NOMTO: Neural Operator-based symbolic Model approximaTion and discOvery

While many physical and engineering processes are most effectively described by non-linear symbolic models, existing non-linear symbolic regression (SR) methods are restricted to a limited set of continuous algebraic functions, thereby limiting their applicability to discover higher order non-linear differential relations. In this work, we introduce the Neural Operator-based symbolic Model approximaTion and discOvery (NOMTO) method, a novel approach to symbolic model discovery that leverages Neural Operators to encompass a broad range of symbolic operations. We demonstrate that NOMTO can successfully identify symbolic expressions containing elementary functions with singularities, special functions, and derivatives. Additionally, our experiments demonstrate that NOMTO can accurately rediscover second-order non-linear partial differential equations. It provides a powerful and flexible tool for model discovery, capable of capturing complex relations in a variety of physical systems. Many physical and engineering processes are most effectively described by concise mathematical expressions derived through meticulous observation and analysis. The accuracy of these models is highly dependent on the quality and quantity of available data. With the emergence of large-scale datasets across diverse physical and engineering domains, deriving compact mathematical models in the form of symbolic expressions has become increasingly attainable. This methodology, known as symbolic regression (SR), aims to identify mathematical expressions that most accurately represent given datasets. SR has become indispensable in fields such as physics, biology, and engineering, where it advances knowledge and fosters innovation by uncovering underlying principles and facilitating the development of interpretable predictive models. In recent years, deep learning-based approaches have significantly advanced the field of SR by leveraging neural networks to identify mathematical expressions directly from data.