Underdamped Langevin Inference

• ULI is a framework built on second-order stochastic dynamics that uses both position and momentum variables to infer force fields and noise amplitudes.
• It corrects biases from finite-difference approximations and measurement noise through smooth basis projections and analytically derived correction terms.
• The approach underpins diverse applications including trajectory reconstruction, variational inference, and particle-based latent-variable estimation in high-dimensional systems.

Underdamped Langevin Inference (ULI) denotes a family of inference procedures built from second-order stochastic dynamics with explicit position and momentum or velocity variables. In its original operational sense, ULI is a framework for reconstructing the stochastic equation of motion of a system whose observed trajectories obey an underdamped Langevin dynamics, using discrete-time position samples contaminated by measurement noise (Brückner et al., 2020). In a broader usage, the same phase-space construction serves as an inference mechanism for sampling, variational inference, unbiased expectation estimation, and latent-variable parameter estimation, because the underdamped dynamics admit an augmented Gibbs law whose position marginal is the target distribution (Geffner et al., 2022).

1. Dynamical formulation and inferential scope

The canonical underdamped model introduces a state variable that contains both coordinates and inertial degrees of freedom. In the trajectory-inference setting, the dynamics are written as

$$\dot{x}_\mu = v_\mu, \qquad \dot{v}_\mu = F_\mu(\mathbf{x},\mathbf{v}) + \sigma_{\mu\nu}(\mathbf{x},\mathbf{v})\,\xi_\nu(t),$$

with Gaussian white noise, and the inferential task is to recover the deterministic drift or force field $F$ and the noise amplitude $\sigma$ from observed trajectories (Brückner et al., 2020). In the Bayesian-neural formulation, the state is written as $$\bm z=[\bm x^{\mathrm T},\bm v^{\mathrm T}]^{\mathrm T}\in\mathbb R^{2d}$$, and the continuous-time underdamped Langevin equation is

$$\dot{\bm x}(t)=\bm v(t),\qquad \dot{\bm v}(t)=\bm\Phi(\bm x,\bm v,t)+\sqrt{2\bm{\mathsf D}(\bm x,\bm v,t)}\,\bm\xi(t),$$

which makes explicit the separation between drift force and diffusion matrix (Bae et al., 2024).

In molecular-dynamics parameter inference, the underdamped model is commonly written as

$$\dot{q} = v, \qquad \dot{v} = -\frac{1}{m}\frac{dF}{dq} - \gamma v + \text{noise},$$

with $q$ the coordinate, $v$ the velocity, $m$ an effective mass, $F(q)$ the free-energy profile, and $F$0 a friction coefficient. The corresponding inference problem is to estimate the free-energy profile $F$1, the friction $F$2 or a friction kernel, and the effective mass or noise amplitude consistent with thermal equilibrium, typically by likelihood maximization on short trajectory segments (Girardier et al., 19 Jun 2025). This setting is also the inertial, memoryless special case of the generalized Langevin equation,

$F$3

which motivates several later extensions toward memory-kernel inference (Girardier et al., 19 Jun 2025).

When ULI is used as a sampler rather than a system-identification tool, the same phase-space structure targets a Gibbs law. A representative form is

$F$4

with invariant density

$F$5

so the marginal invariant law of the position variable is proportional to $F$6 (Wang, 4 Jan 2026). This dual role of underdamped Langevin dynamics—as a generative model for observed inertial trajectories and as an inference mechanism for posterior sampling—organizes much of the subsequent literature.

2. Trajectory-based reconstruction from discrete noisy observations

The framework explicitly named ULI in "Inferring the dynamics of underdamped stochastic systems" (Brückner et al., 2020) addresses a central practical difficulty: experiments usually provide only positions sampled at intervals $F$7, whereas the model is second order in time. Naive finite-difference differentiation produces a discrete acceleration estimator whose stochastic component is correlated with the basis functions used to regress the force, and the resulting estimator remains biased even as $F$8.

