Denoising Non-Equilibrium Structures (DeNS)

• DeNS is a method that denoises off-equilibrium atomistic structures by conditioning on force information to resolve ambiguity away from local energy minima.
• It extends to diverse applications including neural potential pretraining, self-supervised atomistic learning, and molecular relaxation using tailored noise models and Riemannian metrics.
• Empirical results demonstrate improved accuracy in energy and force predictions across catalysts, molecules, and crystalline materials, while often reducing computational costs.

Denoising Non-Equilibrium Structures (DeNS) most specifically denotes a force-conditioned denoising objective for 3D atomistic learning in which a non-equilibrium structure is corrupted and the model predicts the added coordinate noise while conditioning on the forces of the original structure, thereby making denoising well-posed away from local minima (Liao et al., 2024). In a broader research sense, the same premise—treating off-equilibrium, thermally perturbed, or sub-optimal atomistic configurations as denoising targets—also appears in denoise pretraining for neural potentials, self-conditioned atomistic representation learning, Riemannian score-based molecular relaxation, score-based thermal denoising of crystals, and anisotropic denoising for force-field modeling (Wang et al., 2023, Perez et al., 17 Mar 2026, Woo et al., 2024, Hsu et al., 2022, Liu et al., 25 Oct 2025).

1. Origins and scope of the concept

The explicit DeNS formulation arises from the observation that most atomistic data used in simulation and relaxation trajectories are non-equilibrium rather than equilibrium. In the force-field setting, an equilibrium structure is defined by forces near zero, $\|\mathbf{f}_i(S)\| \approx 0$ for all atoms, whereas a non-equilibrium structure has at least one nonzero force and therefore lies away from a local minimum on the potential energy surface (PES) (Liao et al., 2024). A closely related molecular-potential literature uses “nonequilibrium” in a practical PES sense, namely molecular conformations that are not at energy minima; that work explicitly distinguishes this usage from the terminology of statistical mechanics (Wang et al., 2023).

This distinction matters because equilibrium denoising and non-equilibrium denoising are not the same problem. For equilibrium structures, many noisy perturbations map back to a unique local minimum. For non-equilibrium structures, the target is no longer singled out by the condition “forces $\approx 0$,” and the denoising objective becomes ambiguous unless additional information specifies which off-equilibrium configuration is intended (Liao et al., 2024). Much of the subsequent literature can be understood as developing different resolutions of this ambiguity.

The scope of DeNS has expanded beyond a single objective class. In catalyst and molecular force fields, it is used as an auxiliary task or pretraining signal for better energy/force prediction (Liao et al., 2024, Wang et al., 2023). In atomistic self-supervised learning, it becomes a representation-learning objective that can exploit non-equilibrium geometries without force labels (Perez et al., 17 Mar 2026). In molecular structure optimization, the same denoising logic is recast as a deterministic relaxation flow that transports MMFF, ETKDG, or generative conformers toward DFT minima (Woo et al., 2024). In crystalline materials, score-based denoising is used as a structure-space filter that removes thermal vibrations while preserving defects and interfaces (Hsu et al., 2022).

2. Mathematical formulation and the issue of well-posedness

The canonical DeNS setup begins from a non-equilibrium atomistic structure

$$S_{\text{non-eq}} = \{(z_i,\mathbf{p}_i)\}_{i=1}^{N},$$

corrupts atomic positions with Gaussian noise,

$$\tilde{\mathbf{p}}_i = \mathbf{p}_i + \boldsymbol{\epsilon}_i,\qquad \boldsymbol{\epsilon}_i \sim \mathcal{N}(0,\sigma^2 \mathbf{I}_3),$$

and predicts the normalized noise on each corrupted atom. The defining difference from equilibrium denoising is that the model also receives the forces of the original structure $F(S_{\text{non-eq}})$, yielding the DeNS objective

$$\mathcal{L}_{\text{DeNS}} = \mathbb{E}\left[ \sum_{i=1}^{|S_{\text{non-eq}}|} \left\| \frac{\boldsymbol{\epsilon}_i}{\sigma} - \hat{\boldsymbol{\epsilon}}_i(\tilde{S}_{\text{non-eq}}, F(S_{\text{non-eq}})) \right\|_2^2 \right].$$

This conditioning resolves the otherwise ill-posed map $$\tilde{S}_{\text{non-eq}} \mapsto S_{\text{non-eq}}$$ by specifying which off-equilibrium configuration is meant (Liao et al., 2024).

