To estimate proton density fat fraction (PDFF) from chemical shift encoded (CSE) MR images using a deep learning (DL)-based method that is precise and robust to different MR scanners and acquisition echo times (TEs). Variable echo times neural network (VET-Net) is a two-stage framework that first estimates nonlinear variables of the CSE-MR signal model, to posteriorly estimate water/fat signal components using the least-squares method. VET-Net incorporates a vector with TEs as an auxiliary input, therefore enabling PDFF calculation with any TE setting. A single-site liver CSE-MRI dataset (188 subjects, 4146 axial slices) was considered, which was split into training (150 subjects), validation (18), and testing (20) subsets. Testing subjects were scanned using several protocols with different TEs, which we then used to measure the PDFF reproducibility coefficient (RDC) at two regions of interest (ROIs): the right posterior and left hepatic lobes. An open-source multi-site and multi-vendor fat-water phantom dataset was also used for PDFF bias assessment. VET-Net showed RDCs of 1.71% and 1.04% on the right posterior and left hepatic lobes, respectively, across different TEs, which was comparable to a reference graph cuts-based method (RDCs = 1.71% and 0.86%). VET-Net also showed a smaller PDFF bias (-0.55%) than graph cuts (0.93%) when tested on a multi-site phantom dataset. Reproducibility (1.94% and 1.59%) and bias (-2.04%) were negatively affected when the auxiliary TE input was not considered. VET-Net provided unbiased and precise PDFF estimations using CSE-MR images from different hardware vendors and different TEs, outperforming conventional DL approaches. Question Reproducibility of liver PDFF DL-based approaches on different scan protocols or manufacturers is not validated. Findings VET-Net showed a PDFF bias of -0.55% on a multi-site phantom dataset, and RDCs of 1.71% and 1.04% at two liver ROIs. Clinical relevance VET-Net provides efficient, in terms of scan and processing times, and unbiased PDFF estimations across different MR scanners and scan protocols, and therefore it can be leveraged to expand the use of MRI-based liver fat quantification to assess hepatic steatosis.