Deep Computational Mathematics for Partial Differential Equations: Neural Operator Approaches to Solving High-Dimensional Systems

Abstract

This study investigates the role of neural operator learning frameworks in solving complex and high-dimensional partial differential equations (PDEs) that arise in scientific and engineering applications. The research focuses on prominent architectures such as the Fourier Neural Operator (FNO), Deep Operator Network (DeepONet), and physics-informed variants designed to approximate solution mappings between function spaces efficiently. By leveraging data-driven and physics-constrained methodologies, these models demonstrated remarkable accuracy, stability, and computational efficiency compared to traditional numerical solvers. Experimental evaluations revealed that neural operators significantly reduced inference time while maintaining precision across diverse PDE families, including fluid flow, diffusion, and wave propagation problems. The integration of physics-informed regularization further enhanced model generalization under noisy or limited-data conditions. Despite their strong performance, challenges such as spectral bias, overfitting in data-scarce environments, and limited uncertainty quantification capabilities remain open research issues. The study highlights that incorporating multi-resolution and hybrid learning strategies can address these limitations effectively. Overall, neural operator learning represents a paradigm shift in computational modeling, enabling scalable, interpretable, and real-time PDE solutions suitable for scientific simulations, industrial design, and complex system prediction. The findings provide valuable insights for future advancements in operator-based deep learning and its integration with emerging scientific computing frameworks.