A different but related resolution appears in Self-Conditioned Denoising (SCD). There, the ambiguity of denoising non-equilibrium geometries is attributed to the fact that $$\tilde{\mathbf{x}}-\mathbf{x}=\boldsymbol{\varepsilon}$$ does not correspond to a well-defined physical force or “direction to equilibrium,” and a noisy non-equilibrium structure may be hard to distinguish from another legitimate high-energy geometry. SCD makes the target well-posed by conditioning on a global self-embedding of the clean structure,

$\mathbf{c} = \phi_\theta(\mathbf{x})_{L0},$

and optimizing

$$\mathbb{E}_{q_\sigma(\tilde{\mathbf{x}},\mathbf{x})} \Big[ \|\phi_\theta(\tilde{\mathbf{x}}\mid \mathbf{c})-\boldsymbol{\varepsilon}\|_2^2 \Big],$$

with $\approx 0$0 so that the same model produces the conditioning signal and the denoising prediction (Perez et al., 17 Mar 2026).

Riemannian score matching addresses a related geometric failure mode. Standard Euclidean denoising score matching uses additive isotropic noise in Cartesian space, but Cartesian displacements may correlate poorly with energy change. The Riemannian formulation replaces the Euclidean target $\approx 0$1 by the manifold log map $\approx 0$2, where the manifold is defined by physics-informed internal coordinates and a pullback metric from mass-weighted Cartesian coordinates. In that setting, denoising becomes a geodesic relaxation problem on an internal-coordinate manifold rather than a purely Cartesian regression problem (Woo et al., 2024).

3. Major algorithmic variants

In the auxiliary-task DeNS formulation, the denoising branch is coupled directly to supervised energy and force learning. The model alternates between the original task and the DeNS task with probability $\approx 0$3, and during a DeNS step it predicts the energy of the original structure from corrupted coordinates plus encoded forces while simultaneously predicting the noise on corrupted atoms. Partial corruption is used in some datasets, so only a subset of atoms is noised and denoised in each example. Multi-scale noise schedules are used on OC20 and OC22, while MD17 typically uses a fixed small $\approx 0$4 per molecule (Liao et al., 2024).

Denoise pretraining for neural potentials uses a simpler single-step scheme. Nonequilibrium conformations are sampled from ANI-1 and ANI-1x, perturbed by Gaussian noise with $\approx 0$5, and a GNN is pretrained to regress the added noise. The pretrained message-passing layers are then fine-tuned for downstream energy prediction. This formulation is explicitly model-agnostic and was instantiated with SchNet, SE(3)-Transformer, EGNN, and TorchMD-Net (Wang et al., 2023).

SCD introduces a two-pass, backbone-agnostic pretraining algorithm. A first forward pass on the clean structure produces invariant $\approx 0$6 embeddings, which are pooled and passed through a 2-layer MLP to create the self-conditioning vector $\approx 0$7. A second forward pass denoises the corrupted coordinates conditioned on $\approx 0$8. To preserve some unconditional behavior, $\approx 0$9 of self-conditioning embeddings are randomly dropped during pretraining, and downstream usage reverts to a single forward pass without conditioning (Perez et al., 17 Mar 2026).

Riemannian DeNS-like relaxation takes a different operational form. Training uses Brownian motion on a manifold of internal coordinates via

$$S_{\text{non-eq}} = \{(z_i,\mathbf{p}_i)\}_{i=1}^{N},$$0

and inference from arbitrary non-equilibrium inputs uses a deterministic ODE-style flow

$$S_{\text{non-eq}} = \{(z_i,\mathbf{p}_i)\}_{i=1}^{N},$$1

discretized with Euler steps composed with the exponential map. In practice, the method used $$S_{\text{non-eq}} = \{(z_i,\mathbf{p}_i)\}_{i=1}^{N},$$2 denoising steps and was applied to MMFF, ETKDG, and generative conformers as sub-optimal structures (Woo et al., 2024).

