LAtent Phase Inference from Short time sequences using SHallow REcurrent Decoders (LAPIS-SHRED)

Yuxuan Bao; Xingyue Zhang; J. Nathan Kutz

cs.LGApr 1, 2026

LAtent Phase Inference from Short time sequences using SHallow REcurrent Decoders (LAPIS-SHRED)

Authors: Yuxuan Bao, Xingyue Zhang, J. Nathan Kutz

Organizations: Department of Applied Mathematics, University of Washington. · School of Environmental and Forest Sciences, University of Washington. · Autodesk Research, London, UK.

Abstract

Reconstructing full spatio-temporal dynamics from sparse observations in both space and time remains a central challenge in complex systems, as measurements can be spatially incomplete and can be also limited to narrow temporal windows. Yet approximating the complete spatio-temporal trajectory is essential for mechanistic insight and understanding, model calibration, and operational decision-making. We introduce LAPIS-SHRED (LAtent Phase Inference from Short time sequence using SHallow REcurrent Decoders), a modular architecture that reconstructs and/or forecasts complete spatiotemporal dynamics from sparse sensor observations confined to short temporal windows. LAPIS-SHRED operates through a three-stage pipeline: (i) a SHRED model is pre-trained entirely on simulation data to map sensor time-histories into a structured latent space, (ii) a temporal sequence model, trained on simulation-derived latent trajectories, learns to propagate latent states forward or backward in time to span unobserved temporal regions from short observational time windows, and (iii) at deployment, only a short observation window of hyper-sparse sensor measurements from the true system is provided, from which the frozen SHRED model and the temporal model jointly reconstruct or forecast the complete spatiotemporal trajectory. The framework supports bidirectional inference, inherits data assimilation and multiscale reconstruction capabilities from its modular structure, and accommodates extreme observational constraints including single-frame terminal inputs. We evaluate LAPIS-SHRED on six experiments spanning complex spatio-temporal physics: turbulent flows, multiscale propulsion physics, volatile combustion transients, and satellite-derived environmental fields, highlighting a lightweight, modular architecture suited for operational settings where observation is constrained by physical or logistical limitations.

Figures & tables

Fig. 1 : Overview of the LAPIS-SHRED architecture. Stage i (SHRED): Sparse sensor observations $\mathbf{S}=[\mathbf{s}_{0},\ldots,\mathbf{s}_{T}]\in\mathbb{R}^{p\times(T+1)}$ from simulation ensembles are processed via a pre-trained SHRED model into latent representations $\mathbf{z}_{0},\ldots,\mathbf{z}_{T}$ , which are decoded to reconstruct the full spatiotemporal trajectory. A short window of latent codes—from either the initial or terminal phase—is extracted for temporal model training. Stage ii (Temporal Model): Trained on simulation-derived latent trajectories, a BiLSTM architecture learns to map short-window latent codes (augmented with positional encodings) to the unobserved portion of the latent trajectory. Stage iii (LAPIS Inference): At test time, only the short observation window from the ground truth system (reality) is available. The frozen SHRED model extracts the observed-window latent codes, the temporal model infers the unobserved latent trajectory, and the frozen decoder reconstructs the complete spatiotemporal evolution. The architecture shown depicts backward inference from a terminal time window; forward inference from an initial time window proceeds analogously.

Experiment	Mode	Temporal	Dir.	RMSE	SSIM	$\bm{\Delta}$	NRMSE
2D KS	Frame	Seq2Seq	Bwd	0.727	0.524	15.94	0.046
2D KF	Frame	Seq2Seq	Bwd	$\|\mathbf{u}\|$ : 0.179	$\|\mathbf{u}\|$ : 0.887	$\|\mathbf{u}\|$ : 4.10	$\|\mathbf{u}\|$ : 0.044
2D KF	Frame	Seq2Seq	Bwd	$\omega$ : 5.616	$\omega$ : 0.786	$\omega$ : 155.62	$\omega$ : 0.036
2D KVS	Seq2Seq	Seq2Seq	Fwd	9.7e-4	0.564	0.03	0.037
2D KVS	Seq2Seq	Seq2Seq	Bwd	8.7e-4	0.638	0.03	0.033
HF-RDE	Frame	AR	Fwd	0.108	–	0.943	0.114

TABLE I : Summary of LAPIS-SHRED experiments and quantitative results. “Temporal” indicates the latent dynamics model; “Dir.” the inference direction; “Mode” the SHRED operational mode. RMSE and NRMSE are reported for the LAPIS-SHRED reconstruction over the full trajectory. SSIM is reported where computed (“–” otherwise).

