Variable-domain Integral Weak Solution Theory for Deep Learning Methods Solving Differential Equations with Discontinuous Coefficients

The weak solution theory of differential equations has been widely adopted as a classical framework for solving discontinuous coefficient problems in both scientific research and practical engineering applications. While deep learning methods have recently demonstrated remarkable progress in solving differential equations, significant challenges remain when applying conventional weak solution theory to discontinuous coefficient equations within these methods due to the singularities of the equations at discontinuities. This paper presents a novel Variable-domain Integral Weak Solution theory (VIWS) specifically designed for deep learning approaches. Departing from traditional fixed-limit integration by parts methods, the proposed theory establishes a variable-domain integral weak formulation that accommodates the global sampling characteristics essential for deep learning computations. We provide complete derivations for both general and specific forms of representative differential equations, along with a comprehensive deep learning solution framework based on this formulation. Through multiple typical numerical examples, we demonstrate that the proposed theory achieves high accuracy and applicability, thereby establishing a new technical pathway for numerically solving differential equations with discontinuous coefficients.