In crystalline materials, score-based denoising is implemented as iterative subtraction of a learned displacement field,

$$S_{\text{non-eq}} = \{(z_i,\mathbf{p}_i)\}_{i=1}^{N},$$3

where $$S_{\text{non-eq}} = \{(z_i,\mathbf{p}_i)\}_{i=1}^{N},$$4 is trained on synthetically noised perfect lattices. The procedure is deterministic, potential-agnostic, and typically uses $$S_{\text{non-eq}} = \{(z_i,\mathbf{p}_i)\}_{i=1}^{N},$$5–$$S_{\text{non-eq}} = \{(z_i,\mathbf{p}_i)\}_{i=1}^{N},$$6 denoising iterations depending on the system (Hsu et al., 2022).

AniDS extends DeNS with a learned anisotropic corruption model. Instead of isotropic scalar noise, it learns a per-atom full covariance matrix $$S_{\text{non-eq}} = \{(z_i,\mathbf{p}_i)\}_{i=1}^{N},$$7, samples corrupted coordinates from $$S_{\text{non-eq}} = \{(z_i,\mathbf{p}_i)\}_{i=1}^{N},$$8, and trains the denoiser to predict $$S_{\text{non-eq}} = \{(z_i,\mathbf{p}_i)\}_{i=1}^{N},$$9. For non-equilibrium structures, it follows the DeNS logic of force-conditioned denoising and uses partial corruption together with supervised energy, force, and optionally stress losses (Liu et al., 25 Oct 2025).

4. Geometric representations, conditioning signals, and symmetry

A persistent theme in DeNS research is that the denoising geometry should match the geometry of the physical system. In the original force-conditioned DeNS setting, this is handled through equivariant architectures that can ingest forces as vector-valued inputs. Force embeddings are constructed across irreducible representations by combining the force norm with spherical harmonics and SO(3)-equivariant linear layers, then adding these embeddings to the initial node features. This is one reason the method “favors equivariant networks,” especially Equiformer, EquiformerV2, and eSCN (Liao et al., 2024).

Riemannian score matching pushes the same idea further by redefining the ambient geometry itself. Molecular structures are represented by edge-based internal coordinates

$$\tilde{\mathbf{p}}_i = \mathbf{p}_i + \boldsymbol{\epsilon}_i,\qquad \boldsymbol{\epsilon}_i \sim \mathcal{N}(0,\sigma^2 \mathbf{I}_3),$$0

with $$\tilde{\mathbf{p}}_i = \mathbf{p}_i + \boldsymbol{\epsilon}_i,\qquad \boldsymbol{\epsilon}_i \sim \mathcal{N}(0,\sigma^2 \mathbf{I}_3),$$1 and $$\tilde{\mathbf{p}}_i = \mathbf{p}_i + \boldsymbol{\epsilon}_i,\qquad \boldsymbol{\epsilon}_i \sim \mathcal{N}(0,\sigma^2 \mathbf{I}_3),$$2, and the resulting manifold is equipped with a metric

$$\tilde{\mathbf{p}}_i = \mathbf{p}_i + \boldsymbol{\epsilon}_i,\qquad \boldsymbol{\epsilon}_i \sim \mathcal{N}(0,\sigma^2 \mathbf{I}_3),$$3

The paper interprets this through Fukui’s metric and reports that geodesic distance on the manifold correlates strongly with $$\tilde{\mathbf{p}}_i = \mathbf{p}_i + \boldsymbol{\epsilon}_i,\qquad \boldsymbol{\epsilon}_i \sim \mathcal{N}(0,\sigma^2 \mathbf{I}_3),$$4, with Pearson $$\tilde{\mathbf{p}}_i = \mathbf{p}_i + \boldsymbol{\epsilon}_i,\qquad \boldsymbol{\epsilon}_i \sim \mathcal{N}(0,\sigma^2 \mathbf{I}_3),$$5, far better than RMSD or D-MAE (Woo et al., 2024).

