Rapid and accurate hazard forecasting is crucial for prompt evacuations and minimizing casualties during natural disasters. This paper proposes a tsunami forecasting approach using convolutional neural networks (CNNs) for early warning. Numerical experiments for the 2011 Tohoku tsunami demonstrated excellent performance, with minimal errors in maximum tsunami amplitude and arrival time forecasting. The approach is computationally efficient, requiring minimal processing time. Furthermore, the CNN, trained solely on synthetic data, provided reasonable inundation forecasts using real observation data from the 2011 event, even with noisy inputs. This validates the feasibility of AI-enabled tsunami forecasting for rapid and accurate early warnings.