The method therefore represents the force field in a smooth basis,

$F$9

where the orthonormalized basis functions are built from a chosen dictionary $\sigma$0 through

$\sigma$1

With discrete-time observations, the natural estimators

$\sigma$2

lead to the bias expansion

$\sigma$3

For a linear viscous force $\sigma$4, this yields

$\sigma$5

rather than the correct $\sigma$6. ULI removes this bias by subtracting the analytically derived correction term: $\sigma$7

The noise field is inferred by an equally explicit estimator,

$\sigma$8

which allows multiplicative noise to be reconstructed rather than merely a constant diffusion level. Because the correction involves $\sigma$9, the approach requires a smooth basis and is not naturally phrased in terms of top-hat bins.

Measurement noise is treated as a distinct pathology. If the observed signal is

$$\bm z=[\bm x^{\mathrm T},\bm v^{\mathrm T}]^{\mathrm T}\in\mathbb R^{2d}$$0

with time-uncorrelated measurement error, then the second derivative amplifies measurement noise so strongly that the systematic bias can scale like $$\bm z=[\bm x^{\mathrm T},\bm v^{\mathrm T}]^{\mathrm T}\in\mathbb R^{2d}$$1. The measurement-noise-robust variant of ULI therefore replaces pointwise positions and one-sided velocities by a local average position

$$\bm z=[\bm x^{\mathrm T},\bm v^{\mathrm T}]^{\mathrm T}\in\mathbb R^{2d}$$2

and a symmetric velocity

$$\bm z=[\bm x^{\mathrm T},\bm v^{\mathrm T}]^{\mathrm T}\in\mathbb R^{2d}$$3

In this variant the prefactor in the force correction changes from $$\bm z=[\bm x^{\mathrm T},\bm v^{\mathrm T}]^{\mathrm T}\in\mathbb R^{2d}$$4 to $$\bm z=[\bm x^{\mathrm T},\bm v^{\mathrm T}]^{\mathrm T}\in\mathbb R^{2d}$$5, and the noise estimator is replaced by a four-point increment construction designed to cancel the leading measurement-noise terms (Brückner et al., 2020).

A distinctive feature of the original ULI framework is its self-consistent error estimate. Defining the empirical information content

$$\bm z=[\bm x^{\mathrm T},\bm v^{\mathrm T}]^{\mathrm T}\in\mathbb R^{2d}$$6

with $$\bm z=[\bm x^{\mathrm T},\bm v^{\mathrm T}]^{\mathrm T}\in\mathbb R^{2d}$$7 force coefficients, the expected relative inference error satisfies

$$\bm z=[\bm x^{\mathrm T},\bm v^{\mathrm T}]^{\mathrm T}\in\mathbb R^{2d}$$8

This gives both a quantitative uncertainty estimate and a convergence criterion for basis truncation and data length. Empirical demonstrations in the same work include the stochastic damped harmonic oscillator, the Van der Pol oscillator, multiplicative-noise systems, a migrating human breast cancer cell in a two-state micropattern, and a 3D Viscek-like flocking model, where symmetry-aware basis design mitigates the high-dimensionality of the many-body force field (Brückner et al., 2020).

3. Velocity reconstruction, corrected likelihoods, and uncertainty quantification

Later work sharpened the observation that the principal technical obstacle in ULI is not merely missing velocity data, but the statistical structure induced by reconstructing velocities from positions. In the likelihood-maximization setting for molecular trajectories, positions are observed at discrete times $$\bm z=[\bm x^{\mathrm T},\bm v^{\mathrm T}]^{\mathrm T}\in\mathbb R^{2d}$$9, and a generic finite-difference velocity is defined as

$$\dot{\bm x}(t)=\bm v(t),\qquad \dot{\bm v}(t)=\bm\Phi(\bm x,\bm v,t)+\sqrt{2\bm{\mathsf D}(\bm x,\bm v,t)}\,\bm\xi(t),$$0

