  The diffusion bridge is a type of diffusion process that conditions on
hitting a specific state within a finite time period. It has broad applications
in fields such as Bayesian inference, financial mathematics, control theory,
and shape analysis. However, simulating the diffusion bridge for natural data
can be challenging due to both the intractability of the drift term and
continuous representations of the data. Although several methods are available
to simulate finite-dimensional diffusion bridges, infinite-dimensional cases
remain unresolved. In the paper, we present a solution to this problem by
merging score-matching techniques with operator learning, enabling a direct
approach to score-matching for the infinite-dimensional bridge. We construct
the score to be discretization invariant, which is natural given the underlying
spatially continuous process. We conduct a series of experiments, ranging from
synthetic examples with closed-form solutions to the stochastic nonlinear
evolution of real-world biological shape data, and our method demonstrates high
efficacy, particularly due to its ability to adapt to any resolution without
extra training.
