Accurate river prediction is essential for water, food, and energy security, yet extending it reliably from sparse in-situ gauges to the full river network remains an open challenge. Although machine learning has transformed Earth-system modelling, the same sparse and uneven observations have largely confined its advances for rivers to discrete basins rather than connected river systems. Here we introduce GraphRiverCast (GRC), a topology-informed river foundation model for reach-level prediction across the global river network. Exploiting the connectivity and energetics of rivers, it is trained on sparse gauges and predicts daily multivariate hydrodynamics with no prior river conditions. GRC outperforms strong physics-based and machine-learning models. Attribution experiments show that, compared with standard autoregressive models that rely more on temporal recurrence, GRC reconstructs river states from topology and forcing, learning the dynamics themselves and inheriting the self-correcting character of river systems. Its skill therefore barely decays with lead time, even far beyond its training horizon. It also generalises to ungauged reaches and finer-resolution networks, bringing reliable hydrodynamic prediction to poorly monitored regions. More broadly, harnessing the structure and energetics of a system offers a strategy for machine learning in other observation-limited Earth systems.

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