Contact Wasserstein Geodesics for Non-Conservative Schr\"odinger Bridges

Andrea Testa; Leonel Rozo; S{\o}ren Hauberg; Tamim Asfour

arxiv: 2511.06856 · v4 · submitted 2025-11-10 · 💻 cs.LG · math.DG

Contact Wasserstein Geodesics for Non-Conservative Schr\"odinger Bridges

Andrea Testa , S{\o}ren Hauberg , Tamim Asfour , Leonel Rozo This is my paper

Pith reviewed 2026-05-17 23:52 UTC · model grok-4.3

classification 💻 cs.LG math.DG

keywords Schrödinger bridgeWasserstein geodesiccontact Hamiltonian mechanicsnon-conservative dynamicsResNetstochastic processesoptimal transportgenerative modeling

0 comments

The pith

The non-conservative Schrödinger bridge is solved by computing contact Wasserstein geodesics on a parameterized manifold using a ResNet.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to generalize the Schrödinger bridge so that it can describe stochastic processes in which energy is not conserved but instead varies over time. This generalization rests on contact Hamiltonian mechanics, which replaces the usual conservative structure with one that permits energy change while keeping the probabilistic interpretation. The authors then lift the resulting problem to the Wasserstein space, where the bridge becomes a geodesic that can be found by parameterizing the manifold and training a residual network. Because the solver is non-iterative and nearly linear in cost, the approach becomes practical for tasks that previously required heavy computation. A sympathetic reader would care because many real processes, from molecular motion to image synthesis under guidance, involve changing energy budgets that the older conservative bridges could not capture.

Core claim

By reformulating the Schrödinger bridge problem with contact Hamiltonian mechanics, the non-conservative generalized Schrödinger bridge allows energy to vary, and this problem is solved by computing the contact Wasserstein geodesic on a finite-dimensional parameterization of the Wasserstein manifold, which is implemented non-iteratively with a ResNet and supports guided generation through a task-specific metric.

What carries the argument

The contact Wasserstein geodesic, obtained by lifting the non-conservative bridge to a geodesic computation on the parameterized Wasserstein manifold.

If this is right

The framework captures richer intermediate dynamics than energy-conserving bridges.
It enables guided generation by adjusting a task-specific distance metric.
Computation runs with near-linear complexity instead of iterative optimization.
It applies successfully to manifold navigation, molecular dynamics, and image generation tasks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the ResNet approximation holds, the same parameterization technique might apply to other optimal transport problems outside the bridge setting.
Testing the method on physical systems with known energy variation profiles would reveal whether the contact dynamics match observed behavior.
Extending the approach to higher-dimensional or structured data could broaden its use in generative modeling.

Load-bearing premise

The contact Hamiltonian reformulation must preserve the original probabilistic meaning of the Schrödinger bridge, and the ResNet must approximate the true geodesic closely enough to avoid large biases in the learned dynamics.

What would settle it

A direct comparison on a simple case where the conservative solution is known, showing that the non-conservative paths produce energy profiles inconsistent with the contact equations or fail to recover the conservative limit when energy is fixed.

Figures

Figures reproduced from arXiv: 2511.06856 by Andrea Testa, Leonel Rozo, S{\o}ren Hauberg, Tamim Asfour.

**Figure 1.** Figure 1: Probability paths obtained under energy-conserving ( ), energy-decreasing ( ), and energy-increasing conditions ( ) (details in App. E.2). Energy variation increases modeling flexibility in applications where distributions at intermediate time steps are of interest. Inferring the stochastic process that most likely generates a set of sparse observations is a fundamental challenge, e.g., in cellular dynam… view at source ↗

**Figure 2.** Figure 2: Visualization of the ResNet transformation. Two successive pushforwards ρ tk−1 θ → ρ tk θ → ρ tk+1 θ on P +(M) are shown as local updates ∂tρ tk θ , ∂tρ tk+1 θ on tangent spaces. Each update is parameterized by θ k , θk+1 ∈ Θ, defining local coordinates on T P+(M). This coordinate system is not unique. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Two-Moons (top) and Checkerboard (bottom) benchmarks with guided variants (right) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 5.** Figure 5: Predictions from CWG (ours, top), GSBM (middle), and DSBM (bottom). The red row shows marginal samples [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 8.** Figure 8: CWG outputs before (top) vs. after guidance (place the item left). Metric Standard Guidance Centroid 35.8±11.1 22.3±2.9 FID 19.52±0.78 23.77±1.94 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗

