Generative SLiCEs: Continuous-Time Models

• Generative SLiCEs (G-SLiCEs) are continuous-time generative models leveraging structured linear controlled differential equations to achieve universal causal time-series generation.
• They train via flow matching on path space, enabling exact interval transitions and robust performance on arbitrary and irregular observation grids.
• Empirical evaluations reveal superior probabilistic forecasting and generation metrics, outperforming baselines across diverse real-world datasets.

Generative SLiCEs (G-SLiCEs) are continuous-time generative models for time series built on Structured Linear Controlled Differential Equations (SLiCEs) and trained via flow matching on path space. Their defining claim is twofold: maximally expressive SLiCE classes are universal causal time-series generators in distribution, and the resulting continuous-time models improve probabilistic forecasting and downstream generation while retaining the ability to operate on arbitrary observation grids, including irregular ones (Berndt et al., 27 May 2026).

1. Formal definition and model class

G-SLiCEs inherit their backbone from the neural controlled differential equation (NCDE) framework. For an input path $X$, augmented into a control path $\omega^X$, an NCDE evolves a hidden state continuously in physical time $t$: $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t g_\theta(h_s)\,d\omega^X_s,\qquad z_t=r_\psi(h_t).$$

A Linear NCDE restricts the vector field to be linear in the hidden state: $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t\sum_{i=1}^{d_\omega}A_\theta^i h_s\,d\omega_s^{X,i},\qquad z_t=r_\psi(h_t).$$

A SLiCE is a Structured Linear CDE: it uses the same linear NCDE form, but each transition matrix $A_\theta^i$ is restricted to a structured family, such as block-diagonal, diagonal-plus-low-rank, sparse, Walsh-Hadamard structured, or dense. In this sense, SLiCEs sit at the overlap between neural CDEs, state space models, and continuous-time sequence models. The paper defines a model class as maximally expressive if it can uniformly approximate every continuous target map on compact subsets of the relevant path space, and it relies on prior results showing that some structures retain the same approximation power as dense Linear NCDEs, whereas single-layer diagonal selective or non-selective forms abstracting Mamba/S4-style transitions do not (Berndt et al., 27 May 2026).

For piecewise linear controls, the Linear NCDE admits exact interval transitions

$$h_{t_{j+1}}=\Phi_j^\theta(X)h_{t_j}, \qquad \Phi_j^\theta(X)=\exp\!\left(\sum_{i=1}^{d_\omega}A_\theta^i\big(\omega_{t_{j+1}}^{X,i}-\omega_{t_j}^{X,i}\big)\right),$$

so that

$$h_{t_k}=\Phi_{k-1}^\theta(X)\cdots\Phi_0^\theta(X)h_{t_0}.$$

This exact-flow recurrence makes the model look like an input-dependent continuous-time state space model with parallel-in-time computation via associative prefix products.

2. Generative construction and path-space flow matching

The direct generative view is simple: a G-SLiCE consists of a causal SLiCE map $G_\theta:\mathcal X(d_X)\to\mathcal Y(d_y)$ and a prior law $\mu$ on latent paths; sampling is

$\omega^X$0

with generated law

$\omega^X$1

The practical implementation is a flow on path space indexed by a synthetic flow time $\omega^X$2, distinct from physical time $\omega^X$3: $\omega^X$4 The terminal path $\omega^X$5 induces the model law

$\omega^X$6

Training uses conditional flow matching generalized from finite-dimensional vectors to paths. For a coupled pair $\omega^X$7, the interpolation in path space is

$\omega^X$8

and the target velocity is

$\omega^X$9

The training loss is

$t$0

This formulation supports both unconditional generation and conditional probabilistic forecasting. In the conditional case, the prior is conditioned on observed context $t$1, and the paper states that G-SLiCE can be evaluated directly on the requested physical-time grid. That continuous-time property is central to its behavior under arbitrary observation grids and under irregular sampling, where fixed-grid models often require heuristics such as repeating values or oversampling latent Gaussian processes (Berndt et al., 27 May 2026).

