SmooNet: Smooth operator neural network and functional differential equation.

Area of Science:

• Dynamical Systems and Mathematical Modeling
• Computational Neuroscience and Machine Learning

Background:

• Ordinary Differential Equations (ODEs) commonly model dynamical systems but often oversimplify by ignoring system memory.
• This limitation hinders accurate modeling of systems with inherent memory effects.

Purpose of the Study:

• To propose a novel Functional Differential Equation (FDE) framework capable of modeling memory effects in dynamical systems.
• To introduce the Smooth Operator Neural Network (SmooNet) as a tool for approximating the unknown operators within FDEs.

Main Methods:

• Developed a Smooth Operator Neural Network (SmooNet) with a continuous hidden layer ('hidden string') for approximating operators in FDEs.
• Implemented a novel moving window optimization strategy for SmooNet construction and forecasting.
• Established theoretical guarantees for SmooNet's universal approximation capabilities and solution convergence.

Main Results:

• SmooNet demonstrated universal approximation of operators within the FDE framework.
• Solutions from approximate neural FDEs were shown to be uniformly close to solutions of the original FDEs.
• Empirical studies confirmed the model's flexibility and efficiency for studying and forecasting dynamical systems.

Conclusions:

• The proposed FDE model with SmooNets effectively addresses the limitations of ODEs by incorporating memory effects.
• SmooNets provide a powerful and theoretically grounded method for modeling complex dynamical systems.
• The developed framework offers a flexible and efficient tool for scientific forecasting and analysis.