Kinetic Langevin Diffusion for Materials

• The paper presents KLDM, a generative model that reformulates fractional coordinate treatment by introducing auxiliary velocity variables to address periodic boundary conditions.
• It leverages kinetic Langevin dynamics to shift stochasticity to Euclidean velocity space, ensuring a tractable noising process aligned with translation invariance.
• Evaluations on crystal structure prediction and de-novo generation tasks demonstrate competitive performance with improved convergence and stability metrics.

Searching arXiv for KLDM and closely related kinetic Langevin materials papers to ground the article in the cited literature. Kinetic Langevin Diffusion for Materials (KLDM), written in the source paper as KLDiff, is a diffusion-based generative model for crystalline materials that reformulates the treatment of fractional atomic coordinates. Rather than applying diffusion directly on the hypertorus formed by periodic coordinates, KLDM introduces auxiliary Euclidean velocity variables and couples them to coordinates through a kinetic Langevin / trivialized diffusion construction. In this formulation, stochasticity acts in velocity space, while coordinates are transported on the manifold, yielding a tractable noising process and a training objective aligned with periodic translation symmetry. The model is evaluated on Crystal Structure Prediction (CSP) and De-novo Generation (DNG), where it is reported to be competitive with current state-of-the-art models (Cornet et al., 4 Jul 2025).

1. Problem setting and motivation

Crystalline materials in KLDM are represented as

$$\boldsymbol{x}=(\boldsymbol{f},\boldsymbol{l},\boldsymbol{a}),$$

where $\boldsymbol{f}\in[0,1)^{3\times K}$ are fractional coordinates of $K$ atoms, $\boldsymbol{l}\in\mathbb{R}^{3\times 3}$ are lattice vectors, and $\boldsymbol{a}\in\mathbb{Z}^K$ are atom types or compositions. The central geometric difficulty is that fractional coordinates live on a $3K$-dimensional hypertorus,

$$\boldsymbol{f}\in[0,1)^{3\times K}\cong \mathbb{T}^{3\times K}\cong \mathbb{R}^{3\times K}/\mathbb{Z}^{3\times K},$$

so naïve Gaussian noising ignores periodic boundary conditions (Cornet et al., 4 Jul 2025).

The paper situates this problem within a broader symmetry structure. Crystal distributions are highly multimodal and satisfy periodic translation symmetry,

$$p(\boldsymbol{f},\boldsymbol{l},\boldsymbol{a}) = p(g\cdot \boldsymbol{f},\boldsymbol{l},\boldsymbol{a}) \quad \forall g\in \mathbb{T}^3,$$

along with atom permutation symmetries, lattice-basis symmetries, and lattice rotation invariance. A specific issue identified in prior torus-diffusion approaches is that even when the score network is made periodic-translation invariant, the training target can still depend on the chosen translated representative of the same physical crystal. The paper states that this mismatch is especially problematic at low noise and can degrade convergence (Cornet et al., 4 Jul 2025).

Within that context, KLDM is presented as an alternative to two existing strategies: Riemannian diffusion on the hypertorus and wrapped Gaussian noising with a periodic-invariant score network. Its defining move is to generalize Trivialized Diffusion Models (TDM) to crystalline materials by introducing auxiliary velocities, thereby shifting the stochastic process to a flat Euclidean space while preserving periodicity in coordinates (Cornet et al., 4 Jul 2025).

2. Kinetic Langevin formulation on torus-valued coordinates

KLDM models fractional coordinates with a kinetic Langevin diffusion. The paper represents each fractional coordinate as an element of a Lie group isomorphic to the torus, often in an equivalent $SO(2)$-matrix form $\hat{\boldsymbol f}$, and introduces auxiliary velocity variables $\boldsymbol{f}\in[0,1)^{3\times K}$0 in the Lie algebra $\boldsymbol{f}\in[0,1)^{3\times K}$1. The forward process is

$\boldsymbol{f}\in[0,1)^{3\times K}$2

with friction coefficient $\boldsymbol{f}\in[0,1)^{3\times K}$3, set to $\boldsymbol{f}\in[0,1)^{3\times K}$4 in the experiments. Noise is therefore injected into velocities, not directly into coordinates, while coordinates are advected by the current velocity field (Cornet et al., 4 Jul 2025).

