Symbolic Regression via Neural Networks

Abstract

Identifying governing equations for a dynamical system is a topic of critical interest across an array of disciplines, from mathematics to engineering to biology. 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.

References in corpus (15)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Reconciling modern machine learning practice and the bias-variance trade-off
• Deep learning for universal linear embeddings of nonlinear dynamics
• Applied Koopmanism
• Next Generation Reservoir Computing
• Ergodic theory, Dynamic Mode Decomposition and Computation of Spectral Properties of the Koopman operator
• Predicting catastrophes in nonlinear dynamical systems by compressive sensing
• On Mean Absolute Error for Deep Neural Network Based Vector-to-Vector Regression
• Model selection for dynamical systems via sparse regression and information criteria
• On Explaining the Surprising Success of Reservoir Computing Forecaster of Chaos? The Universal Machine Learning Dynamical System with Contrasts to VAR and DMD
• Prediction of Dynamical Systems by Symbolic Regression
• How Entropic Regression Beats the Outliers Problem in Nonlinear System Identification
• Interpretable Polynomial Neural Ordinary Differential Equations
• Finding nonlinear system equations and complex network structures from data: a sparse optimization approach