MixFunn: A Neural Network for Differential Equations with Improved Generalization and Interpretability
This addresses the problem of solving differential equations in physics and engineering with enhanced precision and interpretability, representing a novel method for a known bottleneck.
The paper tackles solving differential equations by introducing MixFunn, a neural network architecture that achieves higher accuracy and improved generalization outside the training domain, with up to four orders of magnitude reduction in parameters compared to conventional methods.
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.