Accurate characterization of soft soil nonlinear consolidation with non-Darcian flow and complex boundary conditions is fundamental to assessing foundation stability. This study presents a Noise-Aware Physics-Informed Neural Network (NA-PINN) framework for one-dimensional nonlinear consolidation problems involving a continuous drainage boundary and non-Darcian flow. Multiple nonlinear couplings may lead to gradient imbalance in standard PINNs. Furthermore, when measurement noise and sparse data are also present, the inversion of the flow index may also become ill-posed as well. To address this problem, the PDE residual weight is prescribed according to the assumed noise level of the observation data, so that the governing equation provides stronger constraints during inverse training under noisy conditions. In addition, a differentiable variable transformation is used to keep the identified flow index within prescribed physical bounds. Benchmark comparisons with finite difference solutions show that, in the tested synthetic cases with 10% measurement noise and 5% spatiotemporal observation density, the relative error of the identified flow index remained below 0.1%. The proposed framework can capture the nonlinear evolution of excess pore water pressure dissipation and average consolidation response under time-dependent loading, and may provide a useful tool for consolidation analysis when monitoring data are sparse or noisy.