3. Universality on path space

The theory is stated on path spaces over a fixed interval $t$2. The input path space is

$t$3

the space of absolutely continuous, time-augmented paths starting from a common point, equipped with the $t$4-variation topology. The output path space is

$t$5

equipped with the supremum metric

$t$6

A path map $t$7 is causal if

$t$8

The paper uses the $t$9-Wasserstein distance on path laws: $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t g_\theta(h_s)\,d\omega^X_s,\qquad z_t=r_\psi(h_t).$$0

A class $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t g_\theta(h_s)\,d\omega^X_s,\qquad z_t=r_\psi(h_t).$$1 is defined to be a universal causal time-series generator if, for every compact $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t g_\theta(h_s)\,d\omega^X_s,\qquad z_t=r_\psi(h_t).$$2, every probability measure $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t g_\theta(h_s)\,d\omega^X_s,\qquad z_t=r_\psi(h_t).$$3 supported on $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t g_\theta(h_s)\,d\omega^X_s,\qquad z_t=r_\psi(h_t).$$4, every continuous causal map $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t g_\theta(h_s)\,d\omega^X_s,\qquad z_t=r_\psi(h_t).$$5, and every $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t g_\theta(h_s)\,d\omega^X_s,\qquad z_t=r_\psi(h_t).$$6, there exists $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t g_\theta(h_s)\,d\omega^X_s,\qquad z_t=r_\psi(h_t).$$7 such that

$$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t g_\theta(h_s)\,d\omega^X_s,\qquad z_t=r_\psi(h_t).$$8

The central lifting theorem states that if a model class can approximate every continuous causal path-to-path map uniformly on compact sets,

$$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t g_\theta(h_s)\,d\omega^X_s,\qquad z_t=r_\psi(h_t).$$9

then it is a universal causal time-series generator. Combined with the path-to-path universality theorem for maximally expressive SLiCEs, this yields the stated corollary that any maximally expressive SLiCE class is a universal causal time-series generator (Berndt et al., 27 May 2026).

The paper also gives a concrete expressivity-gap example based on the recursion

$$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t\sum_{i=1}^{d_\omega}A_\theta^i h_s\,d\omega_s^{X,i},\qquad z_t=r_\psi(h_t).$$0

which generates binary sequences with no consecutive ones. In that setting, a width-2 dense selective exact-flow state space model can approximate the induced law arbitrarily well, whereas dense non-selective exact-flow state space models and width-$$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t\sum_{i=1}^{d_\omega}A_\theta^i h_s\,d\omega_s^{X,i},\qquad z_t=r_\psi(h_t).$$1 diagonal selective exact-flow state space models have explicit lower bounds on approximation error in $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t\sum_{i=1}^{d_\omega}A_\theta^i h_s\,d\omega_s^{X,i},\qquad z_t=r_\psi(h_t).$$2. The stated empirical result is that G-SLiCE learns the target exactly across tested sequence lengths, while the restricted state space models fail. This suggests that the distinction between maximally expressive and non-maximally expressive transition structures is not merely formal but operational (Berndt et al., 27 May 2026).

4. Empirical behavior in forecasting, generation, and grid shift

The empirical study covers conditional probabilistic forecasting, unconditional generation, and generalization across shifted sampling grids. Forecasting experiments are reported on Electricity, Exchange, KDDCup, M4-Hourly, Solar, Traffic, UberTLC-Hourly, Wikipedia, and ETTSmall for grid-shift experiments. Baselines include Seasonal Naive, AutoARIMA, AutoETS, DLinear, DeepAR, TFT, WaveNet, PatchTST, CSDI, SSSD, Biloš et al., TSDiff, and TSFlow (Berndt et al., 27 May 2026).

