Split Complex-Valued Physics-Informed Neural Networks for Forward and Inverse Nonlinear Partial Differential Equations
Abstract
Physics-informed neural networks (PINNs) can solve forward and inverse PDE problems but may struggle with spectral bias and oscillatory, high-frequency, or phase-dependent solutions. We introduce a generalized split complex-valued PINN (SCV-PINN) framework with complex-valued parameters and latent representations. Split activations act separately on real and imaginary components while complex operations preserve their coupling, enabling amplitude--phase representations with practical optimization. Ablation studies evaluate split activation functions and collocation sampling strategies. Forward and inverse benchmarks include the Burgers, Allen--Cahn, Korteweg--de Vries, nonlinear Schrödinger, Helmholtz, and Poisson equations on regular and irregular domains; Kovasznay flow (Re=20); lid-driven cavity flow (Re=100); the Lorenz system; and inverse Burgers and Navier--Stokes problems (Re=100). A three-dimensional Navier--Stokes Beltrami flow further demonstrates applicability to higher-dimensional nonlinear dynamics, with a relative L2 error of 4.07×10−5. Across these tests, SCV-PINN yields high-fidelity solutions and effective parameter identification, improving on real-valued PINNs and selected variants. The results suggest that split complex-valued representations can help address spectral-bias-related difficulties in nonlinear and oscillatory physical systems.
Related
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BibTeX
@misc{barman2026split,
title = {{Split Complex-Valued Physics-Informed Neural Networks for Forward and Inverse Nonlinear Partial Differential Equations}},
author = {Biswanath Barman and Rajendra K. Ray and Debdeep Chatterjee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.15087},
eprint = {2607.15087},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.15087},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Split Complex-Valued Physics-Informed Neural Networks for Forward and Inverse Nonlinear Partial Differential Equations
AU - Biswanath Barman
AU - Rajendra K. Ray
AU - Debdeep Chatterjee
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2607.15087
UR - https://arxiv.org/abs/2607.15087
AB - Physics-informed neural networks (PINNs) can solve forward and inverse PDE problems but may struggle with spectral bias and oscillatory, high-frequency, or phase-dependent solutions. We introduce a generalized split complex-valued PINN (SCV-PINN) framework with complex-valued parameters and latent representations. Split activations act separately on real and imaginary components while complex operations preserve their coupling, enabling amplitude--phase representations with practical optimization. Ablation studies evaluate split activation functions and collocation sampling strategies. Forward and inverse benchmarks include the Burgers, Allen--Cahn, Korteweg--de Vries, nonlinear Schrödinger, Helmholtz, and Poisson equations on regular and irregular domains; Kovasznay flow (Re=20); lid-driven cavity flow (Re=100); the Lorenz system; and inverse Burgers and Navier--Stokes problems (Re=100). A three-dimensional Navier--Stokes Beltrami flow further demonstrates applicability to higher-dimensional nonlinear dynamics, with a relative L_2 error of 4.07×10^−5. Across these tests, SCV-PINN yields high-fidelity solutions and effective parameter identification, improving on real-valued PINNs and selected variants. The results suggest that split complex-valued representations can help address spectral-bias-related difficulties in nonlinear and oscillatory physical systems.
ER -