Beyond Schrödinger Bridges: A Least-Squares Approach for Learning Stochastic Dynamics with Unknown Volatility

Scientists are often interested in inferring the underlying stochastic dynamics of systems from population-level snapshot data, where individual trajectories are unavailable. For example, in biological studies of immune cell activation, mRNA concentrations are measured at a single time point per cell because the measurement process kills the cells. Existing methods based on Schrödinger bridge techniques rely on Kullback–Leibler divergence and assume known, constant volatility, limiting their applicability in realistic settings where volatility may be unknown or varying. In this work, we propose a new framework that directly matches the joint distribution of the state (e.g., mRNA levels) and the time of observation using maximum mean discrepancy. This approach reduces to a least-squares formulation in distributional space and motivates an $R^2$-type goodness-of-fit measure for model inspection and comparison. We show in our experiments that the proposed method outperforms existing Schrödinger bridge–based baselines in forecasting and is robust to unknown volatility and missing observations.