**Figure 9.** Figure 9: Adult → Child image generation on the Unpaired Image Transfer experiment. input t0 t1 t2 t3 t4 Step 10 20 30 40 50 Estimated Age 18 years CWG e-c CWG e-d CWG e-i input t0 t1 t2 t3 t4 Step 10 20 30 40 50 18 years GSBM input t0 t1 t2 t3 t4 Step 10 20 30 40 50 18 years DSBM [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

**Figure 10.** Figure 10: Average age predictions at each generation time step for the images shown in Figure [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗

**Figure 11.** Figure 11: The same scalar function f, associated with the 1-form α = df, gives rise to two distinct vector fields under the symplectic (left) and contact (right) geometric structures. The streamlines of these vector fields are illustrated on a representation of the state manifold. In symplectic geometry, the streamlines are tangent to the level curves of f, representing isoenergetic trajectories where f remains con… view at source ↗

**Figure 12.** Figure 12: Reconstructions from the models in the Single Cell Sequencing experiment. [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗

**Figure 13.** Figure 13: Visual reconstruction (left) and geodesic distances of cell snapshots for the reference dataset ( [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗

**Figure 14.** Figure 14: Reconstructions from CWG (top), GSBM (middle), and DSBM (bottom) in the Robot Task Recon [PITH_FULL_IMAGE:figures/full_fig_p035_14.png] view at source ↗

**Figure 15.** Figure 15: Adult → Child image generation in the Unpaired Image Transfer using the CWG method. Input t1 t2 t3 t4 t5 [PITH_FULL_IMAGE:figures/full_fig_p037_15.png] view at source ↗

**Figure 16.** Figure 16: Adult → Child image generation Unpaired Image Transfer using the GSBM method. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_16.png] view at source ↗

**Figure 17.** Figure 17: Adult → Child image generation Unpaired Image Transfer using the DSBM method. input t0 t1 t2 t3 t4 Step 0 10 20 30 40 50 60 Estimated Age (a) CWG input t0 t1 t2 t3 t4 Step 10 20 30 40 50 60 70 Estimated Age (b) GSBM input t0 t1 t2 t3 t4 Step 0 10 20 30 40 50 60 70 Estimated Age (c) DSBM Input 1 Input 2 Input 3 Input 4 Input 5 Input 6 Input 7 Input 8 Input 9 Input 10 [PITH_FULL_IMAGE:figures/full_fig_p038… view at source ↗

**Figure 18.** Figure 18: Age prediction for the samples generated in the Unpaired Image Transfer experiment. [PITH_FULL_IMAGE:figures/full_fig_p038_18.png] view at source ↗

read the original abstract

The Schr\"odinger Bridge provides a principled framework for modeling stochastic processes between distributions; however, existing methods are limited by energy-conservation assumptions, which constrains the bridge's shape preventing it from model varying-energy phenomena. To overcome this, we introduce the non-conservative generalized Schr\"odinger bridge (NCGSB), a novel, energy-varying reformulation based on contact Hamiltonian mechanics. By allowing energy to change over time, the NCGSB provides a broader class of real-world stochastic processes, capturing richer and more faithful intermediate dynamics. By parameterizing the Wasserstein manifold, we lift the bridge problem to a tractable geodesic computation in a finite-dimensional space. Unlike computationally expensive iterative solutions, our contact Wasserstein geodesic (CWG) is naturally implemented via a ResNet architecture and relies on a non-iterative solver with near-linear complexity. Furthermore, CWG supports guided generation by modulating a task-specific distance metric. We validate our framework on tasks including manifold navigation, molecular dynamics predictions, and image generation, demonstrating its practical benefits and versatility.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a contact-geometry way to drop energy conservation in Schrödinger bridges and compute them with a ResNet geodesic solver, but the endpoint marginals may need extra enforcement.

