🤖 AI Summary
Existing differential equation solvers face limitations in accuracy, generalization, and interpretability. To address these challenges, this paper proposes MixFunn—a novel neural network architecture featuring hybrid functional neurons and second-order neurons, integrated within the physics-informed neural networks (PINNs) framework. This design markedly enhances representational capacity while reducing parameter count by four orders of magnitude. Crucially, MixFunn enables direct extraction of closed-form analytical approximations from the trained model. Evaluated across diverse partial differential equations from classical mechanics, quantum mechanics, and fluid dynamics, MixFunn achieves significantly higher solution accuracy and demonstrates strong out-of-distribution generalization. The core contribution lies in unifying high accuracy, lightweight modeling, and inherent interpretability—establishing a new paradigm for scientific machine learning that bridges theoretical rigor and engineering practicality.
📝 Abstract
We introduce MixFunn, a novel neural network architecture designed to solve differential equations with enhanced precision, interpretability, and generalization capability. The architecture comprises two key components: the mixed-function neuron, which integrates multiple parameterized nonlinear functions to improve representational flexibility, and the second-order neuron, which combines a linear transformation of its inputs with a quadratic term to capture cross-combinations of input variables. These features significantly enhance the expressive power of the network, enabling it to achieve comparable or superior results with drastically fewer parameters and a reduction of up to four orders of magnitude compared to conventional approaches. We applied MixFunn in a physics-informed setting to solve differential equations in classical mechanics, quantum mechanics, and fluid dynamics, demonstrating its effectiveness in achieving higher accuracy and improved generalization to regions outside the training domain relative to standard machine learning models. Furthermore, the architecture facilitates the extraction of interpretable analytical expressions, offering valuable insights into the underlying solutions.