CBS-PINN: Coordinated training to address nonlocality and spectral stiffness in the Calogero-Bogoyavlenskii-Schiff-type equation.

Abstract

The Calogero-Bogoyavlenskii-Schiff-type (CBS-type) equations can be employed to describe the properties of nonlinear wave propagation in hemodynamics, yet their simulation using Physics-Informed Neural Networks (PINNs) has been limited by severe optimization challenges. This work identifies the root cause as a problematic coupling between nonlocal error propagation and spectral stiffness, which renders the standard PINN paradigm unstable. To address this, we propose CBS-PINN, a coordinated framework that orchestrates a sequential and synergistic optimization process. First, Curriculum Prioritization (CP) anchors the solution by enforcing the initial and boundary conditions, thereby suppressing the nonlocal propagation of constraint violations. Subsequently, Curvature-Based Weighting (CBW) leverages the second-order information to mitigate the gradient-scale imbalance induced by spectral stiffness. Extensive experiments demonstrate that CBS-PINN reduces the relative L₂ error to 2.78e-02 on the (2+1)-dimensional CBS-type equation, whereas the errors of the vanilla PINN and other representative baselines are on the order of 1e-01. The framework also exhibits strong robustness in data-scarce settings and excellent generalization to the KdV and Burgers equations, while remaining effective for the (3+1)-dimensional CBS-type equation. Notably, these enhancements are accomplished with minimal additional computational expense, superior stability, and accelerated convergence. By providing a reliable and high-fidelity solver, this work can support more precise pathophysiological analysis in cardiovascular research and provide practical design guidelines for solving CBS-type equations involving non-local terms and higher-order mixed derivatives.