Dual-Scale Neural Operator for Fluid Forecasting

• Dual-Scale Neural Operator (DSO) is a neural architecture that separates local detail preservation from global flow trend processing in fluid dynamics forecasting.
• It leverages depthwise separable convolutions for fine-scale extraction and MLP-Mixer modules for domain-wide aggregation to balance local and global features.
• DSO demonstrates stable long-horizon predictions in 2D turbulence benchmarks, achieving over 88% error reduction compared to related methods.

Dual-Scale Neural Operator (DSO) is a neural operator architecture introduced for long-term autoregressive forecasting of fluid dynamics, particularly 2D turbulence governed by the Navier–Stokes equations. Its defining claim is that local and global information should not be processed uniformly: fine-scale local structure and large-scale global motion are assigned to different computational mechanisms, namely a local pathway based on depthwise separable convolutions and a global pathway based on an MLP-Mixer (Dong et al., 26 Mar 2026). In this formulation, “dual-scale” refers to an explicit separation between short-range, fine-grained spatial interactions and long-range, domain-wide aggregation, rather than merely to multiresolution downsampling or to generic multi-branch composition. Related literature uses similar language in broader or different senses, including coarse-to-fine downscaling operators, dual-scale PDE settings, and global-plus-local neural operators, but these works do not define DSO in the same architectural sense (Lai et al., 2024, Runkel et al., 2024, Liu-Schiaffini et al., 2024, Xu et al., 2024).

1. Definition and conceptual scope

DSO was proposed to address long-term instability in neural-operator-based fluid forecasting. The motivating diagnosis identifies two failure modes in existing architectures: local detail blurring, in which vortex cores, sharp gradients, and other fine structures are progressively smoothed during rollout, and global trend deviation, in which the large-scale motion trajectory drifts from the ground truth over time (Dong et al., 26 Mar 2026). The core argument is that these failures arise because many neural operators treat local and global information processing uniformly, even though their evolution characteristics differ physically.

Within this framework, DSO explicitly decouples computation into two complementary modules. The local module is intended for short-range structure preservation and refinement, while the global module is intended for long-range aggregation and motion coherence (Dong et al., 26 Mar 2026). This places DSO within a broader family of multiscale operator-learning methods, but with a more specific design objective: stable long-horizon autoregressive rollout.

The term “dual-scale” is used differently in adjacent work. “Neural Downscaling” learns a one-way operator from a resolved large-scale subspace to a complementary unresolved small-scale subspace, motivated by inertial manifold and nonlinear Galerkin theory (Lai et al., 2024). Work on dual-scale Stokes–Brinkman flow considers a single Fourier neural operator trained on a PDE with macropore and fibre-bundle regions, but does not introduce an explicitly dual-scale architecture (Runkel et al., 2024). “DPNO” proposes a dual-path architecture, yet its two paths are ResNet-like and DenseNet-like feature propagation patterns rather than coarse/fine scales (Wang et al., 17 Jul 2025). By contrast, DSO defines duality at the level of local versus global physical interactions (Dong et al., 26 Mar 2026).

2. Mathematical setting and motivating physical picture

The problem setting is the 2D incompressible Navier–Stokes system in vorticity form on a periodic domain: $$\partial_t \omega + (u \cdot \nabla)\omega = \nu \Delta \omega + f, \qquad (x,t)\in \mathbb{T}^2 \times (0,T],$$ with initial condition

$\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$

and velocity recovered from the streamfunction $\psi$ by

$$u = \nabla^\perp \psi = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi), \qquad -\Delta \psi = \omega.$$

The operator-learning objective is written as

$$G : C([0,T_{\text{in}}];H^r(\mathbb{T}^2)) \to C((T_{\text{in}},T];H^r(\mathbb{T}^2)),$$

and practically realized as a one-step predictor iterated autoregressively (Dong et al., 26 Mar 2026).