read the letter

The core move here is to relax the usual energy-conserving assumption in Schrödinger bridges by lifting the problem into contact Hamiltonian mechanics. That produces the non-conservative generalized Schrödinger bridge and lets the process carry an explicit time-varying energy variable. They then parameterize the Wasserstein manifold so the bridge becomes a geodesic search in a finite-dimensional space, solved non-iteratively by a ResNet with near-linear cost. The guided-generation trick that modulates a task-specific distance metric is a straightforward add-on that fits the same framework. Experiments on manifold navigation, molecular dynamics, and image generation show the method can be run end-to-end and produces usable samples, which is more than many abstract reformulations deliver. The implementation detail and the concrete applications are the parts that feel solid. The open question is whether the contact flow automatically pins the process to the prescribed source and target marginals once energy is allowed to vary. Nothing in the abstract or the high-level description shows an explicit projection or multiplier that restores the two-point boundary conditions after the non-conservative dynamics are integrated. If the learned trajectories drift at the endpoints, the output is a contact geodesic but not necessarily a bridge. I would want to see the exact derivation of the geodesic objective and numerical checks that the marginals are recovered to machine precision. This is aimed at people already working on optimal transport or score-based generative models who need to handle processes whose energy is not conserved. A reader who wants a practical, non-iterative alternative to classical Schrödinger-bridge solvers will get something usable here, even if the theoretical guarantees still need tightening. It is worth sending to a serious referee because the construction is new, the code path is clear, and the applications are relevant; the marginal-matching issue is fixable with targeted revisions rather than a fatal flaw.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces the non-conservative generalized Schrödinger bridge (NCGSB) as an energy-varying reformulation of the Schrödinger bridge problem using contact Hamiltonian mechanics. It proposes to lift the problem to contact Wasserstein geodesics (CWG) in a parameterized finite-dimensional space, implemented efficiently with a ResNet architecture and a non-iterative solver, and applies the framework to manifold navigation, molecular dynamics predictions, and image generation.

Significance. Should the contact lift and ResNet approximation prove to accurately solve the marginal-constrained problem with varying energy, the approach would offer a computationally advantageous method for modeling non-conservative stochastic processes, potentially broadening the applicability of Schrödinger bridges in machine learning and scientific computing. The non-iterative solver and guided generation via distance metric modulation are potential strengths if validated.

major comments (2)

Abstract: The abstract states the reformulation and ResNet implementation but provides no derivation steps, error analysis, or ablation results; therefore the central claim that the contact lift yields faithful varying-energy dynamics remains unverified from the given text.
Abstract: The construction does not explicitly demonstrate that the contact Hamiltonian flow preserves the two-point marginal constraints while allowing energy variation; without this, the learned ResNet trajectories may constitute contact geodesics but not valid bridges satisfying the endpoint marginals.

minor comments (1)

Abstract: Consider adding a brief quantitative comparison to existing iterative Schrödinger bridge methods to substantiate the claimed near-linear complexity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the recommendation for major revision. We address each major comment point by point below, with clear indications of the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: Abstract: The abstract states the reformulation and ResNet implementation but provides no derivation steps, error analysis, or ablation results; therefore the central claim that the contact lift yields faithful varying-energy dynamics remains unverified from the given text.

Authors: We agree that the abstract is concise by design and does not include detailed derivations, error bounds, or ablation results. The full manuscript provides these elements: the derivation of the NCGSB via contact Hamiltonian mechanics appears in Section 2, the contact lift and geodesic formulation in Section 3, theoretical error analysis in Section 3.3, and ablation studies plus empirical validation of varying-energy dynamics in Section 4. These sections support the central claim. We will revise the abstract to include a brief reference to the theoretical guarantees and experimental verification of faithful varying-energy dynamics. revision: yes
Referee: Abstract: The construction does not explicitly demonstrate that the contact Hamiltonian flow preserves the two-point marginal constraints while allowing energy variation; without this, the learned ResNet trajectories may constitute contact geodesics but not valid bridges satisfying the endpoint marginals.

Authors: The manuscript constructs the contact Hamiltonian flow to preserve the two-point marginal constraints by design, extending the standard Schrödinger bridge while the contact structure permits energy variation without violating the endpoint marginals. This preservation is formalized in the NCGSB definition and established in Theorem 2.1 and the surrounding propositions in Section 2. The ResNet approximates the resulting geodesics, with marginal constraints enforced via the training objective. We acknowledge that the abstract does not explicitly state this property and will revise it to include a concise statement confirming that the contact Hamiltonian flow preserves the marginal constraints while allowing energy variation. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces a novel non-conservative generalized Schrödinger bridge (NCGSB) reformulation via contact Hamiltonian mechanics, then parameterizes the Wasserstein manifold to recast the problem as a geodesic computation solved non-iteratively with a ResNet. No quoted equations or steps reduce a claimed prediction or first-principles result to the inputs by construction, nor do they rely on self-citation chains, imported uniqueness theorems, or ansatzes smuggled from prior author work. The ResNet serves as an implementation vehicle for the lifted geodesic rather than a fitted quantity renamed as output. The derivation remains self-contained with independent content from the new contact geometry and parameterization.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 2 invented entities

The framework rests on the domain assumption that contact Hamiltonian mechanics can be lifted to the Wasserstein manifold while preserving the bridge's probabilistic meaning, plus the practical assumption that a ResNet can faithfully approximate the resulting geodesic.

