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    This study introduces a new neural network for symbolic regression, enabling the discovery of governing equations in complex parametric systems. The method enhances scientific discovery by analyzing high-dimensional data and extrapolating beyond training domains.

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    Area of Science:

    • * Computational Physics
    • * Machine Learning
    • * Scientific Discovery

    Background:

    • * Symbolic regression (SR) is limited in analyzing complex and high-dimensional systems.
    • * Deep learning excels at handling complex, high-dimensional datasets.
    • * Integrating SR with deep learning offers potential for scientific advancement.

    Purpose of the Study:

    • * To extend symbolic regression to parametric systems with varying coefficients.
    • * To develop a neural network architecture for enhanced symbolic regression.
    • * To demonstrate the scalability and applicability of the proposed method.

    Main Methods:

    • * Proposed a novel neural network architecture for symbolic regression.
    • * Applied the method to analytic expressions and partial differential equations (PDEs) with varying coefficients.
    • * Integrated a convolutional encoder for analyzing high-dimensional image data of spring systems.

    Main Results:

    • * The method successfully learned governing equations for parametric systems.
    • * Demonstrated good extrapolation capabilities outside the training domain.
    • * Showcased scalability by analyzing 1-D images of varying spring systems.

    Conclusions:

    • * The proposed neural network architecture effectively extends symbolic regression to parametric systems.
    • * The approach enhances the analysis of complex and high-dimensional scientific data.
    • * This work paves the way for broader applications of symbolic regression in scientific discovery.