Numerical Simulation of Fractional Partial Differential Equations Using Deep Learning-Based Schemes
کلمات کلیدی:
Fractional Differential Models, Deep Neural Approximation, Physics-Informed Solvers, Simulation of Nonlocal Dynamicsچکیده
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
دانلودها
مراجع
[1]Ma, Z., Hou, J., Zhu, W.-H., Peng, Y., & Liu, Y. (2023). PMNN: Physical model-driven neural network for solving
time-fractional differential equations. Chaos, Solitons & Fractals, 177, 114238.
[2]Guo, L., Wu, H., Yu, X., & Zhou, T. (2022). Monte Carlo fPINNs: Deep learning method for forward and inverse problems
involving high-dimensional fractional partial differential equations. Computer Methods in Applied Mechanics and
Engineering, 400, 115523.
[3]Jagtap, A. D., Kharazmi, E., & Karniadakis, G. E. (2020). Conservative physics-informed neural networks on discrete
domains for conservation laws: Applications to forward and inverse problems. Computer Methods in Applied Mechanics and
Engineering, 365, 113028.
[4]Jagtap, A. D., & Karniadakis, G. E. (2020). Extended physics-informed neural networks (XPINNs): A generalized space-time
domain decomposition based deep learning framework for nonlinear PDEs. Communications in Computational Physics,
28(5), Article 115. (Q1)
[5]Grossmann, T. G., Komorowska, U. J., Latz, J., & Schönlieb, C.-B. (2024). Can physics-informed neural networks beat the
finite element method? IMA Journal of Applied Mathematics, 89(1), 143–174. (Q1 applied math)
[6]Farea, A., Yli-Harja, O., & Emmert-Streib, F. (2024). Understanding physics-informed neural networks: Techniques,
applications, trends, and challenges. AI, 5 )مروری/تحلیلی رویکرد ).1557–1534 ,(3)
[7]Wu, C., Zhu, M., Tan, Q., Kartha, Y., & Lu, L. (2023). A comprehensive study of non-adaptive and residual-based adaptive
sampling for PINNs. Computer Methods in Applied Mechanics and Engineering, 403, 115671.
[8]Wang, S., Teng, Y., & Perdikaris, P. (2021). Understanding and mitigating gradient flow pathologies in physics-informed
neural networks. SIAM Journal on Scientific Computing, 43(5), A3055–A3081.
[9]Lu, L., Meng, X., Mao, Z., & Karniadakis, G. E. (2021). DeepXDE: A deep learning library for solving differential equations.
SIAM Review, 63(1), 208–228.
[10]Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework…
Journal of Computational Physics, 378 .707–686 ,
[11]Pang, G., Lu, L., & Karniadakis, G. E. (2019). fPINNs: Fractional physics-informed neural networks. SIAM Journal on
Scientific Computing, 41(4), A2603–A2626.
[12]Serkin, L., & Belyaeva, T. L. (2025). Physics-Informed Neural Networks for Higher-Order Nonlinear Schrödinger Equations:
Soliton Dynamics… Mathematics, 13(11), 1882.
[13]Nuugulu, S. M., Patidar, K. C., & Tarla, D. T. (2024). A PINN approach for solving time fractional Black-Scholes PDEs.
Optimization and Engineering, 1–30.
[14]Ortiz Ortiz, R. D., Martínez Núñez, O., & Marín Ramírez, A. M. (2024). Solving viscous Burgers’ equation: hybrid approach
… PINNs. Mathematics, 12(21), 3430.
[15]Wang, J., Liu, Y., Yan, L., Han, K., Feng, L., & Zhang, R. (2025). Fractional sub-equation neural networks (fSENNs) method
for exact solutions … FPDEs. Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(4).
[16]Zhao, C., Zhang, F., Lou, W., Wang, X., & Yang, J. (2024). A comprehensive review of advances in PINNs and their
applications in complex fluid dynamics. Physics of Fluids, 36(10), 101301.
[17]Fu, J., Xiao, D., Fu, R., Li, C., & Zhu, C. (2023). Physics-data combined machine learning for parametric reduced-order
modelling… Computer Methods in Applied Mechanics and Engineering, 404, 115771.
[18]Yang, C., & Du, Q. (2020). A deep learning approach for solving high-dimensional PDEs. Journal of Computational Physics.
[19]Chauhan, A., Srinivasan, S., & Gupta, A. (2023). Deep learning for numerical partial differential equations. Computers &
Fluids.
[20]Zhang, Y., Xu, Y., & Sun, J. (2021). Deep learning-based numerical solution of high-dimensional parabolic PDEs. Journal of
Computational Physics.
[21]Sirignano, J., MacArt, J. F., & Freund, J. B. (2020). DPM: A deep learning PDE augmentation method… Journal of
Computational Physics, 423, 109811.
[22]Wu, W., Feng, X., & Xu, H. (2022). Improved deep neural networks with domain decomposition for PDEs. Journal of Scientific
Computing, 93(1), 20.
[23]Kabi-Nejad, P. (2024). Investigating Effective Factors in Solving PDEs using Deep Learning Techniques.
[24]Huang, D. Z., Xu, K., Farhat, C., & Darve, E. (2020). Learning constitutive relations from indirect observations using deep
neural networks. Journal of Computational Physics, 416, 109491.
[25]Xiangnan et al. (2025). Data-driven framework for discovering fractional differential equations in complex systems. Nonlinear
Dynamics.
[26]Zhang, T., Zhang, D., Shi, S., & Guo, Z. (2025). Spectral-fPINNs: spectral method based fractional PINNs for solving FPDEs.
Nonlinear Dynamics, 113(11), 12565–12588.
[27]Ma, C., Ma, L., & Wang, Z. (2024). Inverse design of electromagnetic metamaterials: from iterative to deep learning-based
methods. Journal of Micromechanics and Microengineering, 34(5), 053001.
[28]Liang, Gao et al. (2023). Failure-informed adaptive sampling for PINNs. SIAM Journal on Scientific Computing.
[29]Navaneeth, N., & Chakraborty, S. (2023). Physics-informed WNO. Computer Methods in Applied Mechanics and Engineering.
[30]Pantidis, P., Mobasher, M. E. (2023). Integrated finite element neural network (i-fenn) for non-local damage mechanics.
Computer Methods in Applied Mechanics and Engineering.