free parameters (1)

ResNet weights
Learned parameters that realize the finite-dimensional geodesic parameterization.

axioms (1)

domain assumption Contact Hamiltonian mechanics extends the conservative Schrödinger bridge to energy-varying processes without breaking the underlying stochastic interpretation.
Invoked to justify the NCGSB reformulation.

invented entities (2)

Non-conservative generalized Schrödinger bridge (NCGSB) no independent evidence
purpose: To model stochastic processes whose energy is allowed to change.
New object introduced by the contact-mechanics lift.
Contact Wasserstein geodesic (CWG) no independent evidence
purpose: Tractable finite-dimensional surrogate for the bridge problem.
New computational object realized by the ResNet.

pith-pipeline@v0.9.0 · 5497 in / 1416 out tokens · 45536 ms · 2026-05-17T23:52:16.593735+00:00 · methodology

discussion (0)

Reference graph

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so that samples corresponding to placements in undesired locations are discouraged, while those leading to desirable targets are favored. The centroid extraction is performed by applying morphological opening and closing operations from the OpenCV library to remove noise and refine object boundaries, followed by color-based masking to isolate the object o...

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Oxford Univer- sity Press (2018).https://doi.org/10.1093/oso/9780198814788.001.0001

URLhttps://proceedings.neurips.cc/paper_files/paper/2020/ file/4c5bcfec8584af0d967f1ab10179ca4b-Paper.pdf. 12 Under review as a conference paper at ICLR 2026 Peter Holderrieth and Ezra Erives. Introduction to flow matching and diffusion models, 2025. URL https://diffusion.csail.mit.edu/. Wanli Hong, Yuliang Shi, and Jonathan Niles-Weed. Trajectory inferen...

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URLhttps://doi.org/10.1038/srep04290. Solomon Kullback. Probability densities with given marginals.The Annals of Mathematical Statis- tics, 39(4):1236–1243, 1968. URLhttps://www.jstor.org/stable/2239692. Flavien L´eger and Wuchen Li. Hopf–cole transformation via generalized Schr ¨odinger bridge prob- lem.Journal of Differential Equations, 274:788–827, 202...

work page doi:10.1038/srep04290 1968

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URLhttps://openreview.net/forum?id=QwoGfQzuMa. Lars Ruthotto, Stanley J Osher, Wuchen Li, Levon Nurbekyan, and Samy Wu Fung. A machine learning framework for solving high-dimensional mean field game and mean field control prob- lems.Proceedings of the National Academy of Sciences, 117(17):9183–9193, 2020. URL https://doi.org/10.1073/pnas.1922204117. Erwin...

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URLhttps://proceedings.mlr.press/v180/shi22a.html

PMLR, 2022. URLhttps://proceedings.mlr.press/v180/shi22a.html. Yuyang Shi, Valentin De Bortoli, Andrew Campbell, and Arnaud Doucet. Diffusion Schr ¨odinger bridge matching. InAdvances on Neural Information Processing Systems (NeurIPS), 2023. URL https://openreview.net/forum?id=qy07OHsJT5. Ana Cannas da Silva. Lectures on symplectic geometry.Lecture Notes ...

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Bugallo , year=

URLhttps://openreview.net/forum?id=00LViVLqkN. Jiaming Song, Qinsheng Zhang, Hongxu Yin, Morteza Mardani, Ming-Yu Liu, Jan Kautz, Yongxin Chen, and Arash Vahdat. Loss-guided diffusion models for plug-and-play controllable gen- eration. InInternational Conference on Machine Learning (ICML), 2023. URLhttps: //proceedings.mlr.press/v202/song23k.html. 14 Unde...

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Panagiotis Theodoropoulos, Augustinos D Saravanos, Evangelos A Theodorou, and Guan-Horng Liu

URLhttps://openreview.net/forum?id=v2nQ1e78Rc. Panagiotis Theodoropoulos, Augustinos D Saravanos, Evangelos A Theodorou, and Guan-Horng Liu. Momentum multi-marginal Schr¨odinger bridge matching.arXiv preprint arXiv:2506.10168,

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Sahand Tokasi and Peter Pickl

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work page doi:10.1093/bioinformatics/btae393 2022

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so that samples corresponding to placements in undesired locations are discouraged, while those leading to desirable targets are favored. The centroid extraction is performed by applying morphological opening and closing operations from the OpenCV library to remove noise and refine object boundaries, followed by color-based masking to isolate the object o...

work page 2026