Rapid and accurate hazard forecasting is important for prompt evacuations and reducing casualties during natural disasters. In the decade since the 2011 Tohoku tsunami, various tsunami forecasting methods using real-time data have been proposed. However, rapid and accurate tsunami inundation forecasting in coastal areas remains challenging. Here, we propose a tsunami forecasting approach using convolutional neural networks (CNNs) for early warning. Numerical tsunami forecasting experiments for Tohoku demonstrated excellent performance with average maximum tsunami amplitude and tsunami arrival time forecasting errors of similar to 0.4m and similar to 48s, respectively, for 1,000 unknown synthetic tsunami scenarios. Our forecasting approach required only 0.004s on average using a single CPU node. Moreover, the CNN trained on only synthetic tsunami scenarios provided reasonable inundation forecasts using actual observation data from the 2011 event, even with noisy inputs. These results verify the feasibility of AI-enabled tsunami forecasting for providing rapid and accurate early warnings.