Split Complex-Valued Physics-Informed Neural Networks for Forward and Inverse Nonlinear PDEs

Abstract

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.