The reverse-time generative dynamics preserve this asymmetry: $\boldsymbol{f}\in[0,1)^{3\times K}$5 Only the velocity equation requires score correction; the coordinate evolution remains transport-like because there is no direct noise term on $\boldsymbol{f}\in[0,1)^{3\times K}$6 (Cornet et al., 4 Jul 2025).

A notable technical feature is that the joint transition kernel has a closed form,

$\boldsymbol{f}\in[0,1)^{3\times K}$7

where

$\boldsymbol{f}\in[0,1)^{3\times K}$8

$\boldsymbol{f}\in[0,1)^{3\times K}$9

The wrapped normal term

$K$0

is the torus-compatible distribution. As $K$1, the process converges to a prior that is uniform on the torus for $K$2 and standard Gaussian on $K$3 (Cornet et al., 4 Jul 2025).

This construction is the model’s geometric core: the manifold constraint is enforced by transport, while the stochastic component remains Euclidean.

3. Score matching, parameterization, and torus–Euclidean correspondence

Training proceeds through denoising score matching (DSM) with generic objective

$K$4

For the kinetic torus model, the target score is taken with respect to velocity,

$K$5

rather than directly with respect to coordinates (Cornet et al., 4 Jul 2025).

The paper derives the decomposition

$K$6

With zero initial velocities, $K$7, the parameterization simplifies to

$K$8

The paper identifies this simplified form as a main practical contribution and reports that it improves convergence and final performance (Cornet et al., 4 Jul 2025).

The same section of the source paper makes explicit the mapping that links torus coordinates to Euclidean latent variables. A scalar coordinate $K$9 is converted to an angle

$\boldsymbol{l}\in\mathbb{R}^{3\times 3}$0

and then represented as

$\boldsymbol{l}\in\mathbb{R}^{3\times 3}$1

For a Lie-algebra element

$\boldsymbol{l}\in\mathbb{R}^{3\times 3}$2

the matrix exponential is

$\boldsymbol{l}\in\mathbb{R}^{3\times 3}$3

Hence the infinitesimal update

$\boldsymbol{l}\in\mathbb{R}^{3\times 3}$4

is equivalent, on the torus, to periodic translation followed by wrapping back into the unit interval. This is the mechanism by which KLDM performs diffusion-compatible transport on periodic coordinates while keeping the actual noise process in Euclidean velocity space (Cornet et al., 4 Jul 2025).

4. Symmetry handling and multimodal architecture

KLDM’s symmetry treatment is distributed across the geometry, the stochastic process, and the neural architecture. Atom permutation invariance is handled by graph neural networks, lattice rotation invariance by rotation-invariant lattice representations, and periodic translation invariance through torus diffusion together with periodic featurization (Cornet et al., 4 Jul 2025).

A distinctive design constraint is the use of zero-net translation velocity fields,

$\boldsymbol{l}\in\mathbb{R}^{3\times 3}$5

The paper motivates this by noting that a translation-invariant network cannot distinguish global crystal shifts. In practice, velocities are sampled from a mean-free projected Gaussian, noisy coordinates are centered, and the predicted score is projected to the mean-free subspace. The paper reports that this reduces mismatch between the invariant score network and the training target and helps especially at low noise (Cornet et al., 4 Jul 2025).

The model architecture uses a graph neural network backbone similar to DiffCSP, but the input modality is expanded to include fractional coordinates $\boldsymbol{l}\in\mathbb{R}^{3\times 3}$6, auxiliary velocities $\boldsymbol{l}\in\mathbb{R}^{3\times 3}$7, lattice parameters $\boldsymbol{l}\in\mathbb{R}^{3\times 3}$8, and atom types $\boldsymbol{l}\in\mathbb{R}^{3\times 3}$9 for DNG. Message passing relies on node embeddings, periodic sinusoidal embeddings of pairwise fractional differences, velocity inputs, and lattice inputs. The periodic featurization includes

$\boldsymbol{a}\in\mathbb{Z}^K$0

The network predicts separate scores for $\boldsymbol{a}\in\mathbb{Z}^K$1, $\boldsymbol{a}\in\mathbb{Z}^K$2, and, in DNG, $\boldsymbol{a}\in\mathbb{Z}^K$3. The source paper specifies a 2-layer MLP for the velocity score and single-layer MLPs for lattice and atom types (Cornet et al., 4 Jul 2025).

