  Transport processes are ubiquitous. They are, for example, at the heart of
optical flow approaches; or of perfusion imaging, where blood transport is
assessed, most commonly by injecting a tracer. An advection-diffusion equation
is widely used to describe these transport phenomena. Our goal is estimating
the underlying physics of advection-diffusion equations, expressed as velocity
and diffusion tensor fields. We propose a learning framework (YETI) building on
an auto-encoder structure between 2D and 3D image time-series, which
incorporates the advection-diffusion model. To help with identifiability, we
develop an advection-diffusion simulator which allows pre-training of our model
by supervised learning using the velocity and diffusion tensor fields. Instead
of directly learning these velocity and diffusion tensor fields, we introduce
representations that assure incompressible flow and symmetric positive
semi-definite diffusion fields and demonstrate the additional benefits of these
representations on improving estimation accuracy. We further use transfer
learning to apply YETI on a public brain magnetic resonance (MR) perfusion
dataset of stroke patients and show its ability to successfully distinguish
stroke lesions from normal brain regions via the estimated velocity and
diffusion tensor fields.