The forecasting recursion is

$$\hat{\omega}(t+1)=G_\theta\big(\omega(t-T_{\text{in}}+1),\ldots,\omega(t)\big),$$

followed by

$$\hat{\omega}(t+k+1)=G_\theta\big(\hat{\omega}(t+k-T_{\text{in}}+1),\ldots,\hat{\omega}(t+k)\big), \quad k=1,\ldots,T_{\text{pred}}-1.$$

The difficulty is that after the first step the input consists entirely of model-generated states, so small one-step errors can accumulate into unstable long-term trajectories (Dong et al., 26 Mar 2026).

A distinctive feature of the DSO paper is its physical motivation experiment on vortex dynamics. Using a pseudo-spectral solver on a $128\times128$ periodic domain, the study compares a close perturbation at distance $d=0.6$ and a far perturbation at distance $d=2.5$. Local deformation is measured using the maximum vorticity gradient $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$0, and global displacement is measured using the center-of-vorticity position

$\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$1

The reported outcomes are: close perturbation produces local gradient change $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$2 and global position shift $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$3, whereas far perturbation produces local gradient change $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$4 and global position shift $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$5 (Dong et al., 26 Mar 2026). The stated interpretation is qualitative rather than merely numerical: nearby perturbations predominantly affect local vortex structure, while distant perturbations primarily alter global motion trends. This provides the empirical rationale for separating local and global processing.

A plausible implication is that DSO’s “dual-scale” notion is closer to a decomposition by interaction range and dynamical role than to a classical coarse-grid/fine-grid hierarchy. That interpretation is consistent with other global-plus-local operator designs, such as localized-kernel augmentations of FNO and hybrid Fourier–convolution architectures, although those works do not formulate the same local-versus-global forecasting hypothesis (Liu-Schiaffini et al., 2024, Xu et al., 2024).

3. Architecture and operator decomposition

DSO is organized as an encoder–translator–decoder architecture: $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$6 The notation in the paper uses a single input field $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$7, and the reported experiments are described as one-step prediction from one time step to the next, so effectively $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$8 in the reported setting (Dong et al., 26 Mar 2026).

The encoder progressively downsamples spatially and increases channel width using $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$9 convolutional layers: $\psi$0 with $\psi$1. The shallow feature $\psi$2 is preserved for a skip connection into the decoder (Dong et al., 26 Mar 2026).

The decoder uses $\psi$3 transposed-convolution layers: $\psi$4 starting from $\psi$5, where $\psi$6 is the translator output, and the final prediction is

$\psi$7

where $\psi$8 denotes concatenation and $\psi$9 is a $$u = \nabla^\perp \psi = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi), \qquad -\Delta \psi = \omega.$$0 convolution (Dong et al., 26 Mar 2026).

The translator $$u = \nabla^\perp \psi = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi), \qquad -\Delta \psi = \omega.$$1 is the defining component. It consists of $$u = \nabla^\perp \psi = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi), \qquad -\Delta \psi = \omega.$$2 stacked dual-pathway blocks. For block $$u = \nabla^\perp \psi = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi), \qquad -\Delta \psi = \omega.$$3,

$$u = \nabla^\perp \psi = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi), \qquad -\Delta \psi = \omega.$$4

The local and global modules are therefore composed sequentially, not run as parallel branches with explicit fusion. The design order is deliberate: local first, global second (Dong et al., 26 Mar 2026).

The local module is

$$u = \nabla^\perp \psi = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi), \qquad -\Delta \psi = \omega.$$5

where $$u = \nabla^\perp \psi = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi), \qquad -\Delta \psi = \omega.$$6 is depthwise convolution, $$u = \nabla^\perp \psi = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi), \qquad -\Delta \psi = \omega.$$7 is $$u = \nabla^\perp \psi = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi), \qquad -\Delta \psi = \omega.$$8 pointwise convolution, and $$u = \nabla^\perp \psi = (\partial_{x_2}\psi,\,-\partial_{x_1}\psi), \qquad -\Delta \psi = \omega.$$9 is a learnable scaling parameter (Dong et al., 26 Mar 2026). The stated role of this module is local feature extraction and preservation of fine-scale structures such as vortex cores and sharp gradients.

