The DNA binding of most Escherichia coli Transcription Factors (TFs) has not been comprehensively mapped, and few have models that can quantitatively predict binding affinity. We report the global mapping of in vivo DNA binding for 139 E. coli TFs using ChIP-Seq. We use these data to train BoltzNet, a novel neural network that predicts TF binding energy from DNA sequence. BoltzNet mirrors a quantitative biophysical model and provides directly interpretable predictions genome-wide at nucleotide resolution. We use BoltzNet to quantitatively design novel binding sites, which we validate with biophysical experiments on purified protein. We generate models for 124 TFs that provide insight into global features of TF binding, including clustering of sites, the role of accessory bases, the relevance of weak sites, and the background affinity of the genome. Our paper provides new paradigms for studying TF-DNA binding and for the development of biophysically motivated neural networks.