Learning non-equilibrium diffusions with Schrödinger bridges: from exactly solvable to simulation-free
This work addresses the challenge of modeling non-equilibrium dynamics in biological systems, offering an incremental improvement over existing methods by extending beyond potential-driven regimes.
The paper tackles the problem of reconstructing the most likely evolution of stochastic dynamical systems from initial and final ensemble measurements, particularly for non-equilibrium systems with non-conservative forces, by proposing a simulation-free algorithm called mvOU-OTFM that achieves higher accuracy and faster training compared to competing methods on synthetic and real single-cell data.
We consider the Schrödinger bridge problem which, given ensemble measurements of the initial and final configurations of a stochastic dynamical system and some prior knowledge on the dynamics, aims to reconstruct the "most likely" evolution of the system compatible with the data. Most existing literature assume Brownian reference dynamics and are implicitly limited to potential-driven dynamics. We depart from this regime and consider reference processes described by a multivariate Ornstein-Uhlenbeck process with generic drift matrix $\mathbf{A} \in \mathbb{R}^{d \times d}$. When $\mathbf{A}$ is asymmetric, this corresponds to a non-equilibrium system with non-conservative forces at play: this is important for applications to biological systems, which are naturally exist out-of-equilibrium. In the case of Gaussian marginals, we derive explicit expressions that characterise the solution of both the static and dynamic Schrödinger bridge. For general marginals, we propose mvOU-OTFM, a simulation-free algorithm based on flow and score matching for learning the Schrödinger bridge. In application to a range of problems based on synthetic and real single cell data, we demonstrate that mvOU-OTFM achieves higher accuracy compared to competing methods, whilst being significantly faster to train.