Diffuse optical tomography (DOT) has been investigated as an alternative imaging modality for breast cancer detection thanks to its excellent contrast to hemoglobin oxidization level. However, due to the complicated non-linear photon scattering physics and ill-posedness, the conventional reconstruction algorithms are sensitive to imaging parameters such as boundary conditions. To address this, here we propose a novel deep learning approach that learns non-linear photon scattering physics and obtains an accurate three dimensional (3D) distribution of optical anomalies. In contrast to the traditional black-box deep learning approaches, our deep network is designed to invert the Lippman-Schwinger integral equation using the recent mathematical theory of deep convolutional framelets. As an example of clinical relevance, we applied the method to our prototype DOT system. We show that our deep neural network, trained with only simulation data, can accurately recover the location of anomalies within biomimetic phantoms and live animals without the use of an exogenous contrast agent.