The global module is

$$G : C([0,T_{\text{in}}];H^r(\mathbb{T}^2)) \to C((T_{\text{in}},T];H^r(\mathbb{T}^2)),$$0

Here $$G : C([0,T_{\text{in}}];H^r(\mathbb{T}^2)) \to C((T_{\text{in}},T];H^r(\mathbb{T}^2)),$$1 mixes information across all spatial locations for each channel, and $$G : C([0,T_{\text{in}}];H^r(\mathbb{T}^2)) \to C((T_{\text{in}},T];H^r(\mathbb{T}^2)),$$2 mixes information across channels (Dong et al., 26 Mar 2026). This is the mechanism for domain-wide communication and maintenance of correct large-scale motion trends.

The operator can therefore be summarized as

$$G : C([0,T_{\text{in}}];H^r(\mathbb{T}^2)) \to C((T_{\text{in}},T];H^r(\mathbb{T}^2)),$$3

This decomposition is the mathematical core of DSO (Dong et al., 26 Mar 2026).

The distinction between DSO and several neighboring designs is structurally important. DSO is not a self-composing shared-weight operator of the form

$$G : C([0,T_{\text{in}}];H^r(\mathbb{T}^2)) \to C((T_{\text{in}},T];H^r(\mathbb{T}^2)),$$4

which characterizes the depth-scaling framework of self-composing neural operators (He et al., 28 Aug 2025). Nor is it a dual-path feature-reuse wrapper of the form

$$G : C([0,T_{\text{in}}];H^r(\mathbb{T}^2)) \to C((T_{\text{in}},T];H^r(\mathbb{T}^2)),$$5

which characterizes DPNO (Wang et al., 17 Jul 2025). DSO’s duality is instead local-versus-global operator bias inside a single translator block (Dong et al., 26 Mar 2026).

4. Training protocol, evaluation criteria, and physical diagnostics

The reported training protocol uses one-step supervised learning followed by rolling autoregressive evaluation. For NS-Decaying, each 100-step trajectory yields 99 adjacent one-step training pairs, and the trained model is then rolled out across the full horizon at test time (Dong et al., 26 Mar 2026). No teacher forcing beyond standard one-step supervision, no scheduled sampling, no explicit multi-step training loss, and no physics-constrained loss terms are reported in the visible content.

The training environment is specified as follows: 8 NVIDIA A100 GPUs with 40 GB each, PyTorch DistributedDataParallel, Python 3.8, PyTorch 1.8.1, CUDA 11.1, batch size 20, 500 epochs, initial learning rate $$G : C([0,T_{\text{in}}];H^r(\mathbb{T}^2)) \to C((T_{\text{in}},T];H^r(\mathbb{T}^2)),$$6, and random seed 42 (Dong et al., 26 Mar 2026). The optimizer type is not explicitly stated in the provided text.

The primary reported metric is mean squared error: $$G : C([0,T_{\text{in}}];H^r(\mathbb{T}^2)) \to C((T_{\text{in}},T];H^r(\mathbb{T}^2)),$$7 The paper also defines the structural similarity index: $$G : C([0,T_{\text{in}}];H^r(\mathbb{T}^2)) \to C((T_{\text{in}},T];H^r(\mathbb{T}^2)),$$8 MSE is used as the main accuracy measure, while SSIM is used to assess structural fidelity in long rollout (Dong et al., 26 Mar 2026).

The paper further emphasizes physical diagnostics beyond pixelwise error. It evaluates gradient error, associated with deformation features such as strain and vortex stretching, and divergence error, used as a proxy for physical consistency or mass conservation (Dong et al., 26 Mar 2026). The reported interpretation is that DSO’s gains are not limited to low MSE; they extend to physically meaningful structure preservation.

