Advancing river bathymetry mapping through physics-informed neural networks and SWOT satellite observations

River bathymetry—the submerged channel topography invisible to conventional remote sensing and costly to survey at scale—remains unmapped for most of the world's rivers, critically constraining hydrodynamic flood modelling. The Surface Water and Ocean Topography (SWOT) satellite mission now delivers global Water Surface Elevation (WSE) observations, opening a path to infer riverbed elevation from space. Yet recovering bathymetry from WSE alone is fundamentally ill-posed: without additional constraints, infinitely many bed configurations produce identical surface responses. We present a Physics-Informed Neural Network (PINN) framework that mitigates this ill-posedness by assimilating multi-temporal SWOT overpass across diverse flow regimes, treating bed elevation as the sole unknown while prescribing Manning's roughness, discharge, and channel width. A dual-network architecture separates the time-invariant bed elevation from flow-dependent water depth, embedding Gradually Varied Flow (GVF) physics as a differentiable constraint. Synthetic experiments across 24 bed profiles achieve centimetre reconstruction accuracy for smooth morphologies, degrading for abrupt features at the identifiability limits of one-dimensional hydraulics. Validation on the Severn and Thames demonstrates that as few as 12 high-flow overpasses—roughly 10% of the available record—reproduce full-dataset accuracy, with Mean Absolute Errors (MAE) below 0.3 m relative to ground-truth surveys. Critically, reconstruction accuracy is governed primarily by hydraulic parameter uncertainty rather than SWOT observational limitations: Manning's roughness variations alone introduce errors of ±2.4 m, an order of magnitude beyond those from sampling density or measurement noise (±0.8 m). This framework charts a course from sporadic field campaigns to continuous, satellite-driven bathymetric monitoring for operational flood forecasting.