For probabilistic forecasting, the metric is CRPS. G-SLiCE achieves the best CRPS on $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t\sum_{i=1}^{d_\omega}A_\theta^i h_s\,d\omega_s^{X,i},\qquad z_t=r_\psi(h_t).$$3 datasets and beats TSFlow on $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t\sum_{i=1}^{d_\omega}A_\theta^i h_s\,d\omega_s^{X,i},\qquad z_t=r_\psi(h_t).$$4 datasets. Representative results include Electricity: TSFlow $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t\sum_{i=1}^{d_\omega}A_\theta^i h_s\,d\omega_s^{X,i},\qquad z_t=r_\psi(h_t).$$5, G-SLiCE $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t\sum_{i=1}^{d_\omega}A_\theta^i h_s\,d\omega_s^{X,i},\qquad z_t=r_\psi(h_t).$$6; Exchange: TSFlow $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t\sum_{i=1}^{d_\omega}A_\theta^i h_s\,d\omega_s^{X,i},\qquad z_t=r_\psi(h_t).$$7, G-SLiCE $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t\sum_{i=1}^{d_\omega}A_\theta^i h_s\,d\omega_s^{X,i},\qquad z_t=r_\psi(h_t).$$8; KDD: TSFlow $$h_{t_0}=\xi_\theta(X_{t_0}),\qquad h_t=h_{t_0}+\int_{t_0}^t\sum_{i=1}^{d_\omega}A_\theta^i h_s\,d\omega_s^{X,i},\qquad z_t=r_\psi(h_t).$$9, G-SLiCE $A_\theta^i$0; M4-Hourly: TSFlow $A_\theta^i$1, G-SLiCE $A_\theta^i$2; Solar: TSFlow $A_\theta^i$3, G-SLiCE $A_\theta^i$4; Traffic: TSFlow $A_\theta^i$5, G-SLiCE $A_\theta^i$6; Uber: TSFlow $A_\theta^i$7, G-SLiCE $A_\theta^i$8. The paper reports a Friedman test indicating significant rank differences and a paired Wilcoxon test versus TSFlow with $A_\theta^i$9 when excluding the Wiki anomaly (Berndt et al., 27 May 2026).

For unconditional generation, two metrics are reported: $$h_{t_{j+1}}=\Phi_j^\theta(X)h_{t_j}, \qquad \Phi_j^\theta(X)=\exp\!\left(\sum_{i=1}^{d_\omega}A_\theta^i\big(\omega_{t_{j+1}}^{X,i}-\omega_{t_j}^{X,i}\big)\right),$$0-Wasserstein distance between generated and real samples, and Linear Predictive Score (LPS), obtained by training a linear predictor on synthetic data and testing on real data. Examples from the $$h_{t_{j+1}}=\Phi_j^\theta(X)h_{t_j}, \qquad \Phi_j^\theta(X)=\exp\!\left(\sum_{i=1}^{d_\omega}A_\theta^i\big(\omega_{t_{j+1}}^{X,i}-\omega_{t_j}^{X,i}\big)\right),$$1 table are Electricity: TSFlow $$h_{t_{j+1}}=\Phi_j^\theta(X)h_{t_j}, \qquad \Phi_j^\theta(X)=\exp\!\left(\sum_{i=1}^{d_\omega}A_\theta^i\big(\omega_{t_{j+1}}^{X,i}-\omega_{t_j}^{X,i}\big)\right),$$2, G-SLiCE $$h_{t_{j+1}}=\Phi_j^\theta(X)h_{t_j}, \qquad \Phi_j^\theta(X)=\exp\!\left(\sum_{i=1}^{d_\omega}A_\theta^i\big(\omega_{t_{j+1}}^{X,i}-\omega_{t_j}^{X,i}\big)\right),$$3; M4(H): TSFlow $$h_{t_{j+1}}=\Phi_j^\theta(X)h_{t_j}, \qquad \Phi_j^\theta(X)=\exp\!\left(\sum_{i=1}^{d_\omega}A_\theta^i\big(\omega_{t_{j+1}}^{X,i}-\omega_{t_j}^{X,i}\big)\right),$$4, G-SLiCE $$h_{t_{j+1}}=\Phi_j^\theta(X)h_{t_j}, \qquad \Phi_j^\theta(X)=\exp\!\left(\sum_{i=1}^{d_\omega}A_\theta^i\big(\omega_{t_{j+1}}^{X,i}-\omega_{t_j}^{X,i}\big)\right),$$5; Solar: TSFlow $$h_{t_{j+1}}=\Phi_j^\theta(X)h_{t_j}, \qquad \Phi_j^\theta(X)=\exp\!\left(\sum_{i=1}^{d_\omega}A_\theta^i\big(\omega_{t_{j+1}}^{X,i}-\omega_{t_j}^{X,i}\big)\right),$$6, G-SLiCE $$h_{t_{j+1}}=\Phi_j^\theta(X)h_{t_j}, \qquad \Phi_j^\theta(X)=\exp\!\left(\sum_{i=1}^{d_\omega}A_\theta^i\big(\omega_{t_{j+1}}^{X,i}-\omega_{t_j}^{X,i}\big)\right),$$7; Traffic: TSFlow $$h_{t_{j+1}}=\Phi_j^\theta(X)h_{t_j}, \qquad \Phi_j^\theta(X)=\exp\!\left(\sum_{i=1}^{d_\omega}A_\theta^i\big(\omega_{t_{j+1}}^{X,i}-\omega_{t_j}^{X,i}\big)\right),$$8, G-SLiCE $$h_{t_{j+1}}=\Phi_j^\theta(X)h_{t_j}, \qquad \Phi_j^\theta(X)=\exp\!\left(\sum_{i=1}^{d_\omega}A_\theta^i\big(\omega_{t_{j+1}}^{X,i}-\omega_{t_j}^{X,i}\big)\right),$$9. Examples from LPS are Electricity: TSFlow $$h_{t_k}=\Phi_{k-1}^\theta(X)\cdots\Phi_0^\theta(X)h_{t_0}.$$0, G-SLiCE $$h_{t_k}=\Phi_{k-1}^\theta(X)\cdots\Phi_0^\theta(X)h_{t_0}.$$1; M4(H): TSFlow $$h_{t_k}=\Phi_{k-1}^\theta(X)\cdots\Phi_0^\theta(X)h_{t_0}.$$2, G-SLiCE $$h_{t_k}=\Phi_{k-1}^\theta(X)\cdots\Phi_0^\theta(X)h_{t_0}.$$3; Solar: TSFlow $$h_{t_k}=\Phi_{k-1}^\theta(X)\cdots\Phi_0^\theta(X)h_{t_0}.$$4, G-SLiCE $$h_{t_k}=\Phi_{k-1}^\theta(X)\cdots\Phi_0^\theta(X)h_{t_0}.$$5; Traffic: TSFlow $$h_{t_k}=\Phi_{k-1}^\theta(X)\cdots\Phi_0^\theta(X)h_{t_0}.$$6, G-SLiCE $$h_{t_k}=\Phi_{k-1}^\theta(X)\cdots\Phi_0^\theta(X)h_{t_0}.$$7. The reported interpretation is that the generated samples better preserve predictive structure (Berndt et al., 27 May 2026).