SCD uses conditioning of a different kind. Its central claim is that plain coordinate denoising over-trains vector channels and under-trains invariant scalar channels in SE(3)-equivariant GNNs, while also failing to disambiguate non-equilibrium structures. By forcing the $$\tilde{\mathbf{p}}_i = \mathbf{p}_i + \boldsymbol{\epsilon}_i,\qquad \boldsymbol{\epsilon}_i \sim \mathcal{N}(0,\sigma^2 \mathbf{I}_3),$$6 channel to produce a useful global conditioning signal, SCD makes scalar embeddings semantically informative and allows denoising across molecules, materials, proteins, and non-equilibrium geometries (Perez et al., 17 Mar 2026).

AniDS introduces a learned anisotropic metric at the corruption level. Each covariance matrix is parameterized as

$$\tilde{\mathbf{p}}_i = \mathbf{p}_i + \boldsymbol{\epsilon}_i,\qquad \boldsymbol{\epsilon}_i \sim \mathcal{N}(0,\sigma^2 \mathbf{I}_3),$$7

with normalized $$\tilde{\mathbf{p}}_i = \mathbf{p}_i + \boldsymbol{\epsilon}_i,\qquad \boldsymbol{\epsilon}_i \sim \mathcal{N}(0,\sigma^2 \mathbf{I}_3),$$8, guaranteeing symmetry, positive definiteness or semi-definiteness, and SO(3)-equivariance. The construction explicitly suppresses variance along stiff pair directions, such as bonding directions, and thereby replaces the isotropic corruption model with a structure-aware anisotropic one (Liu et al., 25 Oct 2025).

The score-based crystal denoiser is also symmetry-aware, but in a more localized sense. It uses a NequIP-style E(3)-equivariant graph network with a $$\tilde{\mathbf{p}}_i = \mathbf{p}_i + \boldsymbol{\epsilon}_i,\qquad \boldsymbol{\epsilon}_i \sim \mathcal{N}(0,\sigma^2 \mathbf{I}_3),$$9 Å cutoff, spherical harmonics on edge directions, and a compact irrep structure. Its locality is deliberate: the learned field can remove thermal jitter while preserving longer-wavelength structural features such as vacancies, dislocations, and grain boundaries (Hsu et al., 2022).