Lattice vectors are represented as six scalars—three lengths and three angles—transformed with log for lengths and $\boldsymbol{a}\in\mathbb{Z}^K$4 for angles, then diffused with a standard Euclidean VPSDE. For DNG, the paper evaluates three atom-type parameterizations: continuous diffusion on one-hot atom types, continuous diffusion on analog bits, and discrete absorbing diffusion (Cornet et al., 4 Jul 2025).

These choices indicate that KLDM is not a single-process model over all modalities. Fractional coordinates use the kinetic Langevin construction, whereas lattice and atom-type variables continue to use more conventional diffusion parameterizations.

5. Evaluation on crystalline generation tasks

The empirical study covers two tasks: CSP, defined as conditional generation of $\boldsymbol{a}\in\mathbb{Z}^K$5 given $\boldsymbol{a}\in\mathbb{Z}^K$6, and DNG, defined as joint generation of $\boldsymbol{a}\in\mathbb{Z}^K$7. The reported datasets are perov-5, mp-20, mpts-52, and carbon-24 for CSP, with mp-20 also used for DNG (Cornet et al., 4 Jul 2025).

Setting	Datasets	Metrics
CSP	perov-5, mp-20, mpts-52, carbon-24	Match Rate (MR), RMSE, reported at @1 and @20
DNG	mp-20	RMSD, average energy above hull, stability, S.U.N.

For CSP, KLDM is compared against cdvae, DiffCSP, EquiCSP, and FlowMM. The paper states that on perov-5 it is on par at @1 and better at @20, while on mp-20 and mpts-52 it shows strong gains, especially with the predictor-corrector sampler. The best results often come from three design choices highlighted in the ablations: zero initial velocities, simplified score parameterization, and predictor-corrector sampling (Cornet et al., 4 Jul 2025).

Two quantitative CSP results are explicitly emphasized. On mp-20, KLDM reaches 65.83 MR and 0.0517 RMSE with $\boldsymbol{a}\in\mathbb{Z}^K$8-parameterization plus predictor-corrector sampling. On mpts-52, it reaches 23.93 MR and 0.1276 RMSE under the same setting (Cornet et al., 4 Jul 2025).

For DNG on mp-20, KLDM with analog-bit or discrete atom diffusion is reported to outperform DiffCSP on RMSD, energy above hull, and stability, while remaining competitive with Mattergen-style baselines and somewhat behind on S.U.N. in some cases. The paper attributes remaining gaps to a more expressive denoiser in Mattergen, the use of predictor-corrector sampling there, and dataset preprocessing differences (Cornet et al., 4 Jul 2025).

The ablation study further reports that zero initial velocities improve performance and convergence, simplified score parameterization consistently outperforms direct score prediction, and zero-net translation velocity fields yield additional, though smaller, improvements. The paper also states that enforcing zero-net translation stabilizes the Fréchet mean behavior of noisy torus samples, especially at low noise (Cornet et al., 4 Jul 2025).

6. Position within kinetic Langevin theory and related methodology

KLDM belongs to the broader class of underdamped / kinetic Langevin constructions, where positions are coupled to auxiliary velocities and noise acts only in the velocity variable. In the standard Euclidean setting, this takes the form

$\boldsymbol{a}\in\mathbb{Z}^K$9

with invariant density proportional to $3K$0 (Leimkuhler et al., 2023). KLDM adopts the same inertial principle, but transfers it to periodic crystal coordinates through a Lie-group/trivialized-diffusion representation (Cornet et al., 4 Jul 2025).

