A physics-informed piecewise continuity network for solving the non-linear jerk equation: PiPC-Net

In the present study, the Jerk equations, which define the third derivative of position with respect to time, are key to modeling stiff nonlinear dynamical systems in chaotic oscillators, robotic trajectory optimization, vibration damping, and precision control. These high-order ordinary differential equations (ODEs) exhibit stiffness and sensitivity to initial conditions, challenging long-term stability in solvers like Physics-Informed Neural Networks (PINNs). PINNs, despite embedding physical laws, suffer from global errors, spectral bias, and vanishing gradients over extended horizons. This paper introduces the Physics-Informed Piecewise Continuity Network (PiPC-Net), a domain-decomposition method dividing the time domain into subdomains, each approximated by a physics-informed subnetwork. PiPC-Net enforces exact C2 continuity at interfaces by construction, with interface loss terms driving numerical residuals to machine precision. Each subnetwork inherits initial conditions from the preceding subdomain to limit error buildup, enhancing gradient flow and optimization with LBFGS. Evaluated on an oscillatory jerk and a damped cubic jerk with variations of parameters, PiPC-Net outperforms PINNs. It achieves 2.3 to 2.6 times faster convergence, reducing residual losses, with variance below 0.20. MAE reductions reach 72%–87% globally, peaking at 421.6 times for second problem with η=1.5 at t=20. RMSEs range from 0.000437-0.0135 (PSNR 37.4 to 54.2 dB) for oscillatory and 0.000837-0.00634 (PSNR 44.0-61.5 dB) for damped cases, with errors under 2.5% amplitude and 0.06 rad phase lag. Error profiles show a sawtooth pattern, capping peaks at 0.0181-.0399 with 68% lower variance than PINNs.