5. Empirical behavior across molecular, catalytic, and materials domains

On catalyst and molecular force-field benchmarks, force-conditioned DeNS improves both accuracy and training efficiency. On the OC20 S2EF-2M subset, EquiformerV2 with DeNS improved force MAE from $F(S_{\text{non-eq}})$0 to $F(S_{\text{non-eq}})$1 meV/Å at $F(S_{\text{non-eq}})$2 epochs, and training with DeNS for $F(S_{\text{non-eq}})$3 epochs outperformed the baseline trained for $F(S_{\text{non-eq}})$4 epochs while increasing training time by only $F(S_{\text{non-eq}})$5 and parameters from $F(S_{\text{non-eq}})$6M to $F(S_{\text{non-eq}})$7M. On OC22, EquiformerV2 with DeNS improved S2EF-Total energy MAE by approximately $F(S_{\text{non-eq}})$8–$F(S_{\text{non-eq}})$9 and force MAE by approximately $$\mathcal{L}_{\text{DeNS}} = \mathbb{E}\left[ \sum_{i=1}^{|S_{\text{non-eq}}|} \left\| \frac{\boldsymbol{\epsilon}_i}{\sigma} - \hat{\boldsymbol{\epsilon}}_i(\tilde{S}_{\text{non-eq}}, F(S_{\text{non-eq}})) \right\|_2^2 \right].$$0–$$\mathcal{L}_{\text{DeNS}} = \mathbb{E}\left[ \sum_{i=1}^{|S_{\text{non-eq}}|} \left\| \frac{\boldsymbol{\epsilon}_i}{\sigma} - \hat{\boldsymbol{\epsilon}}_i(\tilde{S}_{\text{non-eq}}, F(S_{\text{non-eq}})) \right\|_2^2 \right].$$1, and reduced IS2RE-Total energy MAE by $$\mathcal{L}_{\text{DeNS}} = \mathbb{E}\left[ \sum_{i=1}^{|S_{\text{non-eq}}|} \left\| \frac{\boldsymbol{\epsilon}_i}{\sigma} - \hat{\boldsymbol{\epsilon}}_i(\tilde{S}_{\text{non-eq}}, F(S_{\text{non-eq}})) \right\|_2^2 \right].$$2 meV on ID and $$\mathcal{L}_{\text{DeNS}} = \mathbb{E}\left[ \sum_{i=1}^{|S_{\text{non-eq}}|} \left\| \frac{\boldsymbol{\epsilon}_i}{\sigma} - \hat{\boldsymbol{\epsilon}}_i(\tilde{S}_{\text{non-eq}}, F(S_{\text{non-eq}})) \right\|_2^2 \right].$$3 meV on OOD splits relative to EquiformerV2 without DeNS. On MD17, Equiformer with DeNS beat the $$\mathcal{L}_{\text{DeNS}} = \mathbb{E}\left[ \sum_{i=1}^{|S_{\text{non-eq}}|} \left\| \frac{\boldsymbol{\epsilon}_i}{\sigma} - \hat{\boldsymbol{\epsilon}}_i(\tilde{S}_{\text{non-eq}}, F(S_{\text{non-eq}})) \right\|_2^2 \right].$$4 baseline on all molecules and achieved similar or better performance than the $$\mathcal{L}_{\text{DeNS}} = \mathbb{E}\left[ \sum_{i=1}^{|S_{\text{non-eq}}|} \left\| \frac{\boldsymbol{\epsilon}_i}{\sigma} - \hat{\boldsymbol{\epsilon}}_i(\tilde{S}_{\text{non-eq}}, F(S_{\text{non-eq}})) \right\|_2^2 \right].$$5 baseline while requiring $$\mathcal{L}_{\text{DeNS}} = \mathbb{E}\left[ \sum_{i=1}^{|S_{\text{non-eq}}|} \left\| \frac{\boldsymbol{\epsilon}_i}{\sigma} - \hat{\boldsymbol{\epsilon}}_i(\tilde{S}_{\text{non-eq}}, F(S_{\text{non-eq}})) \right\|_2^2 \right].$$6 less training time (Liao et al., 2024).