For the VEC integrator this quantity is not equal to the true integration velocity $$\dot{\bm x}(t)=\bm v(t),\qquad \dot{\bm v}(t)=\bm\Phi(\bm x,\bm v,t)+\sqrt{2\bm{\mathsf D}(\bm x,\bm v,t)}\,\bm\xi(t),$$1, but instead obeys, to leading order,

$$\dot{\bm x}(t)=\bm v(t),\qquad \dot{\bm v}(t)=\bm\Phi(\bm x,\bm v,t)+\sqrt{2\bm{\mathsf D}(\bm x,\bm v,t)}\,\bm\xi(t),$$2

so the mismatch is minimized at $$\dot{\bm x}(t)=\bm v(t),\qquad \dot{\bm v}(t)=\bm\Phi(\bm x,\bm v,t)+\sqrt{2\bm{\mathsf D}(\bm x,\bm v,t)}\,\bm\xi(t),$$3 (Girardier et al., 19 Jun 2025). The key conclusion is not simply that $$\dot{\bm x}(t)=\bm v(t),\qquad \dot{\bm v}(t)=\bm\Phi(\bm x,\bm v,t)+\sqrt{2\bm{\mathsf D}(\bm x,\bm v,t)}\,\bm\xi(t),$$4 is noisy, but that it is structurally correlated with the same stochastic increments that drive the coordinate update. As a result, a likelihood based on the assumption of conditionally clean velocities becomes systematically biased.

The corrected molecular-dynamics likelihood is written in Gaussian form in terms of short-time propagator moments and covariance matrix entries,

$$\dot{\bm x}(t)=\bm v(t),\qquad \dot{\bm v}(t)=\bm\Phi(\bm x,\bm v,t)+\sqrt{2\bm{\mathsf D}(\bm x,\bm v,t)}\,\bm\xi(t),$$5

with

$$\dot{\bm x}(t)=\bm v(t),\qquad \dot{\bm v}(t)=\bm\Phi(\bm x,\bm v,t)+\sqrt{2\bm{\mathsf D}(\bm x,\bm v,t)}\,\bm\xi(t),$$6

and $$\dot{\bm x}(t)=\bm v(t),\qquad \dot{\bm v}(t)=\bm\Phi(\bm x,\bm v,t)+\sqrt{2\bm{\mathsf D}(\bm x,\bm v,t)}\,\bm\xi(t),$$7 (Girardier et al., 19 Jun 2025). The correction rewrites the moments

$$\dot{\bm x}(t)=\bm v(t),\qquad \dot{\bm v}(t)=\bm\Phi(\bm x,\bm v,t)+\sqrt{2\bm{\mathsf D}(\bm x,\bm v,t)}\,\bm\xi(t),$$8

so that the finite-difference relations are explicitly propagated into the inference objective. The resulting method is designed for short, non-ergodic trajectories, where one cannot rely on long-time averaging to wash out discretization artifacts. A practical validation diagnostic reconstructs the effective noise

$$\dot{\bm x}(t)=\bm v(t),\qquad \dot{\bm v}(t)=\bm\Phi(\bm x,\bm v,t)+\sqrt{2\bm{\mathsf D}(\bm x,\bm v,t)}\,\bm\xi(t),$$9

which, for a correct inference, should satisfy

$$\dot{q} = v, \qquad \dot{v} = -\frac{1}{m}\frac{dF}{dq} - \gamma v + \text{noise},$$0

The paper validates the corrected likelihood on a benchmark case and on a fullerene dimer in water, and reports that the corrected likelihood produces consistent free-energy profiles across time resolutions (Girardier et al., 19 Jun 2025).

A complementary development replaces hand-crafted basis projections by Bayesian neural estimators while preserving the same leading-order bias corrections. In "Inferring the Langevin Equation with Uncertainty via Bayesian Neural Networks" (Bae et al., 2024), the observable trajectory is only $$\dot{q} = v, \qquad \dot{v} = -\frac{1}{m}\frac{dF}{dq} - \gamma v + \text{noise},$$1, the velocity is reconstructed as

$$\dot{q} = v, \qquad \dot{v} = -\frac{1}{m}\frac{dF}{dq} - \gamma v + \text{noise},$$2

and the augmented state is taken as

$$\dot{q} = v, \qquad \dot{v} = -\frac{1}{m}\frac{dF}{dq} - \gamma v + \text{noise},$$3

The crucial derivation is that the increment of the estimated velocity does not follow the true underdamped Langevin equation. Instead,

$$\dot{q} = v, \qquad \dot{v} = -\frac{1}{m}\frac{dF}{dq} - \gamma v + \text{noise},$$4