Grid-shift robustness is presented as a particularly strong result. In cross-frequency generalization, training at $$h_{t_k}=\Phi_{k-1}^\theta(X)\cdots\Phi_0^\theta(X)h_{t_0}.$$8 min and testing at $$h_{t_k}=\Phi_{k-1}^\theta(X)\cdots\Phi_0^\theta(X)h_{t_0}.$$9 h yields CRPS $G_\theta:\mathcal X(d_X)\to\mathcal Y(d_y)$0 for G-SLiCE and $G_\theta:\mathcal X(d_X)\to\mathcal Y(d_y)$1 for direct TSFlow. For irregular-grid generalization, ETTSmall $G_\theta:\mathcal X(d_X)\to\mathcal Y(d_y)$2-min data are subsampled to $G_\theta:\mathcal X(d_X)\to\mathcal Y(d_y)$3 irregular timestamps using a Gamma renewal process, with shape parameter $G_\theta:\mathcal X(d_X)\to\mathcal Y(d_y)$4 controlling regularity. G-SLiCE remains stable across irregular and regular test grids, with CRPS roughly $G_\theta:\mathcal X(d_X)\to\mathcal Y(d_y)$5–$G_\theta:\mathcal X(d_X)\to\mathcal Y(d_y)$6 and NRMSE roughly $G_\theta:\mathcal X(d_X)\to\mathcal Y(d_y)$7–$G_\theta:\mathcal X(d_X)\to\mathcal Y(d_y)$8, whereas TSFlow can show similar CRPS but much worse NRMSE; one example at $G_\theta:\mathcal X(d_X)\to\mathcal Y(d_y)$9 gives TSFlow CRPS $\mu$0, NRMSE $\mu$1, and G-SLiCE CRPS $\mu$2, NRMSE $\mu$3 (Berndt et al., 27 May 2026).

