From geometry to dynamics: Learning overdamped Langevin dynamics from sparse observations with geometric constraints

Dimitra Maoutsa

REVIEW 3 major objections 4 minor 9 references

From geometry to dynamics: Learning overdamped Langevin dynamics from sparse observations with geometric constraints

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper establishes that geometry-guided path augmentation recovers drift from sparse samples of non-conservative Langevin systems.

desk verdict Genuinely new integration, well-specified, but the geometric premise is unproven and likely false for non-conservative systems; deserves a rigorous referee with major revision. read the letter →

arxiv 2512.23566 v2 pith:NF6TDTGZ submitted 2025-12-29 math.DS cond-mat.stat-mechcs.LGmath.OCstat.ML

Dimitra Maoutsa This is my paper

classification math.DScond-mat.stat-mechcs.LGmath.OCstat.ML MSC 62M0560H1062G05

keywords overdampedLangevinsystemsdriftestimationsparseobservationspathaugmentationinvariantdensitygeometrystochasticoptimalcontrolGaussianprocessinferenceRiemannianmetriclearning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the drift of an overdamped Langevin system can be recovered from very sparsely sampled trajectories, if the missing inter-observation paths are inferred with the help of the geometry of the system's invariant density. The method learns a Riemannian metric from the observed density, constructs geodesics between successive observations, and solves a stochastic control problem to generate diffusion bridges constrained to stay near those geodesics. Gaussian-process drift inference on the augmented paths, iterated within an expectation-maximization loop, is reported to recover non-conservative force fields even when only one observation per oscillation period is available. If correct, this would extend nonparametric system identification into a regime where both temporal methods (which need fine sampling) and geometric methods (which need conservative forces) fail.

What carries the argument

The central mechanism is the geometry-guided diffusion bridge: the solution of a stochastic control problem whose objective penalizes deviation from the geodesic connecting consecutive observations under the learned metric, alongside the usual observation constraints. The metric is learned nonparametrically as the inverse of the weighted local diagonal covariance of the observed states; the geodesic is the energy-minimizing curve under that metric. The optimal control is expressed as the difference of log-gradients of a forward filtering density and a time-reversed density, which yields a tractable particle-based bridge sampler. This bridge is what injects the geometric inductive bias into t

What would settle it

Run the method on a two-dimensional system whose invariant density is rotationally symmetric (so the learned metric is near-isotropic and the geodesics between observation pairs are nearly straight chords) while the true most-probable transition paths of the SDE are strongly curved arcs. If geometry-guided augmentation no longer beats Ornstein-Uhlenbeck augmentation, the reported gains depend on the empirical geometry coinciding with the true transition geometry; if it still wins, the benefit comes from the control formulation itself, not from geodesic alignment.

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Extended reading notes

Core claim

On its own terms, the central discovery is that the curvature of the invariant density carries enough information about the unobserved continuous path to make drift inference possible from sparse observations, even for non-conservative systems. The method learns a nonparametric Riemannian metric from the inverse local diagonal covariance of the observations, computes geodesics between consecutive observations as energy-minimizing curves, and solves a stochastic control problem to generate diffusion bridges that pass through the observations while staying near those geodesics. The augmented paths feed a Gaussian-process expectation-maximization loop that estimates the drift. The paper reports

Load-bearing premise

The load-bearing premise is that the geodesics of the invariant density learned from observations are a reliable guide to the true unobserved transition paths of the Euclidean SDE; the paper itself concedes that the observed process is not a diffusion on the learned Riemannian manifold and offers only a speculative inverse-Lamperti-transform argument, not a proof.

Editorial extensions

If this is right

Drift inference for overdamped Langevin systems becomes possible from data with effectively one sample per oscillation period, without assuming a parametric model for the force field.
Non-conservative systems with stationary probability currents, which are inaccessible to potential-based geometric methods, fall within the scope of geometric augmentation.
The geometric constraint acts as a regularizer that reduces the variance of drift estimates, as evidenced by the reported smaller error bars relative to Ornstein-Uhlenbeck bridging.
Incorrectly specified diffusion constants (within a modest range) do not destroy the accuracy of the inferred drift after two augmentation iterations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct but untested extension is to state-dependent noise: since the control formulation only requires a forward/reverse generator pair, the same geodesic constraint should apply when the diffusion coefficient is known but position-dependent.
The paper's justification via an inverse Lamperti transform hints at a deeper equivalence: the learned metric may equal the invariant geometry of a related diffusion with state-dependent noise, in which case geodesics would be the Onsager-Machlup most-probable paths of that process; testing this equivalence on a system with a known Lamperti transform is a concrete next step.
The paper does not state that the need for a well-resolved invariant density implies the method is best suited to ergodic data with long observation windows; for non-stationary or transient dynamics the geodesic prior is likely to be misleading, so a check that the observed density has stabilized should accompany practical use.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: this paper has a genuinely new integration—metric learning from the invariant density, geodesic-constrained bridge sampling, and GP drift inference inside an EM loop—and it is specified well enough to implement from the text alone. But the load-bearing premise, that geodesics of the learned metric approximate the true transition paths of the Euclidean SDE, is not proved and is generically false for non-conservative systems. The paper's own Table 2 shows the method losing to LatentSDE on exactly the out-of-equilibrium system it claims to target.