On neural-potential pretraining benchmarks, denoising nonequilibrium molecules yields broad transfer gains. Pretraining on ANI-1 and ANI-1x reduced energy RMSE/MAE on ANI-1 by about $$\mathcal{L}_{\text{DeNS}} = \mathbb{E}\left[ \sum_{i=1}^{|S_{\text{non-eq}}|} \left\| \frac{\boldsymbol{\epsilon}_i}{\sigma} - \hat{\boldsymbol{\epsilon}}_i(\tilde{S}_{\text{non-eq}}, F(S_{\text{non-eq}})) \right\|_2^2 \right].$$7–$$\mathcal{L}_{\text{DeNS}} = \mathbb{E}\left[ \sum_{i=1}^{|S_{\text{non-eq}}|} \left\| \frac{\boldsymbol{\epsilon}_i}{\sigma} - \hat{\boldsymbol{\epsilon}}_i(\tilde{S}_{\text{non-eq}}, F(S_{\text{non-eq}})) \right\|_2^2 \right].$$8 on average and on ANI-1x by about $$\mathcal{L}_{\text{DeNS}} = \mathbb{E}\left[ \sum_{i=1}^{|S_{\text{non-eq}}|} \left\| \frac{\boldsymbol{\epsilon}_i}{\sigma} - \hat{\boldsymbol{\epsilon}}_i(\tilde{S}_{\text{non-eq}}, F(S_{\text{non-eq}})) \right\|_2^2 \right].$$9 and $$\tilde{S}_{\text{non-eq}} \mapsto S_{\text{non-eq}}$$0, respectively. The same pretraining transferred to SPICE, where pretrained GNNs showed $$\tilde{S}_{\text{non-eq}} \mapsto S_{\text{non-eq}}$$1 lower RMSE and MAE on average, and to large MD22 systems, where pretrained models achieved average improvements of $$\tilde{S}_{\text{non-eq}} \mapsto S_{\text{non-eq}}$$2 and $$\tilde{S}_{\text{non-eq}} \mapsto S_{\text{non-eq}}$$3 MAE per atom on buckyball catcher and double-walled nanotube. In a data-efficiency test on ANI-1x, pretrained EGNN with only $$\tilde{S}_{\text{non-eq}} \mapsto S_{\text{non-eq}}$$4 of the data reached MAE $$\tilde{S}_{\text{non-eq}} \mapsto S_{\text{non-eq}}$$5 kcal/mol versus $$\tilde{S}_{\text{non-eq}} \mapsto S_{\text{non-eq}}$$6 for non-pretrained EGNN, and the pretrained $$\tilde{S}_{\text{non-eq}} \mapsto S_{\text{non-eq}}$$7 model outperformed the non-pretrained $$\tilde{S}_{\text{non-eq}} \mapsto S_{\text{non-eq}}$$8 model (Wang et al., 2023).

In molecular structure optimization, Riemannian denoising substantially improves both geometry and energy. On QM9 starting from MMFF geometries, median $$\tilde{S}_{\text{non-eq}} \mapsto S_{\text{non-eq}}$$9 decreased from $$\tilde{\mathbf{x}}-\mathbf{x}=\boldsymbol{\varepsilon}$$0 kcal/mol for MMFF to $$\tilde{\mathbf{x}}-\mathbf{x}=\boldsymbol{\varepsilon}$$1 for Euclidean DSM and $$\tilde{\mathbf{x}}-\mathbf{x}=\boldsymbol{\varepsilon}$$2 for R-DSM, while median RMSD fell from $$\tilde{\mathbf{x}}-\mathbf{x}=\boldsymbol{\varepsilon}$$3 Å to $$\tilde{\mathbf{x}}-\mathbf{x}=\boldsymbol{\varepsilon}$$4 Å and median D-MAE from $$\tilde{\mathbf{x}}-\mathbf{x}=\boldsymbol{\varepsilon}$$5 Å to $$\tilde{\mathbf{x}}-\mathbf{x}=\boldsymbol{\varepsilon}$$6 Å. Because chemical accuracy is defined as $$\tilde{\mathbf{x}}-\mathbf{x}=\boldsymbol{\varepsilon}$$7, R-DSM reached chemical accuracy on average. Subsequent DFT optimization from R-DSM-refined structures also became much cheaper: mean force calls dropped from $$\tilde{\mathbf{x}}-\mathbf{x}=\boldsymbol{\varepsilon}$$8 to $$\tilde{\mathbf{x}}-\mathbf{x}=\boldsymbol{\varepsilon}$$9, mean SCF cycles from $\mathbf{c} = \phi_\theta(\mathbf{x})_{L0},$0 to $\mathbf{c} = \phi_\theta(\mathbf{x})_{L0},$1, and SCF time from $\mathbf{c} = \phi_\theta(\mathbf{x})_{L0},$2 s to $\mathbf{c} = \phi_\theta(\mathbf{x})_{L0},$3 s when both initializations converged to the same final minimum (Woo et al., 2024).

