Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery

--Symbolic regression is a powerful technique that can discover analytical equations that describe data, which can lead to explainable models and generalizability outside of the training data set. In contrast, neural networks have achieved amazing levels of accuracy on image recognition and natural language processing tasks, but are often seen as black-box models that are difficult to interpret and typically extrapolate poorly. Here we use a neural network-based architecture for symbolic regression that we call the Sequential Equation Learner (SEQL) network and integrate it with other deep learning architectures such that the whole system can be trained end-to-end through backpropagation. T o demonstrate the power of such systems, we study their performance on several substantially different tasks. First, we show that the neural network can perform symbolic regression and learn the form of several functions. Next, we present an MNIST arithmetic task where a separate part of the neural network extracts the digits. Finally, we demonstrate prediction of dynamical systems where an unknown parameter is extracted through an encoder . We find that the EQL-based architecture can extrapolate quite well outside of the training data set compared to a standard neural network-based architecture, paving the way for deep learning to be applied in scientific exploration and discovery. Many complex phenomena in science and engineering can be reduced to general models that can be described in terms of relatively simple mathematical equations. For example, classical electrodynamics can be described by Maxwell's equations and non-relativistic quantum mechanics can be described by the Schr odinger equation. These models elucidate the underlying dynamics of a particular system and can provide general predictions over a very wide range of conditions. On the other hand, modern machine learning techniques have become increasingly powerful for many tasks including image recognition and natural language processing, but the neural network-based architectures in these state-of-the-art techniques are black-box models that often make them difficult for use in scientific exploration.