Bathymetry Estimations Using Self Adaptive-Physics Informed Neural Networks on Two Dimensional Shallow Water Equations

Accurate bathymetry estimation is fundamental for applications in marine navigation, coastal hazard forecasting, and geophysical research. Conventional approaches, including shipborne sonar and satellite remote sensing, while effective, remain expensive, data-intensive, and limited in regions with sparse or noisy measurements. In this study, we develop a Self-Adaptive Physics-Informed Neural Network (SA-PINN) framework for the inverse problem of reconstructing bathymetry from the two-dimensional shallow water equations. The model integrates both data fidelity and physical constraints through dynamically weighted loss terms, which evolve during training to balance the contributions of supervised and partial differential equation (PDE) residuals. To enhance and better capture the oscillatory bathymetric patterns, we have utilised Fourier feature embeddings in the input space, allowing the network to capture multiscale spatial structures with improved resolution. Our numerical experiments demonstrate the capability of SA-PINN to recover intricate bathymetric features from limited measurements of free-surface elevation and velocity fields. We construct synthetic training data via superposition of multiple linear wave modes, generating spatially complex interference patterns representative of realistic shallow-water dynamics. Beyond reconstruction accuracy, we perform a high-order perturbation analysis using finite-difference retraining to quantify the local sensitivity and analyticity of the inverse operator. The observed staircase decay of Taylor expansion norms provides empirical evidence of local stability and weak nonlinearity in the bathymetry inversion, while highlighting stronger nonlinear effects in velocity-driven couplings. An adaptive-weight sensitivity study further reveals that accurate bathymetry recovery is confined to a narrow, dynamically learned region in loss-weight space, confirming that fixed-weight PINNs are fundamentally unstable for this inverse problem. Overall, this work establishes SA-PINNs as a robust, data-efficient framework for shallow-water bathymetry inversion, while providing theoretical insight into identifiability, stability, and sensitivity through perturbation-based diagnostics.