In atomistic self-supervised learning, SCD makes non-equilibrium data useful for pretraining rather than a source of ambiguity. On QM9 HOMO prediction, fine-tuning after SCD gave MAE $\mathbf{c} = \phi_\theta(\mathbf{x})_{L0},$4 meV from GEOM1, $\mathbf{c} = \phi_\theta(\mathbf{x})_{L0},$5 meV from GEOM10, and $\mathbf{c} = \phi_\theta(\mathbf{x})_{L0},$6 meV from OMol25, compared with $\mathbf{c} = \phi_\theta(\mathbf{x})_{L0},$7 without pretraining. On Matbench mp_gap, SCD pretraining on OMol25 yielded $\mathbf{c} = \phi_\theta(\mathbf{x})_{L0},$8 eV MAE versus $\mathbf{c} = \phi_\theta(\mathbf{x})_{L0},$9 baseline, and on LBA id60 it achieved $$\mathbb{E}_{q_\sigma(\tilde{\mathbf{x}},\mathbf{x})} \Big[ \|\phi_\theta(\tilde{\mathbf{x}}\mid \mathbf{c})-\boldsymbol{\varepsilon}\|_2^2 \Big],$$0 RMSE versus $$\mathbb{E}_{q_\sigma(\tilde{\mathbf{x}},\mathbf{x})} \Big[ \|\phi_\theta(\tilde{\mathbf{x}}\mid \mathbf{c})-\boldsymbol{\varepsilon}\|_2^2 \Big],$$1 baseline. The same paper emphasizes that pretraining with non-periodic, non-equilibrium molecule geometries from OMol25 conveyed a $$\mathbb{E}_{q_\sigma(\tilde{\mathbf{x}},\mathbf{x})} \Big[ \|\phi_\theta(\tilde{\mathbf{x}}\mid \mathbf{c})-\boldsymbol{\varepsilon}\|_2^2 \Big],$$2 benefit to bandgap prediction in periodic materials (Perez et al., 17 Mar 2026).

In structural identification, score-based denoising acts as a highly effective thermal filter. For high-temperature Cu crystals, adaptive CNA initially misclassified $$\mathbb{E}_{q_\sigma(\tilde{\mathbf{x}},\mathbf{x})} \Big[ \|\phi_\theta(\tilde{\mathbf{x}}\mid \mathbf{c})-\boldsymbol{\varepsilon}\|_2^2 \Big],$$3–$$\mathbb{E}_{q_\sigma(\tilde{\mathbf{x}},\mathbf{x})} \Big[ \|\phi_\theta(\tilde{\mathbf{x}}\mid \mathbf{c})-\boldsymbol{\varepsilon}\|_2^2 \Big],$$4 of atoms as disordered, but after one denoising step the misclassification rates dropped to $$\mathbb{E}_{q_\sigma(\tilde{\mathbf{x}},\mathbf{x})} \Big[ \|\phi_\theta(\tilde{\mathbf{x}}\mid \mathbf{c})-\boldsymbol{\varepsilon}\|_2^2 \Big],$$5 for BCC, $$\mathbb{E}_{q_\sigma(\tilde{\mathbf{x}},\mathbf{x})} \Big[ \|\phi_\theta(\tilde{\mathbf{x}}\mid \mathbf{c})-\boldsymbol{\varepsilon}\|_2^2 \Big],$$6 for FCC, and $$\mathbb{E}_{q_\sigma(\tilde{\mathbf{x}},\mathbf{x})} \Big[ \|\phi_\theta(\tilde{\mathbf{x}}\mid \mathbf{c})-\boldsymbol{\varepsilon}\|_2^2 \Big],$$7 for HCP; after $$\mathbb{E}_{q_\sigma(\tilde{\mathbf{x}},\mathbf{x})} \Big[ \|\phi_\theta(\tilde{\mathbf{x}}\mid \mathbf{c})-\boldsymbol{\varepsilon}\|_2^2 \Big],$$8–$$\mathbb{E}_{q_\sigma(\tilde{\mathbf{x}},\mathbf{x})} \Big[ \|\phi_\theta(\tilde{\mathbf{x}}\mid \mathbf{c})-\boldsymbol{\varepsilon}\|_2^2 \Big],$$9 steps the lattices became essentially perfect. On the DC3 benchmark at the melting point, after $\approx 0$00 denoising steps a-CNA reached $\approx 0$01 accuracy on all systems except BCC Fe at $\approx 0$02, and PTM reached $\approx 0$03 on all systems with Fe at $\approx 0$04, where the remaining discrepancies corresponded to real Frenkel pairs (Hsu et al., 2022).

