Recovering 3D phase features of complex biological samples traditionally sacrifices computational efficiency and processing time for physical model accuracy and reconstruction quality. Here, we overcome this challenge using an approximant-guided deep learning framework in a high-speed intensity diffraction tomography system. Applying a physics model simulator-based learning strategy trained entirely on natural image datasets, we show our network can robustly reconstruct complex 3D biological samples. To achieve highly efficient training and prediction, we implement a lightweight 2D network structure that utilizes a multi-channel input for encoding the axial information. We demonstrate this framework on experimental measurements of weakly scattering epithelial buccal cells and strongly scattering C. elegans worms. We benchmark the network's performance against a state-of-the-art multiple-scattering model-based iterative reconstruction algorithm. We highlight the network's robustness by reconstructing dynamic samples from a living worm video. We further emphasize the network's generalization capabilities by recovering algae samples imaged from different experimental setups. To assess the prediction quality, we develop a quantitative evaluation metric to show that our predictions are consistent with both multiple-scattering physics and experimental measurements.