This emphasis on derivative- and structure-sensitive evaluation situates DSO within a broader multiscale operator-learning trend. Other works addressing multiscale PDEs also highlight the inadequacy of plain fieldwise $$G : C([0,T_{\text{in}}];H^r(\mathbb{T}^2)) \to C((T_{\text{in}},T];H^r(\mathbb{T}^2)),$$9 supervision when fine scales matter. HANO introduces an empirical $$\hat{\omega}(t+1)=G_\theta\big(\omega(t-T_{\text{in}}+1),\ldots,\omega(t)\big),$$0 loss to emphasize high-frequency components (Liu et al., 2022); localized-kernel operator learning reports major gains precisely where local detail is critical (Liu-Schiaffini et al., 2024); and dual-scale permeability prediction shows that the best field loss need not be the best objective for macroscopic observables (Runkel et al., 2024). This suggests that DSO’s evaluation philosophy is aligned with a larger methodological shift toward structure-aware assessment, although DSO itself is trained with one-step MSE in the visible text (Dong et al., 26 Mar 2026).

5. Benchmarks and empirical performance

DSO is evaluated on two turbulence benchmarks. NS-Forced uses viscosity

$$\hat{\omega}(t+1)=G_\theta\big(\omega(t-T_{\text{in}}+1),\ldots,\omega(t)\big),$$1

persistent low-wavenumber forcing, and dataset shape

$$\hat{\omega}(t+1)=G_\theta\big(\omega(t-T_{\text{in}}+1),\ldots,\omega(t)\big),$$2

NS-Decaying models freely decaying 2D turbulence with no external forcing and dataset shape

$$\hat{\omega}(t+1)=G_\theta\big(\omega(t-T_{\text{in}}+1),\ldots,\omega(t)\big),$$3

The train/validation/test split is $$\hat{\omega}(t+1)=G_\theta\big(\omega(t-T_{\text{in}}+1),\ldots,\omega(t)\big),$$4 (Dong et al., 26 Mar 2026).

On NS-Forced, the reported all-step MSE values are: DSO $$\hat{\omega}(t+1)=G_\theta\big(\omega(t-T_{\text{in}}+1),\ldots,\omega(t)\big),$$5, FNO $$\hat{\omega}(t+1)=G_\theta\big(\omega(t-T_{\text{in}}+1),\ldots,\omega(t)\big),$$6, LSM $$\hat{\omega}(t+1)=G_\theta\big(\omega(t-T_{\text{in}}+1),\ldots,\omega(t)\big),$$7, SimVP $$\hat{\omega}(t+1)=G_\theta\big(\omega(t-T_{\text{in}}+1),\ldots,\omega(t)\big),$$8, UNO $$\hat{\omega}(t+1)=G_\theta\big(\omega(t-T_{\text{in}}+1),\ldots,\omega(t)\big),$$9, CNO $$\hat{\omega}(t+k+1)=G_\theta\big(\hat{\omega}(t+k-T_{\text{in}}+1),\ldots,\hat{\omega}(t+k)\big), \quad k=1,\ldots,T_{\text{pred}}-1.$$0, U-Net $$\hat{\omega}(t+k+1)=G_\theta\big(\hat{\omega}(t+k-T_{\text{in}}+1),\ldots,\hat{\omega}(t+k)\big), \quad k=1,\ldots,T_{\text{pred}}-1.$$1, Swin $$\hat{\omega}(t+k+1)=G_\theta\big(\hat{\omega}(t+k-T_{\text{in}}+1),\ldots,\hat{\omega}(t+k)\big), \quad k=1,\ldots,T_{\text{pred}}-1.$$2, ConvLSTM $$\hat{\omega}(t+k+1)=G_\theta\big(\hat{\omega}(t+k-T_{\text{in}}+1),\ldots,\hat{\omega}(t+k)\big), \quad k=1,\ldots,T_{\text{pred}}-1.$$3, ResNet $$\hat{\omega}(t+k+1)=G_\theta\big(\hat{\omega}(t+k-T_{\text{in}}+1),\ldots,\hat{\omega}(t+k)\big), \quad k=1,\ldots,T_{\text{pred}}-1.$$4, and PastNet $$\hat{\omega}(t+k+1)=G_\theta\big(\hat{\omega}(t+k-T_{\text{in}}+1),\ldots,\hat{\omega}(t+k)\big), \quad k=1,\ldots,T_{\text{pred}}-1.$$5 (Dong et al., 26 Mar 2026). At horizon-specific checkpoints, DSO and FNO both report $$\hat{\omega}(t+k+1)=G_\theta\big(\hat{\omega}(t+k-T_{\text{in}}+1),\ldots,\hat{\omega}(t+k)\big), \quad k=1,\ldots,T_{\text{pred}}-1.$$6 at 1-step, while at 10-step DSO achieves $$\hat{\omega}(t+k+1)=G_\theta\big(\hat{\omega}(t+k-T_{\text{in}}+1),\ldots,\hat{\omega}(t+k)\big), \quad k=1,\ldots,T_{\text{pred}}-1.$$7 versus FNO $$\hat{\omega}(t+k+1)=G_\theta\big(\hat{\omega}(t+k-T_{\text{in}}+1),\ldots,\hat{\omega}(t+k)\big), \quad k=1,\ldots,T_{\text{pred}}-1.$$8, and at 19-step DSO achieves $$\hat{\omega}(t+k+1)=G_\theta\big(\hat{\omega}(t+k-T_{\text{in}}+1),\ldots,\hat{\omega}(t+k)\big), \quad k=1,\ldots,T_{\text{pred}}-1.$$9 versus FNO $128\times128$0, UNO $128\times128$1, and CNO $128\times128$2 (Dong et al., 26 Mar 2026).