Hence the observed velocity dynamics are not Markovian at finite $$\dot{q} = v, \qquad \dot{v} = -\frac{1}{m}\frac{dF}{dq} - \gamma v + \text{noise},$$5, and naive overdamped-style regression is biased even as $$\dot{q} = v, \qquad \dot{v} = -\frac{1}{m}\frac{dF}{dq} - \gamma v + \text{noise},$$6. The leading-order unbiased estimators become

$$\dot{q} = v, \qquad \dot{v} = -\frac{1}{m}\frac{dF}{dq} - \gamma v + \text{noise},$$7

$$\dot{q} = v, \qquad \dot{v} = -\frac{1}{m}\frac{dF}{dq} - \gamma v + \text{noise},$$8

so ULI requires two separate drift networks and a $$\dot{q} = v, \qquad \dot{v} = -\frac{1}{m}\frac{dF}{dq} - \gamma v + \text{noise},$$9 rescaling for diffusion (Bae et al., 2024). The Bayesian posterior over weights is mean-field Gaussian, training uses Bayes-by-backprop with the reparameterization trick and Adam, and epistemic uncertainty is quantified by posterior predictive variance,

$q$0

In the reported underdamped examples—the stochastic van der Pol oscillator and the nonstationary Brownian Carnot engine—the uncertainty maps shrink as trajectory length increases and are larger in under-sampled regions (Bae et al., 2024).

4. Path-space variational inference and reverse-time constructions

A distinct but closely related line of work recasts ULI as variational inference on augmented trajectories. "Langevin Diffusion Variational Inference" (Geffner et al., 2022) starts from the standard variational objective

$q$1

with $q$2 known up to normalization, and constructs a rich variational family by numerically simulating the underdamped Langevin diffusion process and its time reversal. Rather than approximate only a terminal marginal, the method defines an augmented variational path measure

$q$3

and a matching augmented target

$q$4

leading to the augmented objective

$q$5

The forward diffusion is the underdamped Langevin SDE

$q$6

where $q$7 interpolates between an initial approximation and the target. Its reverse-time dynamics are

$q$8

Under exact simulation, these reverse dynamics yield the optimal backward transitions for the auxiliary-variable bound. The intractable term $q$9 is approximated as

$v$0

where $v$1 is a learnable score network. Setting $v$2 recovers a simpler class of Langevin variational methods; retaining the score network gives what the paper calls a more powerful backward augmentation (Geffner et al., 2022).

The numerical implementation is not an arbitrary discretization but a specific splitting construction. The forward SDE is decomposed into $v$3 for deterministic transport in position, $v$4 for deterministic force update from $v$5, and $v$6 for Ornstein–Uhlenbeck momentum refresh. The resulting leapfrog-style forward step is

$v$7

$v$8

while the backward step uses the inverse leapfrog followed by reverse momentum resampling. A key tractability result is

$v$9

With

$m$0

and Gaussian $m$1 obtained by Euler-Maruyama simulation of the OU-like momentum component, the ELBO remains tractable (Geffner et al., 2022).

The paper’s central structural claim is that several earlier methods are special cases of a single diffusion-theoretic construction. ULA is recovered by using overdamped transitions, no score network, and the high-friction limit; MCD uses overdamped dynamics with score-network-based improved backward transitions; UHA uses underdamped transitions with no score network and exact momentum resampling; LDVI combines underdamped Langevin transitions with score-network-based reverse augmentation (Geffner et al., 2022). Empirically, the reported comparisons on logistic regression on ionosphere and sonar, Brownian motion latent time-series, the Lorenz system, and random effect regression on seeds show a consistent pattern: underdamped dynamics help, score-based backward augmentation helps, and combining both usually yields the best ELBO. The same experiments also show that a simpler Euler-Maruyama discretization performs worse than the splitting-based simulation scheme, making the numerical integrator part of the inferential design rather than a secondary implementation detail (Geffner et al., 2022).