Credit where due: the variational derivation (Eqs. 7-11) is internally coherent, the GP update is standard, and the algorithmic pipeline is detailed to the level of pseudocode. The comparison set is broad, including recent Schr\"odinger-bridge and flow-matching baselines. For the Van der Pol system, the method substantially improves over GP and OU augmentation at large inter-observation intervals. The circularity burden is low: the drift is estimated from augmented paths and scored against ground truth, so it is not a fit-then-predict artifact.

The soft spots are significant. The stress-test note is right: for a fixed invariant density \pi, any divergence-free current C with \nabla\cdot(C\pi)=0 gives a different drift f with the same \pi, so the learned metric H is blind to the non-conservative part. The geodesic constraint cannot select among these drifts. This is not a minor technical gap; it is the core of the method. Section C concedes as much, offering only a hypothetical inverse Lamperti transform and no proof. The paper's own out-of-equilibrium results confirm the failure: Geometric (ours) is consistently worse than LatentSDE, and at tau=150 worse than plain GP. The abstract's claim of "outperforming existing methods" is not supported by the full tables. A lesser but real weakness is the lack of code or data, which makes the synthetic benchmarks hard to reproduce.

My take: this is a promising recipe for systems with sharply concentrated invariant densities, not a general non-conservative inference tool. Send it to a serious referee, but expect major revision: a rigorous derivation or a falsifiable condition for when the geodesic constraint is valid, honest reporting of full benchmark tables, and code release. As it stands, the method is a useful engineering contribution with an overclaimed theoretical foundation.

Referee Report

3 major / 4 minor

Summary. The paper proposes a nonparametric method for estimating the drift of an overdamped Langevin SDE from sparsely sampled trajectories. The method has three components: (i) a Riemannian metric learned from the observed invariant density, Eq. (4); (ii) geodesics between consecutive observations computed under that metric, Eq. (5), used as soft constraints in a variational path-augmentation scheme, Eqs. (7)-(11); and (iii) Gaussian-process drift inference from the augmented paths, Eq. (12), iterated in an EM framework. The central claim is that geometry-driven path augmentation recovers the drift even for large inter-observation intervals and for non-conservative systems, where standard Gaussian-likelihood or Ornstein-Uhlenbeck augmentation fails. Experiments on Van der Pol, an out-of-equilibrium harmonic-plus-circulation system, Hopf, and Selkov models are compared with several baselines.

Significance. If the central premise holds, the paper offers a useful synthesis of geometric and temporal approaches to stochastic system identification, with a concrete algorithmic pipeline (particle-based stochastic control, sparse GP drift inference) that is described at implementation level and should be reproducible. The variational derivation and the GP update are internally coherent, and the experiments include several non-conservative benchmarks. However, the load-bearing geometric premise is not proven and is questionable for non-conservative systems; moreover, the paper's own Table 2 shows that the proposed method is not uniformly superior in the non-conservative setting it emphasizes. The significance of the contribution therefore depends on whether the geometric constraint can be justified as a principled approximation or only as an ad hoc regularizer.

major comments (3)