The ablation on transition structure ties these gains to expressivity. Across configurations, the dense variant wins in $\mu$4 cases, the best configuration uses a dense block on $\mu$5 datasets, and the average relative improvement over TSFlow is $\mu$6 when only diagonal blocks are selected versus $\mu$7 when dense blocks are selected. The paper’s explicit conclusion is that extra expressivity is not always needed, but when selected it tends to bring larger improvements (Berndt et al., 27 May 2026).

5. Relation to other “slice” and “sliced” models

The term “G-SLiCEs” can be misunderstood because several unrelated research programs also use “slice,” “sliced,” or “SLICES.” The overlap is terminological rather than taxonomic.

Term	Domain	Core idea
G-SLiCEs	Time-series generation	Structured linear CDEs with path-space flow matching
SliceGAN	Materials microstructure	3D generation from 2D slices via a 3D generator and 2D discriminators
SINF / SIG / GIS	Density estimation and generation	Iterative 1D optimal transport on informative slices
SLICES-PLUS	Crystal generation	Symmetry-aware string representation for autoregressive crystal generation
C-SliceGen	Abdominal CT	Conditional slice harmonization toward a target vertebral level

SliceGAN is an unconditional, slice-statistics-constrained 3D generation framework in which a 3D generator is judged through 2D cross-sections; it is directly about dimensionality expansion from 2D supervision, not about continuous-time path laws (Kench et al., 2021). Its AdaIN-based extension adds controllable grain-size steering to 3D microstructure synthesis from 2D-derived training data, but still without a causal path-space formulation (Chung et al., 2021). Sliced Iterative Normalizing Flows develop a greedy invertible transport model by matching one-dimensional marginals along informative directions; that work is close in spirit on the “reduce to 1D slices” axis, but its object is high-dimensional probability transport rather than causal continuous-time sequence generation (Dai et al., 2020). “Sliced generative models” use one-dimensional projected discrepancies to regularize latent distributions in autoencoder-based generation, again through projections rather than controlled differential equations (Knop et al., 2019). In a different application area, “C-SliceGen” maps an arbitrary abdominal CT slice to a predefined vertebral level in order to reduce positional variance, which makes it a conditional slice standardization method rather than a path-space generative process (Yu et al., 2022). “SLICES-PLUS” belongs to crystalline materials and denotes a symmetry-aware string representation for autoregressive crystal generation, not a controlled differential equation framework (Wang et al., 2024).

A plausible implication is that G-SLiCEs belong to a broader family of models that exploit low-dimensional or structured views of complex objects, but their distinctive contribution lies in causal path-space generation, continuous-time handling of arbitrary grids, and the specific universality result in $\mu$8 (Berndt et al., 27 May 2026).

6. Limitations, boundary conditions, and interpretation

The paper states several limitations explicitly. First, the current implementation uses a first-order approximation of the exponential in the exact-flow recurrence rather than efficient exact structured kernels. Second, the universality theory applies to the direct pushforward model, whereas the practical ODE-flow implementation adds invertibility and flow constraints that are not fully characterized theoretically. Third, practical training still depends on discretization and interpolation. Fourth, non-compact Gaussian process priors used in practice admit only a high-probability approximation interpretation on compact sets carrying arbitrarily large mass rather than a global compact-support $\mu$9 statement. The practical interpretation section also identifies possible failure modes, including underfitting when too restrictive a transition structure is chosen, optimization difficulties in path-space flow learning, and mismatch between ODE flow constraints and the target distributional map (Berndt et al., 27 May 2026).

These limitations clarify a common misconception. The universality theorem does not mean that every practical parameterization or every efficient structured transition is equally expressive; the paper repeatedly distinguishes maximally expressive SLiCE classes from restricted alternatives. Nor does continuous time remove numerical approximation issues; rather, it changes the modeling geometry by allowing direct evaluation on the requested physical-time grid and by making irregular-grid generalization natural. In that sense, G-SLiCEs are best understood not as a generic “slice-based” generator, but as a specific continuous-time generative formalism in which structured linear controlled differential equations furnish both the expressive backbone and the theoretical route from pathwise approximation to generative approximation (Berndt et al., 27 May 2026).