Learning non-equilibrium diffusions with Schrödinger bridges: from exactly solvable to simulation-free

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 modelling systems driven by the gradient of a potential energy. 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 in which non-gradient forces are at play: this is important for applications to biological systems, which naturally exist out-of-equilibrium. In the case of Gaussian marginals, we derive explicit expressions that characterise exactly 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 an approximation to 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.

💡 Research Summary

The paper tackles the Schrödinger Bridge Problem (SBP) from a non‑equilibrium perspective by adopting a multivariate Ornstein‑Uhlenbeck (mvOU) process as the reference dynamics. Traditional SBP literature largely assumes a Brownian reference, which restricts the drift to the gradient of a potential energy and thus only models equilibrium, gradient‑driven systems. In contrast, the authors consider a generic drift matrix A∈ℝ^{d×d} in the SDE

 dXₜ = A Xₜ dt − m dt + σ dBₜ,

allowing A to be asymmetric. An asymmetric drift introduces non‑conservative forces, yielding irreversible dynamics and a non‑equilibrium steady state—features that are essential for realistic biological systems such as single‑cell dynamics.

The contributions are twofold. First, for Gaussian endpoint distributions ρ₀ and ρ₁, the authors derive closed‑form expressions for the Gaussian Schrödinger Bridge (GSB) under the mvOU reference. Using the disintegration identity that links the dynamic SBP to its static counterpart, they obtain an explicit SDE for the conditioned process (Theorem 1) and, consequently, analytic formulas for the time‑dependent mean μₜ, covariance Σₜ, score sₜ(x)=∇ₓ log pₜ|x₀,x₁(x), and probability flow uₜ(x)=A x−m+cₜ−½ σσᵀ∇ₓ log pₜ|x₀,x₁(x) (Theorem 2). The control term cₜ depends on a single one‑dimensional integral Λₜ⁻¹∫₀ᵗΛₛ ds, which is independent of the endpoints, making the computation scalable to high dimensions. The Brownian bridge appears as the special case A=0, confirming that the framework strictly generalizes the classical SBP.

Second, for arbitrary (non‑Gaussian) marginals, the paper introduces mvOU‑OTFM, a simulation‑free learning algorithm that combines conditional flow matching and score matching. The static SBP is solved as an entropy‑regularized optimal transport problem using the Sinkhorn‑Knopp algorithm, yielding the optimal coupling π. With the analytical expressions for the mvOU bridge’s score and flow in hand, the method minimizes a joint loss

 L(θ,φ)=E_{t,x₀,x₁}