[Sec. C (Eq. 57) and Discussion, last paragraph] The central premise—that geodesics of the metric H learned from the invariant density identify the most probable unobserved transition paths of the Euclidean SDE (Eq. 1)—is unproved and generically false for non-conservative systems. For a fixed invariant density there are many drifts producing the same stationary distribution (adding any divergence-free current C with ∇·(Cπ)=0 leaves π unchanged). The learned metric H in Eq. (4) is identical for all members of this family, so geodesic constraints cannot select the correct drift. The Onsager-Machlup Lagrangian in Eq. (57) depends on f and ∇·f, not on H; geodesics of H characterize a different diffusion on (R^d,H). Section C explicitly concedes this and invokes only a hypothetical inverse Lamperti transform, with no proof. The Discussion's claim that geodesics are 'the most probable path ... in the Onsager-Machlup sense' is therefore not
[Table 2 and Sec. 4, 'For most settings... outperformed'] The numerical results for the out-of-equilibrium system directly contradict the paper's general claim of superiority for non-conservative systems. In Table 2, for τ=150/200/250, Geometric (ours) gives wRMSE 2.762/3.034/2.693, while LatentSDE gives 2.348/2.340/2.356; at τ=150, plain GP achieves 2.632, i.e., the proposed method is also worse than the Gaussian baseline. Thus, in the very setting that motivates the paper—non-conservative dynamics with a stationary probability current—the geodesic constraint does not outperform simpler alternatives. The paper should either explain this failure, restrict its claims to systems whose invariant density is strongly concentrated (e.g., Van der Pol), or provide a non-conservative benchmark where the method is actually superior.
[Sec. H.2, Eqs. (93)-(94)] The theoretical analysis shows that Euler-Maruyama-based drift inference neglects terms involving flow curvature, specifically ∇(Jf f)·f and ∇(½Δ_D f)·f, producing a bias of O(τ). However, the analysis does not establish that geodesics of the learned metric H correct this bias. The bias terms are functionals of the true drift f and its derivatives; they are not expressed in terms of the invariant-density metric H, and no theorem or calculation connects the geodesic constraint U_G(x,t)=‖Γ_t−x‖² to these remainder terms. The paper asserts rather than derives this link. Without such a derivation, the geometric augmentation remains an externally imposed regularizer whose success is system-dependent. The authors should either supply the missing connection or present a targeted experiment that isolates the curvature bias and demonstrates that geodesic constraints remove it.

minor comments (4)

[Sec. 5 and Sec. C] The transform is called both 'Lamperti' and 'Lamberti'; the standard spelling is Lamperti. Please correct.
[Algorithms A2-A4] The algorithm listing contains undefined placeholder notation: 'BALL2' and 'gbALL2' appear without explanation, and the comment in Algorithm A2 refers to 'BALL2' as if it were a variable. Please clean up the pseudocode.
[Fig. 3 caption] The caption says 'one observation per oscillation period' for large τ, but the corresponding parameter values in Sec. J are not explicitly tied to oscillation periods. Please clarify the relationship between τ and the system's characteristic time scale, since this claim is used to interpret the results.
[Eq. (7)] In the second line of Eq. (7) the observation term U_O(x,t) is written with a delta function in the text but is omitted from the displayed integral; make the display consistent with the SI version in Eq. (37).

Circularity Check

0 steps flagged · score 2.0 of 10

No meaningful circularity; the drift estimate is benchmarked against independent ground truth. The geometric premise is unproved but not circular.

full rationale

The paper's derivation chain does not reduce to its inputs. The Riemannian metric H (Eq. 4) is learned from the observed invariant density; geodesics are computed from H; these geodesics enter the control objective as U_G (Eqs. 7 and 37); and the drift is updated from the controlled path measure via the GP objective (Eqs. 12 and 42). No fitted parameter is later relabeled as a prediction: the drift estimates are compared to separately specified ground-truth drifts in synthetic benchmarks (Van der Pol, out-of-equilibrium, Hopf, Selkov), so the reported wRMSE values are independent checks. The main self-citations (Maoutsa & Opper 2021/2022 for the particle control solver and the author's own workshop reports) are methodological; the cited solver supplies the bridge construction machinery, but the geometric-augmentation idea is implemented and tested here rather than assumed from those citations. The load-bearing premise that geodesics of the learned metric are the most probable paths of the Euclidean SDE is not established: Sec. C states that justifying the approach 'may be possible' only through an inverse Lamperti transform and that 'The existence of such a transformation would justify the proposed method,' with no proof given. That is a correctness or validation gap, not circularity, because the method's output is not defined in terms of the ground truth or in terms of the premise. Table 2's worse out-of-equilibrium results are likewise an empirical limitation rather than circular reasoning. There is therefore no exhibited reduction of a claimed prediction to an input by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

Free parameters are hand-set weights, bandwidths, and GP/numerical hyperparameters; none are optimized against the ground-truth drift. The main axiom burden is the geodesic-as-most-probable-path premise (Sec. C), which the author explicitly labels as 'may be possible' rather than proven. No invented physical entities: the metric and geodesic objects are data-driven constructs, not new postulated forces or particles.

free parameters (5)

