A recent PNAS paper reveals that several popular deep reconstruction networks are unstable. Specifically, three kinds of instabilities were reported: (1) strong image artefacts from tiny perturbations, (2) small features missed in a deeply reconstructed image, and (3) decreased imaging performance with increased input data. Here, we propose an analytic compressed iterative deep (ACID) framework to address this challenge. ACID synergizes a deep network trained on big data, kernel awareness from compressed sensing (CS)-inspired processing, and iterative refinement to minimize the data residual relative to real measurement. Our study demonstrates that the ACID reconstruction is accurate, is stable, and sheds light on the converging mechanism of the ACID iteration under a bounded relative error norm assumption. ACID not only stabilizes an unstable deep reconstruction network but also is resilient against adversarial attacks to the whole ACID workflow, being superior to classic sparsity-regularized reconstruction and eliminating the three kinds of instabilities. •Deep reconstruction solution to the instabilities identified by a recent PNAS paper•Analytic compressed iterative deep (ACID) method is for hybrid reconstruction•ACID framework combines benefits from deep learning and compressed sensing•ACID is accurate and stable, superior to sparsity-regularized reconstruction alone Tomographic image reconstruction with deep learning has been a rapidly emerging field since 2016. Recently, a PNAS paper revealed that several well-known deep reconstruction networks are unstable for computed tomography (CT) and magnetic resonance imaging (MRI), and, in contrast, compressed sensing (CS)-inspired reconstruction methods are stable because of their theoretically proven property known as “kernel awareness.” Therefore, for deep reconstruction to realize its full potential and become a mainstream approach for tomographic imaging, it is critically important to stabilize deep reconstruction networks. Here, we propose an analytic compressed iterative deep (ACID) framework to synergize deep learning and compressed sensing through iterative refinement. We anticipate that this integrative model-based data-driven approach will promote the development and translation of deep tomographic image reconstruction networks. We propose an analytic compressed iterative deep (ACID) framework for accurate yet stable deep reconstruction. ACID synergizes a deep reconstruction network trained on big data, kernel awareness from compressed sensing-inspired processing, and iterative refinement to minimize the data residual relative to real measurement. We anticipate that this integrative model-based data-driven approach will promote the development and translation of deep tomographic image reconstruction networks.