5. ULI as an inference engine: latent variables, particles, unbiased estimators, and stochastic gradients

A large algorithmic literature uses underdamped Langevin dynamics directly as an inference engine once the target distribution or marginal likelihood has been specified. In latent-variable maximum marginal likelihood, "Kinetic Interacting Particle Langevin Monte Carlo" (Oliva et al., 2024) introduces the Kinetic Interacting Particle Langevin Diffusion

$m$2

whose stationary $m$3-marginal is

$m$4

Thus the number of particles $m$5 acts like an inverse temperature, and under strong convexity the stationary concentration obeys

$m$6

Two explicit discretizations are given: KIPLMC1, an exponential-integrator kinetic scheme using only first derivatives, and KIPLMC2, an explicit OBABO-type splitting scheme that is second-order, explicit, and does not require Hessians. The theoretical bounds separate mixing or optimization error, discretization error of order $m$7, and concentration bias of order $m$8, while the experiments on synthetic and Wisconsin cancer logistic regression indicate that KIPLMC2 is markedly more stable than KIPLMC1 and momentum particle gradient descent at larger step sizes (Oliva et al., 2024).

A mean-field analogue appears in "Mean-field underdamped Langevin dynamics and its spacetime discretization" (Fu et al., 2023). There the continuous-time mean-field underdamped dynamics are

$m$9

for entropy-regularized mean-field optimization problems such as mean-field neural network training, maximum mean discrepancy minimization, and kernel Stein discrepancy minimization. The practical $F(q)$0-particle algorithm, NULA, updates each particle through

$F(q)$1

$F(q)$2

with

$F(q)$3

The continuous-time theory gives an improved hypocoercive rate with $F(q)$4, and the discrete-time theory provides global TV guarantees for both the mean-field and $F(q)$5-particle algorithms (Fu et al., 2023).

ULI has also been coupled to debiasing methodology. "Unbiased Estimation using Underdamped Langevin Dynamics" (Ruzayqat et al., 2022) considers targets on phase space with density

$F(q)$6

simulated through Euler-discretized underdamped Langevin dynamics on levels $F(q)$7. The final estimator removes both finite-time bias and discretization bias by combining coupled meeting-time debiasing at each fixed level with randomization over levels in the telescoping decomposition

$F(q)$8

For each level, the lag-1 unbiased estimator is

$F(q)$9

and the randomized single-term estimator is weighted by $F$00. Under assumptions (A1)–(A5), the method is unbiased and has finite variance, with reported experiments on Bayesian logistic regression, a 100-dimensional double-well potential, and a $F$01 Ginzburg–Landau model (Ruzayqat et al., 2022).

In large-data Bayesian inference, stochastic gradients introduce another form of inferential distortion. "Langevin Markov Chain Monte Carlo with stochastic gradients" (Matthews et al., 2018) develops the Noisy Gradient Integrator for underdamped Langevin dynamics with mini-batch gradients satisfying

$F$02

The method folds the stochastic-gradient noise into the Ornstein–Uhlenbeck part of a splitting integrator through a calibrated damping matrix

$F$03

and is second-order weakly consistent when

$F$04

For Gaussian targets with Gaussian gradient noise, the $F$05-marginal is preserved exactly: $F$06 so there is no sampling bias in $F$07 under the stated condition $F$08 (Matthews et al., 2018).

6. Convergence theory, metastability, and broader extensions

The theoretical analysis of underdamped Langevin inference has progressively moved beyond strong log-concavity and Wasserstein-only arguments. "Improved Discretization Analysis for Underdamped Langevin Monte Carlo" (Zhang et al., 2023) introduces a Girsanov-based discretization analysis in Rényi divergence for the standard underdamped diffusion

$F$09

and its frozen-gradient discretization. The main advances are the removal of the Lipschitz Hessian requirement from earlier KL analyses, extension to distributions satisfying a Poincaré inequality, and the ability to handle weakly smooth potentials $F$10. In the strongly convex case, the paper proves a KL guarantee with

$F$11

and states explicitly that it does not obtain full discrete-time acceleration, leaving open the gap to the ideal $F$12 dependence (Zhang et al., 2023).

