To reduce CO₂ emissions in response to global climate change, shale reservoirs could be ideal candidates for long-term carbon geo-sequestration involving multi-scale transport processes. However, most current CO₂ sequestration models do not adequately consider multiple transport mechanisms. Moreover, the evaluation of CO₂ storage processes usually involves laborious and time-consuming numerical simulations unsuitable for practical prediction and decision-making. In this paper, an integrated model involving gas diffusion, adsorption, dissolution, slip flow, and Darcy flow is proposed to accurately characterize CO₂ storage in depleted shale reservoirs, supporting the establishment of a training database. On this basis, a hybrid physics-informed data-driven neural network (HPDNN) is developed as a deep learning surrogate for prediction and inversion. By incorporating multiple sources of scientific knowledge, the HPDNN can be configured with limited simulation resources, significantly accelerating the forward and inversion processes. Furthermore, the HPDNN can more intelligently predict injection performance, precisely perform reservoir parameter inversion, and reasonably evaluate the CO₂ storage capacity under complicated scenarios. The validation and test results demonstrate that the HPDNN can ensure high accuracy and strong robustness across an extensive applicability range when dealing with field data with multiple noise sources. This study has tremendous potential to replace traditional modeling tools for predicting and making decisions about CO₂ storage projects in depleted shale reservoirs.