closed-form expressions thatpoints from computed solutions to the fitted models. Varican be conducive to interpretation and analysis.