Score Matching via Differentiable Physics. (arXiv:2301.10250v1 [cs.LG])

Diffusion models based on stochastic differential equations (SDEs) gradually
perturb a data distribution $p(mathbf{x})$ over time by adding noise to it. A
neural network is trained to approximate the score $nabla_mathbf{x} log
p_t(mathbf{x})$ at time $t$, which can be used to reverse the corruption
process. In this paper, we focus on learning the score field that is associated
with the time evolution according to a physics operator in the presence of
natural non-deterministic physical processes like diffusion. A decisive
difference to previous methods is that the SDE underlying our approach
transforms the state of a physical system to another state at a later time. For
that purpose, we replace the drift of the underlying SDE formulation with a
differentiable simulator or a neural network approximation of the physics. We
propose different training strategies based on the so-called probability flow
ODE to fit a training set of simulation trajectories and discuss their relation
to the score matching objective. For inference, we sample plausible
trajectories that evolve towards a given end state using the reverse-time SDE
and demonstrate the competitiveness of our approach for different challenging
inverse problems.