On NS-Decaying, the separation is much larger. The all-step MSE values are: DSO $128\times128$3, SimVP $128\times128$4, FNO $128\times128$5, LSM $128\times128$6, U-Net $128\times128$7, Swin $128\times128$8, ConvLSTM $128\times128$9, CNO $d=0.6$0, ResNet $d=0.6$1, PastNet $d=0.6$2, and UNO NaN collapse (Dong et al., 26 Mar 2026). At specific horizons, DSO reports $d=0.6$3 at 1-step, $d=0.6$4 at 50-step, and $d=0.6$5 at 99-step, whereas FNO reports $d=0.6$6 at 50-step and $d=0.6$7 at 99-step, and SimVP reports $d=0.6$8 at 50-step and $d=0.6$9 at 99-step (Dong et al., 26 Mar 2026).

The paper’s headline claim of over 88% error reduction refers to NS-Decaying all-step MSE: DSO’s $d=2.5$0 is about $d=2.5$1 of the next-best result, SimVP’s $d=2.5$2, corresponding to roughly $d=2.5$3 lower error (Dong et al., 26 Mar 2026). The paper emphasizes that this is a long-horizon stability result, not merely a short-term accuracy result.

The ablation study isolates the effect of each pathway on NS-Decaying. Full DSO gives all-step $d=2.5$4, 1-step $d=2.5$5, 50-step $d=2.5$6, and 99-step $d=2.5$7. Removing the convolution branch yields all-step $d=2.5$8, 1-step $d=2.5$9, 50-step $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$00, and 99-step $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$01. Removing the MLP-Mixer branch degrades performance much more severely, to all-step $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$02, 1-step $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$03, 50-step $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$04, and 99-step $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$05 (Dong et al., 26 Mar 2026). The paper’s conclusion is that both branches matter, but global processing is especially critical for long-horizon stability.

The visual interpretation reported in the paper is equally specific. FNO preserves rough global behavior but smooths out turbulent details; LSM preserves more local structure but drifts globally; ConvLSTM is unstable in long rollout; DSO remains visually close to the ground truth even at step 99 (Dong et al., 26 Mar 2026). This corresponds directly to the two identified failure modes: blur and drift.

