Title
Integrating Schrödinger Half-Bridges
Abstract
Schrödinger half bridges minimise a cost functional consisting of a running cost and a terminal cost. The initial particle distribution is fixed, and the terminal cost penalises distance from an assigned reference distribution. Modelling this set-up in physically realistic dynamics, such as those in nanoscale information processing devices, is highly non-linear and generally unsolvable with conventional numerical methods.
Inspired by generative diffusion models, we use a neural network to parametrise the drift and perform gradient descent over the cost functional. By using a Monte Carlo method to estimate the distance from the reference distribution, we eliminate the need for any discretisation in space, which makes our method generalisable to higher dimensions. Our approach means that we can obtain solutions for parameter values even in regimes where semi-analytic results are no longer valid, which also includes weighting the cost functional such that we approach a full bridge (where the final particle distribution is fixed). We share proof-of-concept results that are ready for scaling up.