β (geodesic-constraint weight) = 0.5 in Tables 1-2; 1.0 in Fig. 2
Weight of the geometric term U^G in the free energy (Eq. 7/37). The paper notes β=1 gives a better approximation of the transition density than β=0.5, so results depend on this hand-set constant.
σ_M (metric kernel bandwidth)
Bandwidth in weights w_k(x)=exp(-||O_k-x||^2/(2σ_M^2)) in Eq. 4/34; characterizes the curvature of the empirical manifold and must be chosen per dataset.
ε (metric regularization)
Small constant ensuring non-zero diagonal covariance in Eq. 4/34; contributes to the scale of the learned metric.
GP kernel lengthscales (ℓ1, ℓ2, ℓ3)
Hyperparameters of the sparse Gaussian process drift estimator (Algorithm A3); given as shared across dimensions but not selected or reported.
Discretization sizes N, S, M = N=100 particles, S=300 GP inducing points, M=40 score-estimator points
Numerical choices for the particle control solver and sparse GP; trade accuracy against computational cost and are set by hand.

assumptions (6)

domain assumption The invariant density of the observed SDE induces a Riemannian metric that is well approximated by the inverse of the weighted local diagonal covariance (Eq. 4); observations are discrete samples of this empirical manifold.
Core geometric premise. Sec. G motivates low-dimensional concentration for dissipative systems and small-noise exponential concentration, but the specific metric choice follows Arvanitidis et al. 2019 and is not derived from the SDE.
ad hoc to paper Geodesics with respect to the learned metric identify the most probable unobserved paths and can serve as control constraints in path augmentation.
Sec. C shows the Onsager-Machlup Lagrangian for a Euclidean diffusion (Eq. 57) differs from the Riemannian geodesic Lagrangian (Eq. 58); the reconciliation via an inverse Lamperti transform is explicitly conjectural ('may be possible').
domain assumption The drift is time-homogeneous and the diffusion coefficient is known, constant, and state-independent.
Required by Eq. 1 and by the control formulation. The paper tests robustness to ±0.1 noise misestimation, but not unknown or state-dependent diffusion.
standard math The path-integral control solution from Maoutsa & Opper (2022) correctly constructs the conditioned diffusion bridge.
The control equations (Eqs. 8-11 and 38-41) are taken from this prior peer-reviewed work and are not independently re-derived in the manuscript.
domain assumption Observations are exact, i.e. no measurement noise is present, so the observation potential U^O can use a Dirac likelihood.
The free energy in Eq. 7 uses δ-function observation constraints; the paper does not treat observation error.
domain assumption The observation window is long enough for the empirical manifold to resolve the invariant geometry.
Stated in the Limitations section: sufficiently long observation windows are required to accurately characterize the invariant density.

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Cite this review

Pith. "Pith review of From geometry to dynamics: Learning overdamped Langevin dynamics from sparse observations with geometric constraints." pith.science (2026). https://pith.science/paper/NF6TDTGZ

@misc{pith2026251223566,
  author       = {Pith},
  title        = {Pith review of: From geometry to dynamics: Learning overdamped Langevin dynamics from sparse observations with geometric constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NF6TDTGZ}},
  note         = {Machine review of arXiv:2512.23566}
}

read the original abstract

How can we learn the laws underlying the dynamics of stochastic systems when their trajectories are sampled sparsely in time? Existing methods either require temporally resolved high-frequency observations, or rely on geometric arguments that apply only to conservative systems, limiting the range of dynamics they can recover. Here, we present a new framework that reconciles these two perspectives by reformulating inference as a stochastic control problem. Our method uses geometry-driven path augmentation, guided by the geometry in the system's invariant density to reconstruct likely trajectories and infer the underlying dynamics without assuming specific parametric models. Applied to overdamped Langevin systems, our approach accurately recovers stochastic dynamics even from extremely undersampled data, outperforming existing methods in synthetic benchmarks. This work demonstrates the effectiveness of incorporating geometric inductive biases into stochastic system identification methods.

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Reference graph

Works this paper leans on

9 extracted references · 1 linked inside Pith

[1]

Gaussian process regression without state estimation (GP)
[2]

path augmentation with Ornstein-Uhlenbeck dynamics with Gaussian process inference (OU) [Batz et al., 2018]

2018
[3]

sparse variational inference with state estimation (SVISE) [Course and Nair, 2023a]
[4]

the drift function (KM- basis) [Nabeel et al., 2025]

basis function approximation of Kramers-Moyal coefficients, i.e. the drift function (KM- basis) [Nabeel et al., 2025]

2025
[5]

We further compared our method with recent Schroedinger bridge generating frameworks that primary aim to infer population dynamics from snapshot data

latent SDE inference with amortized reparameterization with (LatentSDE+GP-pre) and without pre-training (LatentSDE) [Course and Nair, 2023b]. We further compared our method with recent Schroedinger bridge generating frameworks that primary aim to infer population dynamics from snapshot data. In particular we considered the following frameworks: I. Metric ...

