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What's in here: the theory side of optimal transport. OT as energy minimization over distributions, the chain entropic OT ↔ Schrödinger bridge ↔ path-level energy-based model, and concrete sketches for running discrete OT as block-Gibbs sampling on stochastic hardware (THRML/Extropic-style). Its applied twin is Wasserstein distances and optimal transport — different page, not a duplicate.

Short answer: yes, in a very nontrivial, “these two worlds secretly share the same math” way. Let’s unpack it.

You’re basically asking:

Can energy-based / stochastic models like THRML + geodesic flows be used to

implement or approximate optimal transport

Conceptually:

• Optimal transport = “cheapest way to morph one probability distribution into another.”
• EBMs/THRML = “define an energy on configurations, then explore / relax / sample low-energy configurations via stochastic dynamics and block Gibbs on graphs.”
• The bridge: Wasserstein gradient flows, Schrödinger bridges, and “entropic OT.”

So:

1. OT as energy minimization on probability distributions

In standard OT:

• You have two distributions \mu_0, \mu_1.
• You want a coupling or a flow that minimizes:

  • Monge–Kantorovich cost:

    \min_{\gamma} \int c(x, y) \, d\gamma(x,y)

  • Or dynamic Benamou–Brenier formulation:

    \min_{\rho_t, v_t} \int_0^1 \int \frac{1}{2} \|v_t(x)\|^2 \rho_t(x) \, dx\, dt

    subject to continuity equation:

    \partial_t \rho_t + \nabla \cdot (\rho_t v_t) = 0,\quad \rho_0 = \mu_0, \ \rho_1 = \mu_1.

This is literally “geodesics” in Wasserstein space: probability distributions as points on a Riemannian manifold, with geodesics = optimal mass transport paths.

That already smells like:

• An energy functional over distributions.