Score matching for bridges without learning time-reversals

We propose a new algorithm for learning bridged diffusion processes using score-matching methods. Our method relies on reversing the dynamics of the forward process and using this to learn a score function, which, via Doob's $h$-transform, yields a bridged diffusion process; that is, a process conditioned on an endpoint. In contrast to prior methods, we learn the score term $\nabla_x \log p(t, x; T, y)$ directly, for given $t, y$, completely avoiding first learning a time-reversal. We compare the performance of our algorithm with existing methods and see that it outperforms using the (learned) time-reversals to learn the score term. The code can be found at https://github.com/libbylbaker/forward_bridge.