Physics-informed neural networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by incorporating physical laws into the training of neural networks. However, their effectiveness in solving integral equations (IEs) remains constrained by suboptimal manual architecture design and computational inefficiencies in existing neural architecture search (NAS) methods. This paper introduces PINNAS (Physics-Informed Neural Network Architecture Search), a novel framework that integrates gradient-based NAS with PINNs to automate architecture optimization for solving IEs. Key innovations include: 1) a differentiable neural architecture search (DNAS) mechanism that relaxes discrete search spaces into continuous domains, enabling joint optimization of network weights and architecture parameters, 2) dynamic masking techniques to resolve tensor shape mismatches in variable-width layers, and 3) domain-specific search spaces tailored for integral operators. Extensive experiments on six IE types (one-dimensional or two-dimensional, linear or nonlinear, Volterra or Fredholm) demonstrate that PINNAS-optimized architectures achieve 15%–30% lower mean squared error (MSE) than manually designed networks while using 40% fewer parameters. Crucially, we reveal that non-uniform layer widths outperform uniform configurations, challenging conventional NAS practices. This work bridges the gap between automated machine learning and scientific computing, offering a scalable strategy for IE solutions.

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.