MixFunn: A Neural Architecture for Differential Equations with Mixed Functions and Second-Order Interactions

Abstract We introduce MixFunn, a neural network architecture designed for modeling solutions of differential equations. The proposed architecture combines two main components: (i) mixed-function neurons, which employ a set of predefined activation functions to enrich the functional representation, and (ii) second-order neurons, which explicitly incorporate interactions between different neurons. Together, these elements aim to increase representational flexibility while maintaining a compact model structure. We evaluate MixFunn on a set of differential equation problems, including the damped and forced harmonic oscillator, the one-dimensional viscous Burgers equation, and the quantum infinite square well. The results indicate that the proposed architecture can achieve accurate approximations using fewer parameters compared to standard multilayer perceptrons. Additionally, the structure of the network enables the extraction of analytical expressions that approximate the learned solutions, providing a degree of interpretability. These findings suggest that MixFunn offers a viable alternative for neural modeling of differential equations, particularly in settings where compact representations and interpretability are desirable.