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    Area of Science:

    • Artificial Intelligence
    • Machine Learning
    • Scientific Discovery

    Background:

    • Symbolic regression discovers explainable analytic equations from data.
    • Neural networks excel in accuracy but lack interpretability and extrapolate poorly.
    • Bridging these techniques offers a path to interpretable and predictive AI.

    Purpose of the Study:

    • To integrate a neural network-based symbolic regression architecture, the equation learner (EQL) network, with deep learning systems.
    • To demonstrate the EQL network's capability in performing symbolic regression and learning function forms.
    • To evaluate the EQL network's performance on diverse tasks, including function learning, image-based arithmetic, and dynamical systems prediction.

    Main Methods:

    • Developed and implemented a neural network-based symbolic regression architecture: the equation learner (EQL) network.
    • Integrated the EQL network with other deep learning architectures for end-to-end training via backpropagation.
    • Tested the system on symbolic regression tasks, an MNIST arithmetic task using convolutional networks, and dynamical system prediction with encoder-based parameter extraction.

    Main Results:

    • The EQL network successfully performed symbolic regression, learning the form of various functions.
    • Demonstrated effective digit extraction using convolutional networks for an MNIST arithmetic task.
    • Showcased accurate prediction of dynamical systems, including extracting unknown parameters.
    • The EQL-based architecture exhibited superior extrapolation capabilities compared to standard neural networks.

    Conclusions:

    • The integrated EQL network effectively combines the strengths of symbolic regression and deep learning.
    • This approach enhances model interpretability and predictive accuracy, particularly in extrapolation.
    • The EQL architecture shows significant promise for advancing deep learning applications in scientific exploration and discovery.