用于正向和反向非线性偏微分方程的分裂复值物理信息神经网络
Split Complex-Valued Physics-Informed Neural Networks for Forward and Inverse Nonlinear PDEs
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中文总结 AI 辅助
研究针对传统实值PINNs的不足，提出分裂复值物理信息神经网络SCV-PINN。其在复域定义参数和表示，采用分裂复值激活函数增强逼近能力。经大量实验验证，在多种PDE基准上比现有方法有更低误差和更准参数识别，为非线性PDEs提供强大通用扩展。
中文摘要 AI 辅助
物理信息神经网络（PINNs）已成为求解正向和反向偏微分方程（PDEs）的强大框架，但传统实值PINNs（RV-PINNs）常存在频谱偏差、表达能力有限以及对高频、振荡和相位相关动力学精度降低等问题。本文提出广义分裂复值物理信息神经网络（SCV-PINN），其网络参数和潜在表示在复域中定义。该框架采用分裂复值激活函数，通过对实部和虚部分别应用标准实值激活函数，提供数值稳定性、计算效率和改进的逼近能力。通过大量消融研究评估了不同的分裂激活函数和配置采样策略。在包括Burgers、Allen-Cahn、Korteweg-de Vries、非线性Schrodinger、Helmholtz、Poisson、Kovasznay流（Re = 20）、顶盖驱动腔流（Re = 100）、Lorenz系统、反向Burgers、反向Navier-Stokes（Re = 100）和三维Navier-Stokes Beltrami流等正向和反向PDE基准上对所提出框架进行了验证。对于Beltrami基准，SCV-PINN实现了4.07 x 10^-5的相对L2误差。数值结果一致表明，SCV-PINN比RV-PINNs和几种现有的PINN变体具有更低的相对L2误差和更准确的参数识别。所提出的SCV-PINN为复值、多尺度、振荡、高维和实值非线性PDEs提供了标准PINNs的强大且通用的扩展。
英文摘要
Physics-informed neural networks (PINNs) have emerged as a powerful framework for solving forward and inverse partial differential equations (PDEs), but conventional real-valued PINNs (RV-PINNs) often suffer from spectral bias, limited expressivity, and reduced accuracy for high-frequency, oscillatory, and phase-dependent dynamics. In this work, we propose a generalized split complex-valued physics-informed neural network (SCV-PINN), in which network parameters and latent representations are defined in the complex domain. The framework employs split complex-valued activation functions by independently applying standard real-valued activations to the real and imaginary components, providing numerical stability, computational efficiency, and improved approximation capability. This formulation enables simultaneous learning of amplitude and phase information, enhancing the representation of nonlinear and oscillatory systems. Extensive ablation studies evaluate different split activation functions and collocation sampling strategies. The proposed framework is validated on forward and inverse PDE benchmarks including Burgers, Allen-Cahn, Korteweg-de Vries, nonlinear Schrodinger, Helmholtz, Poisson, Kovasznay flow (Re = 20), lid-driven cavity flow (Re = 100), the Lorenz system, inverse Burgers, inverse Navier-Stokes (Re = 100), and a three-dimensional Navier-Stokes Beltrami flow. For the Beltrami benchmark, SCV-PINN achieves a relative L2 error of 4.07 x 10^-5. Numerical results consistently demonstrate lower relative L2 errors and more accurate parameter identification than RV-PINNs and several existing PINN variants. The proposed SCV-PINN provides a robust and generalized extension of standard PINNs for complex-valued, multiscale, oscillatory, high-dimensional, and real-valued nonlinear PDEs.
