Symbolic regression is a powerful technique to discover analytic equations that describe data, which can lead to explainable models and the ability to predict unseen data. In contrast, neural networks have achieved amazing levels of accuracy on image recognition and natural language processing tasks, but they are often seen as black-box models that are difficult to interpret and typically extrapolate poorly. In this article, we use a neural network-based architecture for symbolic regression called the equation learner (EQL) network and integrate it with other deep learning architectures such that the whole system can be trained end-to-end through backpropagation. To 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 convolutional network extracts the digits. Finally, we demonstrate the 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 with a standard neural network-based architecture, paving the way for deep learning to be applied in scientific exploration and discovery.

A black-box method for modeling the time-domain response of integrated circuits (ICs) based on echo state networks is proposed. The number and value of the input and feedback delays required for modeling nonlinear systems with memory are detected automatically, and the training procedure is very fast and robust. The resulting models can be implemented in any hardware description language. The proposed modeling approach is applied to three test cases that cover a wide range of analog behavior. The achieved accuracy is comparable to the reference modeling method, and, depending on the complexity of the modeled IC, the simulation speed-up factor is in the order between 10 and 100.