Two-Stage Bayesian Physics-Informed Neural Networks for Groundwater Contaminant Source Identification

Groundwater contaminant source identification faces inherent challenges due to system concealment, transport process hysteresis, and data uncertainty. Although physics-informed neural networks (PINNs) enhance computational efficiency by embedding governing equations into learning frameworks, their susceptibility to observational noise and inadequate uncertainty quantification hinders practical implementation. Here, we propose a two-stage Bayesian physics-informed neural network (TSBPINN) that synergistically integrates physical constraints with Bayesian inference to address these limitations. The framework operates through two phase-locked learning stages: initial enforcement of physical consistency via boundary condition constraints establishes foundational concentration fields, followed by precise characterization of advection-diffusion dynamics through partial differential equation (PDE) embeddings. A Bayesian probabilistic architecture models observational noise as Gaussian likelihood functions, enabling concurrent inversion of source parameters (position/intensity) and uncertainty propagation analysis through posterior distributions. Across noise levels of 1%, 5%, 10%, and 25%, TSBPINN consistently outperformed conventional PINN, Bayesian physics-informed neural network (BPINN), and two-stage physics-informed neural network (TSPINN) in predicting the contaminant source concentration and position. At 10% noise, for example, TSBPINN yielded C0=3.90 (true: 4.0) and x0=0.87 (true: 1.0), surpassing all baselines. Moreover, TSBPINN achieved empirical coverage probabilities of 94%–95% for 95% credible intervals—substantially higher than BPINN (23%–38%)—and produced narrower uncertainty bands. These results demonstrate that TSBPINN enhances both noise robustness and uncertainty quantification, offering a reliable framework for groundwater source inversion in noisy environments.