A computationally efficient parallel training framework for solving integral equations using deep learning methods

Solving integral equations via deep learning encounters significant computational bottlenecks when order-reduction techniques transform problems into strongly coupled differential systems requiring multi-network collaborative training. While achieving high accuracy, existing distributed training paradigms exhibit fundamental limitations. Data parallelism suffers from prohibitive gradient synchronization overhead in multi-network coupling scenarios, while pipeline parallelism struggles with bubble inefficiencies in the shallow architectures typical of scientific computing. To overcome these challenges, we proposed the computationally efficient parallel training framework (CEPTF), which introduces three key innovations. A unified computational efficiency metric balancing acceleration gains with resource costs, mathematical-aware dynamic partitioning that adapts to equation structure and hardware constraints, and hybrid parallelism integrating optimized communication topologies with constraint-preserving synchronization. Comprehensive validation across linear/nonlinear Fredholm/Volterra equations demonstrates that CEPTF achieves 3.18 $\times$ to 6.32 $\times$ acceleration (318.6%–632.9% speedup ratio) while maintaining solution accuracy of ${10}^{-7}$ to ${10}^{-9}$ magnitude, outperforming established parallel paradigms by 1.8 $\times$ to 3.2 $\times$ in speedup ratios and 42%–67% in computational efficiency. The framework's adaptability to heterogeneous computing environments and robust performance under challenging conditions, including singular kernels and irregular domains, establishes a new paradigm for scalable scientific machine learning.