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<p>Time-fractional partial differential equations (TFPDEs) provide a powerful framework for modeling anomalous diffusion, memory effects, and non-local transport phenomena, but their non-local nature poses significant challenges for traditional numerical methods and standard physics-informed neural networks (PINNs). To solve time-fractional PDEs, we employ four key concepts in PINNs method and named it PINNs-MSFF. First, we use automatic differentiation to compute integer-order spatial derivatives. Second, we apply L1 finite-difference discretization for the Caputo time-fractional derivative. Third, we incorporate Fourier-feature embeddings to mitigate spectral bias. Finally, we utilize a trainable multi-scale subnetwork ensemble to effectively capture both global and localized solution structures. The proposed framework is validated on benchmark problems, including the time-fractional Diffusion (TFDE), Burgers (TFBE), and Klein–Gordon (TFKGE) equations. To ensure a rigorous assessment, the methodology is benchmarked systematically against the standard fPINN architecture and a high-fidelity finite-difference method utilizing the L1 discretization scheme (L1-FDM) as the reference baseline. Extensive numerical experiments demonstrate that PINNs-MSFF achieves superior accuracy, stability, and convergence, effectively capturing complex fractional dynamics, sharp localized gradients, and dispersive phase transitions where standard PINNs often fail. Its mesh-free nature, adaptability to higher-dimensional domains, complex geometries, and capacity for inverse parameter identification highlight PINNs-MSFF as a versatile and robust tool for solving a wide range of fractional-order models in scientific and engineering applications.</p>

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Multi-scale physics-informed neural networks with Fourier features for approximating time-fractional PDEs

  • 期刊论文

  • 发表日期：

    2026年7月23日

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  • Harender Kumar

摘要

Time-fractional partial differential equations (TFPDEs) provide a powerful framework for modeling anomalous diffusion, memory effects, and non-local transport phenomena, but their non-local nature poses significant challenges for traditional numerical methods and standard physics-informed neural networks (PINNs). To solve time-fractional PDEs, we employ four key concepts in PINNs method and named it PINNs-MSFF. First, we use automatic differentiation to compute integer-order spatial derivatives. Second, we apply L1 finite-difference discretization for the Caputo time-fractional derivative. Third, we incorporate Fourier-feature embeddings to mitigate spectral bias. Finally, we utilize a trainable multi-scale subnetwork ensemble to effectively capture both global and localized solution structures. The proposed framework is validated on benchmark problems, including the time-fractional Diffusion (TFDE), Burgers (TFBE), and Klein–Gordon (TFKGE) equations. To ensure a rigorous assessment, the methodology is benchmarked systematically against the standard fPINN architecture and a high-fidelity finite-difference method utilizing the L1 discretization scheme (L1-FDM) as the reference baseline. Extensive numerical experiments demonstrate that PINNs-MSFF achieves superior accuracy, stability, and convergence, effectively capturing complex fractional dynamics, sharp localized gradients, and dispersive phase transitions where standard PINNs often fail. Its mesh-free nature, adaptability to higher-dimensional domains, complex geometries, and capacity for inverse parameter identification highlight PINNs-MSFF as a versatile and robust tool for solving a wide range of fractional-order models in scientific and engineering applications.