<p>Rapid response to ocean disturbance that has tsunamigenic potential requires characterisation of that disturbance. Traditional methods assume seismic sources, which can lead to suboptimal accuracy for non-seismic events. Some more recent approaches use simulations based on a fixed set of offshore sensors to infer coastal effects at selected stations. This study introduces a different approach to tsunami forecasting from offshore sensor data by modelling the Green’s functions, which describe the impulse response at a given ocean location due to a disturbance at another location. Achieving a reasonable level of resolution requires a large number of Green’s functions for the inversion process, and significant storage requirements. We address these issues by using neural networks as compressed function representations, and by an inversion based on iterative constrained optimization. Our approach is robust to changes in the set of off-shore sensors, thus accommodating malfunctioning sensors, and it also allows for the inversion of non-seismic events. Results using simulations of both historical and hypothetical non-seismic tsunamis near Japan validate our approach, which offers a promising approach to tsunami forecasting. Our inversion reduces tsunami forecasting uncertainty by constraining the initial condition.</p>