6. Relation to broader neural-operator research

DSO belongs to a larger research program on multiscale, global-local, and structure-aware neural operators, but it occupies a specific niche. Its closest conceptual relatives are methods that argue a single operator mechanism is insufficient when the target map mixes long-range/global effects with short-range/local structure.

One direct comparison point is work on localized integral and differential kernels. That line of research augments FNO or SFNO with local branches designed to preserve discretization-invariant local operators, motivated by the claim that global spectral mixing can over-smooth and miss local detail (Liu-Schiaffini et al., 2024). Another is DCNO, which combines Fourier layers for low-frequency global structure with dilated convolution layers for high-frequency localized correction, interleaved in an alternating sequence (Xu et al., 2024). A plausible implication is that DSO, DCNO, and localized-kernel operator learning all instantiate the same broad hypothesis: global and local interactions require different inductive biases. DSO differs in expressing that hypothesis in an autoregressive fluid-forecasting setting with a specific local-then-global block based on depthwise separable convolution and MLP-Mixer (Dong et al., 26 Mar 2026).

HANO is relevant for a different reason. It addresses spectral bias in multiscale operator learning through hierarchical attention and an empirical $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$06 loss, effectively coupling fine-level local interactions with coarser aggregated interactions in a multilevel hierarchy (Liu et al., 2022). Relative to HANO, DSO is simpler and more specialized: it uses a dual-scale decomposition rather than a full multilevel hierarchy, and it emphasizes long-term rollout stability in turbulence rather than general multiscale PDE approximation (Dong et al., 26 Mar 2026).

Several neighboring papers are related only indirectly and should not be conflated with DSO. “Self-Composing Neural Operators with Depth and Accuracy Scaling via Adaptive Train-and-Unroll Approach” proposes recurrent self-composition of a shared backbone block,

$\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$07

with adaptive train-and-unroll depth scaling; its multi-scale aspect appears only in a multigrid-inspired Helmholtz backbone for a specific application (He et al., 28 Aug 2025). “DPNO” is dual-path in the sense of ResNet-like and DenseNet-like connectivity, not dual-scale in the sense of local and global physics (Wang et al., 17 Jul 2025). “Neural Downscaling” is dual-scale in a different mathematical sense, learning a one-way map $\omega(x,0)=\omega_0(x), \qquad x\in \mathbb{T}^2,$08 from large-scale modes to small-scale modes on complementary spectral subspaces (Lai et al., 2024). Work on permeability prediction in a dual-scale flow problem studies a dual-scale PDE but uses a single FNO or TFNO rather than a dedicated dual-scale architecture (Runkel et al., 2024).

The main misconception surrounding DSO is therefore terminological. In the DSO paper, “dual-scale” does not denote a generic two-branch neural operator, a dual-resolution multigrid architecture, a coarse/fine spectral decomposition, or a recurrent self-composition scheme. It denotes explicit decoupling of local structure processing and global trend processing inside a learned fluid-dynamics update operator (Dong et al., 26 Mar 2026). That definition is narrower than some uses of “multiscale” and broader than a simple encoder–decoder hierarchy.

A plausible synthesis across the cited literature is that DSO represents one concrete endpoint in a continuum of multiscale operator designs: from one-way coarse-to-fine slaving (Lai et al., 2024), to global-plus-local operator augmentation (Liu-Schiaffini et al., 2024), to hybrid spectral-local multiscale processors (Xu et al., 2024), to explicit local-versus-global autoregressive forecasting blocks (Dong et al., 26 Mar 2026). Within that continuum, DSO’s distinctive contribution is the claim that long-term fluid forecasting fails when local detail preservation and global trajectory control are not treated as distinct modeling problems, and that a dual-scale operator can address both simultaneously (Dong et al., 26 Mar 2026).