Supplementary Research on Several Issues of Variable-domain Integral Weak Solution Theory (VIWS) for Deep Learning Methods

Solving differential equations with discontinuous coefficients is crucial for both academic research and engineering applications. Currently, using deep learning to solve these equations is a major research focus. The Variable Integration Weak Solution (VIWS) theory provides a new technical approach to address challenges in this field. Previous studies have established the basic framework of this theory and verified it on standard cases with regular geometric domains. Based on previous work, this paper derives specific VIWS forms for domains with irregular complex boundaries and internal porous media. It also discusses the continuity of higher-order differential equations and the conditions for determining solutions. Furthermore, to handle complex cases involving discontinuous numerical solutions and shock waves, a method is proposed that uses neural networks to map variables to flux combination function. Finally, the applicability of the VIWS theory under these different conditions is verified through several typical examples. This study refines the VIWS theory and presents application methods that are closer to engineering reality, laying a foundation for transitioning this theoretical research into practical engineering applications.