AniDS further improves DeNS-style force learning by replacing isotropic noise with learned anisotropic covariances. On MD17, average force MAE improved from $\approx 0$05 for DeNS ($\approx 0$06) to $\approx 0$07 for AniDS ($\approx 0$08), an $\approx 0$09 relative improvement. On OC22 S2EF-Total, EquiformerV2 + DeNS achieved force MAE $\approx 0$10 meV/Å, whereas EquiformerV2 + AniDS reached $\approx 0$11, a further $\approx 0$12 improvement. On MPtrj $\approx 0$13 ALEXANDRIA, force MAE improved from $\approx 0$14 eV/Å to $\approx 0$15 (Liu et al., 25 Oct 2025).

6. Physical interpretation, misconceptions, and limitations

Many DeNS formulations are motivated by score-matching arguments. For nonequilibrium-molecule pretraining, the denoising target is interpreted as a local approximation to $\approx 0$16, making noise prediction a proxy for a score function or pseudo-force field under a local Gaussian assumption (Wang et al., 2023). In the Riemannian setting, the score vector is explicitly said to “play the role of a force field,” and the use of a physics-informed internal-coordinate metric is intended to align denoising directions with $\approx 0$17 on the PES (Woo et al., 2024). AniDS makes the same relation more directional: the denoiser regresses $\approx 0$18, which is derived as proportional to the force field under the Gaussian-mixture approximation (Liu et al., 25 Oct 2025).

A common misconception is that denoising should always be read as direct force learning. The SCD literature explicitly argues the opposite for its setting: noise vectors are poorly correlated with DFT forces even at equilibrium, and adding denoising as an auxiliary loss to force training can degrade force prediction. There, denoising is interpreted instead as enforcing local smoothness and learning global semantic structure via the self-embedding (Perez et al., 17 Mar 2026). The literature therefore supports a more limited statement: denoising can approximate physically meaningful vector fields in some formulations, but the strength and meaning of that approximation depend on the corruption model, conditioning signal, and downstream objective.

Another misconception is that denoised structures are always physically relaxed states. This is true only for some formulations. Riemannian score matching is explicitly designed as a molecular optimizer that transports MMFF or ETKDG geometries toward DFT minima (Woo et al., 2024). By contrast, the crystal denoiser is described as producing “mathematical constructs” with no guarantee of lying on the mechanical equilibrium manifold of any interatomic potential; its purpose is analysis, not replacing dynamics or performing relaxation (Hsu et al., 2022).

The limitations are correspondingly heterogeneous. Force-conditioned DeNS requires force labels and is therefore not applicable to purely unlabeled corpora (Liao et al., 2024). SCD is single-step rather than a full diffusion model and was not used for generative sampling (Perez et al., 17 Mar 2026). Riemannian denoising was evaluated on small organic molecules with fixed molecular graphs, pairwise-distance internal coordinates, and reduced robustness for severely distorted inputs (Woo et al., 2024). The crystal denoiser is trained on ideal lattices plus synthetic noise, so its score field is strongly shaped near crystalline manifolds and denoised states should be treated as filtered representations rather than direct physical observables (Hsu et al., 2022). AniDS still relies on a mixture-of-Gaussians approximation, uses per-atom pairwise-based covariances rather than explicit higher-order collective correlations, and does not explicitly parameterize temperature or external driving conditions (Liu et al., 25 Oct 2025).

Taken together, these works suggest that DeNS is best understood not as a single algorithm but as a research program organized around one question: how to exploit off-equilibrium atomistic structures through denoising without losing the geometric and physical information that makes those structures useful. The answers currently in the literature include force conditioning, self-conditioning, Riemannian metrics, and anisotropic learned noise models, each trading off label requirements, physical fidelity, and transferability across domains (Liao et al., 2024).