  The goal of diffusion-weighted magnetic resonance imaging (DWI) is to infer
the structural connectivity of an individual subject's brain in vivo. To
statistically study the variability and differences between normal and abnormal
brain connectomes, a mathematical model of the neural connections is required.
In this paper, we represent the brain connectome as a Riemannian manifold,
which allows us to model neural connections as geodesics. This leads to the
challenging problem of estimating a Riemannian metric that is compatible with
the DWI data, i.e., a metric such that the geodesic curves represent individual
fiber tracts of the connectomics. We reduce this problem to that of solving a
highly nonlinear set of partial differential equations (PDEs) and study the
applicability of convolutional encoder-decoder neural networks (CEDNNs) for
solving this geometrically motivated PDE. Our method achieves excellent
performance in the alignment of geodesics with white matter pathways and
tackles a long-standing issue in previous geodesic tractography methods: the
inability to recover crossing fibers with high fidelity.
