Objective: We aim to evaluate a method for estimating 1D physical dose deposition profiles in carbon ion therapy via analysis of dynamic PET images using a deep residual learning convolutional neural network (CNN). The method is validated using Monte Carlo simulations of 12C ion spread-out Bragg peak (SOBP) profiles, and demonstrated with an experimental PET image. Approach: A set of dose deposition and positron annihilation profiles for monoenergetic 12C ion pencil beams in PMMA are first generated using Monte Carlo simulations. From these, a set of random polyenergetic dose and positron annihilation profiles are synthesised and used to train the convolutional neural network. Performance is evaluated by generating a second set of simulated 12C ion SOBP profiles (one 116 mm SOBP profile and ten 60 mm SOBP profiles), and using the trained neural network to estimate the dose profile deposited by each beam and the position of the distal edge of the SOBP. Next, the same methods are used to evaluate the network using an experimental PET image, obtained after irradiating a PMMA phantom with a 12C ion beam at QST's HIMAC facility in Chiba, Japan. The performance of the CNN is compared to that of a recently published iterative technique using the same simulated and experimental 12C SOBP profiles. Main results: The CNN estimated the simulated dose profiles with a mean relative error of 0.7%±1.0% and the distal edge position with an accuracy of 0.1~mm±0.2 mm, and estimate the dose delivered by the experimental 12C ion beam with a mean relative error of 3.7%, and the distal edge with an accuracy of 1.7 mm. Significance: The CNN was able to produce estimates of the dose distribution with comparable or improved accuracy and computational efficiency compared to the iterative method and other similar PET-based direct dose quantification techniques.