2024
[1917]

Vari- ational inference for diffusion processes.Advances in Neural Information Processing Sys- tems, 20:17–24, 2007.(cited on page: 39) Xuechen Li, Ting-Kam Leonard Wong, Ricky T

PMLR, 2024.(cited on page: 39) C´edric Archambeau, Manfred Opper, Yuan Shen, Dan Cornford, and John Shawe-Taylor. Vari- ational inference for diffusion processes.Advances in Neural Information Processing Sys- tems, 20:17–24, 2007.(cited on page: 39) Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, and David Duvenaud. Scalable gradients for Stochastic ...

arXiv 2024
[2018]

However, during inference the true dynamics are unknown and the local linearisations on inaccurate drift estimates employed in Batz et al

provides a good approximation of the underlying transition density, when the underlying linear process employed for each bridge has a drift that comes from the local linearisation of theground truthdrift function. However, during inference the true dynamics are unknown and the local linearisations on inaccurate drift estimates employed in Batz et al. [201...

2018
[2020]

following the drift fµ(x) =−Ω µνxν +αe −x2/2σ2 xµ withΩ= 2 2 −2 2 , (116) forα= 10 and simulated the stochastic system with noise amplitudeσ= 0.5 on a time grid of dt= 0.01 steps, observed at inter-observation intervalsτ={150, 200, 250} ×dtand for total durationT= 1000 time units. For theHopf systemwe used the drift f1(x1,x 2) =z 2, (117) f2(x1,x 2) =−z 1...

1968

Show all 9 references

[2022]

Metric flow matching [Kapusniak et al., 2024] interpolates data distributions that respect the geodesic interpolants computed according to the metric induced by the observations

proposed a score-based generative model that models target densities with support on prescribed Riemannian manifolds in terms of a time-reversal of Langevin dynamics. Metric flow matching [Kapusniak et al., 2024] interpolates data distributions that respect the geodesic interp...

2024

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Gaussian process regression without state estimation (GP)

[2] [2]

path augmentation with Ornstein-Uhlenbeck dynamics with Gaussian process inference (OU) [Batz et al., 2018]

2018

[3] [3]

sparse variational inference with state estimation (SVISE) [Course and Nair, 2023a]

[4] [4]

the drift function (KM- basis) [Nabeel et al., 2025]

basis function approximation of Kramers-Moyal coefficients, i.e. the drift function (KM- basis) [Nabeel et al., 2025]

2025

[5] [5]

We further compared our method with recent Schroedinger bridge generating frameworks that primary aim to infer population dynamics from snapshot data

latent SDE inference with amortized reparameterization with (LatentSDE+GP-pre) and without pre-training (LatentSDE) [Course and Nair, 2023b]. We further compared our method with recent Schroedinger bridge generating frameworks that primary aim to infer population dynamics from snapshot data. In particular we considered the following frameworks: I. Metric ...

2024

[6] [1917]

Vari- ational inference for diffusion processes.Advances in Neural Information Processing Sys- tems, 20:17–24, 2007.(cited on page: 39) Xuechen Li, Ting-Kam Leonard Wong, Ricky T

PMLR, 2024.(cited on page: 39) C´edric Archambeau, Manfred Opper, Yuan Shen, Dan Cornford, and John Shawe-Taylor. Vari- ational inference for diffusion processes.Advances in Neural Information Processing Sys- tems, 20:17–24, 2007.(cited on page: 39) Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, and David Duvenaud. Scalable gradients for Stochastic ...

arXiv 2024

[7] [2018]

However, during inference the true dynamics are unknown and the local linearisations on inaccurate drift estimates employed in Batz et al

provides a good approximation of the underlying transition density, when the underlying linear process employed for each bridge has a drift that comes from the local linearisation of theground truthdrift function. However, during inference the true dynamics are unknown and the local linearisations on inaccurate drift estimates employed in Batz et al. [201...

2018

[8] [2020]

following the drift fµ(x) =−Ω µνxν +αe −x2/2σ2 xµ withΩ= 2 2 −2 2 , (116) forα= 10 and simulated the stochastic system with noise amplitudeσ= 0.5 on a time grid of dt= 0.01 steps, observed at inter-observation intervalsτ={150, 200, 250} ×dtand for total durationT= 1000 time units. For theHopf systemwe used the drift f1(x1,x 2) =z 2, (117) f2(x1,x 2) =−z 1...

1968