Numerical Simulation of Fractional Partial Differential Equations Using Deep Learning-Based Schemes
Keywords:
Fractional Differential Models, Deep Neural Approximation, Physics-Informed Solvers, Simulation of Nonlocal DynamicsAbstract
Fractional partial differential equations (FPDEs) have emerged as powerful mathematical models for capturing memory-dependent and nonlocal phenomena across a broad range of applications such as anomalous diffusion, viscoelasticity, and quantitative finance. Despite their modeling potential, numerical solutions to FPDEs are often hindered by the complexity introduced through nonlocal operators in time and space. This study presents a novel computational strategy driven by deep learning, utilizing physics-integrated neural architectures to model space-time fractional PDEs without the use of conventional discretization frameworks. The proposed method integrates a tailored loss formulation that embeds the underlying physics into the learning process, enabling mesh-free training and efficient resolution of high-dimensional problems. Experimental evaluations across various benchmark scenarios reveal substantial improvements over conventional solvers in terms of precision, convergence behavior, and resilience to noise. Furthermore, the framework demonstrates strong generalization capabilities across diverse geometries and boundary settings. The outcomes emphasize the capacity of neural operator-based frameworks to overcome the inherent computational barriers in fractional models, offering new pathways for their integration into real-time simulations, parameter inference, and predictive modeling tasks across engineering and scientific disciplines
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