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Speed-of-sound has been shown as a potential biomarker for breast cancer imaging, successfully differentiating malignant tumors from benign ones. Speed-of-sound images can be reconstructed from time-of-flight measurements from ultrasound images acquired using conventional handheld ultrasound transducers. Variational Networks (VN) have recently been shown to be a potential learning-based approach for optimizing inverse problems in image reconstruction. Despite earlier promising results, these methods however do not generalize well from simulated to acquired data, due to the domain shift. In this work, we present for the first time a VN solution for a pulse-echo SoS image reconstruction problem using diverging waves with conventional transducers and single-sided tissue access. This is made possible by incorporating simulations with varying complexity into training. We use loop unrolling of gradient descent with momentum, with an exponentially weighted loss of outputs at each unrolled iteration in order to regularize training. We learn norms as activation functions regularized to have smooth forms for robustness to input distribution variations. We evaluate reconstruction quality on ray-based and full-wave simulations as well as on tissue-mimicking phantom data, in comparison to a classical iterative (L-BFGS) optimization of this image reconstruction problem. We show that the proposed regularization techniques combined with multi-source domain training yield substantial improvements in the domain adaptation capabilities of VN, reducing median RMSE by 54% on a wave-based simulation dataset compared to the baseline VN. We also show that on data acquired from a tissue-mimicking breast phantom the proposed VN provides improved reconstruction in 12 milliseconds.