Reusable Schrödinger Bridges from Coarse Endpoint Interactions

Schrödinger bridges connect prescribed distributions through reference dynamics and provide a common framework for control, generation, and scientific inference. When many endpoint pairs share the same dynamics, can their bridges also share a smaller optimization problem? We show that fine endpoint information and the interaction governing endpoint pairing can be represented separately. Because the marginals are fixed, additive single-endpoint terms leave the optimizer unchanged. Approximating the remaining interaction on coarse states therefore gives a reusable solver that preserves the original fine marginals exactly. For positive finite references, a uniform log-interaction residual $\eta$ bounds each Kullback-Leibler (KL) direction by $$\begin{gathered} min \\ 2 \\ eta, \\ frac \\ et{a}^{2}2 \end{gathered}$$, with a sharp quadratic coefficient. A sparse-query converse then leads to a reference-only partition rule with an equal-budget approximation guarantee. We further decompose query error and quantify the dependence recovered by nested refinement, while fast mixing explains why coarse interactions arise dynamically. Our experiments verify the predicted error structure and demonstrate that reusing coarse endpoint interactions provides a valid and practical way to solve repeated bridge queries in molecular and stochastic-dynamics settings.