A subsequent development removes explicit ambient-dimension dependence from KL guarantees. "Dimension-Independent Convergence of Underdamped Langevin Monte Carlo in KL Divergence" (Zhang et al., 2 Mar 2026) refines the KL local error framework so that bounds depend on an upper Hessian matrix $F$13 through $F$14 rather than $F$15. For standard ULMC in the strongly convex regime, with $F$16,

$F$17

yield

$F$18

For the randomized midpoint discretization, the strongly convex complexity improves to

$F$19

The resulting picture is that underdamped schemes are provably preferable when $F$20, but the gain is geometry-dependent rather than uniform (Zhang et al., 2 Mar 2026).

Several works emphasize that acceleration depends sensitively on dynamical design choices. "Optimal friction matrix for underdamped Langevin sampling" (Chak et al., 2021) treats the friction matrix $F$21 as an observable-dependent hyperparameter and gives the asymptotic-variance gradient

$F$22

where $F$23 solves the Poisson equation $F$24. In Bayesian logistic regression, the learned $F$25 decreases from $F$26 toward a very small scalar and reduces empirical asymptotic variance by about an order of magnitude relative to $F$27 in both full-gradient and minibatch settings (Chak et al., 2021). "Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions" (Duncan et al., 2017) adds skew-symmetric perturbations in both the position and momentum equations,

$F$28

without changing the invariant measure, and shows for Gaussian targets that suitable perturbations reduce asymptotic variance, while unsuitable perturbations can worsen it (Duncan et al., 2017). "Gradient-adjusted underdamped Langevin dynamics for sampling" (Zuo et al., 2024) modifies the kinetic equations by adding a gradient adjustment in the position equation,

$F$29

and for Gaussian targets reports continuous-time and discrete-time convergence improvements, including an Euler–Maruyama mixing-time scaling of order $F$30 toward a biased target under the stated parameter choices (Zuo et al., 2024).

The long-time geometry of ULI also matters. "Eyring-Kramers Law for the Underdamped Langevin Process" (Lee et al., 16 Mar 2025) studies the low-temperature dynamics

$F$31

in a double-well potential and proves the asymptotic mean transition time

$F$32

with

$F$33

This identifies the precise metastable barrier-crossing scale of underdamped samplers in the small-noise regime (Lee et al., 16 Mar 2025). In a complementary nonequilibrium direction, "Bounds on the precision of currents in underdamped Langevin dynamics" (Dechant, 2022) shows that the overdamped thermodynamic uncertainty relation does not extend unchanged to finite mass. The current precision bound becomes

$F$34

where the additional positive terms encode local mean acceleration and velocity-fluctuation structure (Dechant, 2022). This is directly relevant when ULI is used to infer dissipation or dynamical precision from currents.

Finally, recent work has expanded the conceptual reach of underdamped inference. "Learning Relationship between Quantum Walks and Underdamped Langevin Dynamics" (Wang, 4 Jan 2026) proves that a coined quantum walk with randomization is asymptotically equivalent to underdamped Langevin dynamics in Le Cam deficiency distance, whereas the non-randomized walk is not asymptotically equivalent because of a high-frequency oscillatory component. "Accelerated massive MIMO detector based on annealed underdamped Langevin dynamics" (Zilberstein et al., 2022) adapts annealed underdamped Langevin inference to massive MIMO detection by smoothing a discrete posterior over symbol constellations and then running a splitting-based underdamped sampler in the spectral domain. In the reported low-complexity regimes, the method yields lower symbol error rate than an overdamped Langevin detector with lower runtime, while high-complexity regimes make the two essentially similar (Zilberstein et al., 2022).

Taken together, these developments show that ULI is not a single algorithm but a structured family of inference methods built around inertial stochastic dynamics. Across trajectory reconstruction, corrected likelihoods, Bayesian neural estimation, path-space variational inference, particle-based latent-variable optimization, unbiased Monte Carlo, and nonasymptotic sampling theory, the recurring lesson is that inertia, reverse-time structure, and discretization must be treated jointly. The same momentum variable that improves exploration and mixing also creates the characteristic inferential difficulties of ULI: unobserved velocities, correlated finite-difference noise, discretization-sensitive reverse dynamics, observable-dependent friction, and metastable phase-space transport.