EPINN: Enhanced Physics-Informed Neural Network for Solving Continuous Integral Equations

Abstract

Background: Integral equations play a crucial role in modeling complex systems across various scientific disciplines. However, traditional numerical methods and existing physics-informed neural networks (PINNs) face substantial challenges, including the curse of dimensionality, uncontrolled error propagation, and limited generalization capabilities. Objectives: This paper aims to overcome these limitations by developing a robust and scalable solver for high-dimensional and nonlinear integral equations. The primary goal is to achieve higher accuracy and efficiency compared to traditional methods and existing deep learning approaches. Methods: We present the enhanced physics-informed neural network (EPINN), a novel framework that incorporates three key innovations: 1) a variable-order operator decomposition theory that transforms integral equations into well-posed differential systems, thereby mitigating error accumulation, 2) a differentiable primal function projection layer that ensures physical consistency within the Sobolev spaces, and 3) a boundary-aware multi-objective training paradigm that improves generalization. Results: Experimental validation across five benchmark cases spanning two to four dimensions, including linear/nonlinear Volterra/Fredholm and hybrid Volterra-Fredholm integral equations, demonstrates the superior performance of EPINN. Compared with traditional methods, EPINN reduces relative errors by 1 to 2 orders of magnitude, while achieving over 92% accuracy with limited training data. When compared with existing deep learning solvers, EPINN provides significant improvements in computational efficiency (with a speedup factor of 3 to 6 times) and accuracy (error reduction of 23% to 85%). Conclusions: These advancements establish EPINN as a robust and scalable solver for high-dimensional and nonlinear integral equations, with wide-ranging applications in computational physics and engineering. The success of EPINN suggests that integrating physical principles with neural networks can lead to substantial improvements in solving complex mathematical problems.