Several lines of kinetic Langevin theory provide context for this design. In smooth strongly log-concave settings, continuous-time kinetic Langevin diffusion is geometrically ergodic in $3K$1, and the corresponding discretizations yield nonasymptotic Wasserstein guarantees for KLMC and KLMC2, including improved dependence on the condition number and refined second-order discretization when the Hessian is Lipschitz (Dalalyan et al., 2018). For discretized underdamped dynamics, contraction analyses in weighted norms identify explicit stepsize restrictions and show that BAOAB and OBABO are $3K$2-limit convergent (GLC), meaning that they converge to an overdamped discretization in the high-friction limit with stepsize restrictions independent of the friction parameter (Leimkuhler et al., 2023). Extensions to BBK, SPV, SVV, SES/EB, and stochastic-gradient variants preserve $3K$3-order contraction up to estimator-noise penalties, providing a scheme-by-scheme stability guide for molecular-dynamics and machine-learning settings (Leimkuhler et al., 2023).

At the continuous-time level, hypocoercive analysis gives a second layer of context. For the kinetic Langevin process with Hamiltonian

$3K$4

entropic convergence to the Boltzmann–Gibbs equilibrium can be established without assuming bounded $3K$5, by combining a generalized entropy multipliers method with a weighted logarithmic Sobolev inequality. The weighted condition $3K$6 bounded allows polynomially growing confining potentials and thus relaxes the classical bounded-Hessian hypothesis (Cattiaux et al., 2017). This suggests that velocity-coupled diffusion can remain analytically controlled even when the energy landscape is not globally quadratic.

Other adjacent developments broaden the scope of the kinetic viewpoint. Langevin dynamics with general kinetic energies preserve the target position marginal while altering phase-space transport, and have been used to reduce metastability and improve numerical stability through Metropolized splitting schemes (Stoltz et al., 2016). In nonconvex optimization, kinetic Langevin dynamics with logarithmic cooling $3K$7 admit simulated-annealing convergence guarantees in both continuous and discrete time, though the source paper emphasizes that momentum does not improve the asymptotic annealing exponent relative to the overdamped case (He et al., 2022). For coupling theory, the hypoelliptic noise structure makes total-variation analysis more subtle than in elliptic diffusions: in the quadratic case, no Markovian coupling captures the asymptotic TV decay rate, whereas explicit non-Markovian coalescence constructions do (Bou-Rabee et al., 29 May 2026).

Taken together, this literature places KLDM within a well-developed family of hypoelliptic, velocity-augmented diffusion processes. Its novelty is not the use of kinetic Langevin dynamics in isolation, but the transfer of that machinery to fractional coordinates on the hypertorus in a way that directly addresses crystal periodicity and score-target mismatch (Cornet et al., 4 Jul 2025).

7. Limitations, implementation scope, and open directions

The source paper explicitly notes several limitations. KLDM does not explicitly model lattice permutation invariance in the way some competing methods do; the velocity noising process could potentially be improved; lattice and atom types still rely on standard Euclidean diffusion; better periodic-aware architectures may further improve results; incorporating space-group information or Wyckoff positions is left for future work; and full validation with DFT is not yet performed for all generated samples (Cornet et al., 4 Jul 2025).

The implementation details also delimit the current scope. The experiments use a time horizon $3K$8 for the kinetic process, friction $3K$9, a discrete-time implementation for apples-to-apples comparisons, and either an exponential integrator for the velocity/fractional-coordinate dynamics plus Euler–Maruyama for other modalities, or a predictor-corrector scheme for velocities with Euler–Maruyama for lattice and atom types. Training uses AdamW, no gradient clipping, and early stopping based on validation MR or valid structures. The paper also states that training on a single GPU is sufficient and reports RTX 3090 / RTX A5000 class hardware (Cornet et al., 4 Jul 2025).

A plausible implication is that KLDM currently serves as a modular framework rather than a closed endpoint. Its present form isolates the periodic-coordinate problem and solves it with a kinetic-Langevin lift to velocity space; further gains may depend on extending the same symmetry-aware treatment to lattice representations, compositional generation, and crystallographic constraints that are only partially addressed in the current model (Cornet et al., 4 Jul 2025).