Fig. 2 : LAPIS-SHRED backward reconstruction on the 2D Kuramoto–Sivashinsky system. Ground truth (top row), LAPIS-SHRED reconstruction from the terminal 10% of the trajectory (second row), and SHRED baseline using the full sensor time-series (third row), shown at selected time steps.

Fig. 3 : LAPIS-SHRED backward reconstruction on 2D Kolmogorov flow: vorticity field. Ground truth (top row), LAPIS-SHRED reconstruction from the terminal 10% of the trajectory (second row), and SHRED baseline using the full sensor time-series (third row), shown at selected time steps.

Fig. 4 : LAPIS-SHRED forward prediction on the 2D von Karman vortex street. Ground truth (top row), LAPIS-SHRED prediction from the initial $10\%$ of the trajectory (second row), and SHRED baseline using the full sensor time-series (third row), shown at selected time steps.

Fig. 5 : Forward-LAPIS-SHRED inference with multi-scale architecture on the 1D rotating detonation engine (RDE) dataset. Top row: spatiotemporal field of the high-fidelity data (left), multi-scale reconstruction (center), and the Forward-LAPIS-SHRED inference (right), which concatenates the training window, a held-out test region, and a 200-step autoregressive forward rollout into the unobserved regime. The purple dashed line marks the training data boundary and the green dashed line marks the observation boundary. Bottom row: spatial snapshots at selected time steps comparing the ground truth, Forward-LAPIS-SHRED inference, and the low-fidelity-only baseline.

Fig. 6 : LAPIS-SHRED backward reconstruction on the 1D RDE ignition stage. Ground truth (top row), LAPIS-SHRED reconstruction from the terminal 20% of the trajectory (second row), and SHRED baseline using the full sensor history (third row), shown at selected time steps. Pressure field is shown in red; temperature in blue.

Fig. 7 : LAPIS-SHRED forward inference on the 2D NDSI dataset. Ground truth for the 2025 snow season (top row), LAPIS-SHRED prediction from the initial 5 frames (second row), and SHRED baseline using the full sensor time-series (third row), shown at selected time steps.

Fig. 8 : LAPIS-SHRED backward inference on the 2D NDSI dataset from a single terminal frame using static padding. Ground truth (top row), LAPIS-SHRED reconstruction (second row), and SHRED baseline using the full sensor time-series (third row), shown at selected time steps.

Panel of Figure 10 — Fig. 9 : Ablation study on the NDSI dataset. Panels (a)–(c) report both forward and backward NRMSE and SSIM; panels (d) and (e) are direction-specific. Solid lines denote the mean over three random seeds; shaded bands indicate $\pm 1$ standard deviation.

Fig. 10 : LAPIS-SHRED (short observation window) versus SHRED with full sensor access across all experiments. (a) NRMSE; lower is better. (b) SSIM (where computed); higher is better. LAPIS-SHRED closely approaches the full-observation baseline despite observing only 7–20% of the temporal domain.

Expt.	Method	RMSE	NRMSE	SSIM
HF- RDE	SHRED-ROM (AR)	0.110	0.117	—
HF- RDE	LAPIS-SHRED	0.108	0.114	—
NDSI Fwd	SHRED-ROM (AR)	0.322	0.214	0.325
	SHRED-ROM (int.)	0.304	0.191	0.391
	LAPIS-SHRED	0.266	0.167	0.405

TABLE II : SHRED-ROM baseline on HF-RDE and NDSI forward experiments. AR: autoregressive forecast; int.: ensemble-interpolated sensors. All methods use identical sensors and observation windows.

Fig. 11 : SHRED-ROM baseline on the HF-RDE dataset. Top row: ground truth (left), SHRED-ROM reconstruction (center), and autoregressive forward rollout (right). Bottom row: spatial snapshots. Compare with Figure 5 .

Fig. 12 : SHRED-ROM baseline on the NDSI forward experiment. Ground truth (top), autoregressive forecast from 5 observed frames (middle), and SHRED-ROM with full sensor time-series access (bottom). Compare with Figure 7 .

Fig. 13 : SHRED-ROM baseline (ensemble-interpolated) on the NDSI forward experiment. Ground truth (top), reconstruction from ensemble-interpolated sensor series (middle), and SHRED-ROM with full sensor time-series access (bottom). Compare with Figure 7 .

Method	Sparse spatial sensing	Temporal seq. encoding	Forward inference	Backward inference
LAPIS-SHRED	✓	✓	✓	✓
SHRED-ROM †	✓	✓	✓	✗
PDS	✓	✗	✗	✗
POD-AE-SE	✓	✗	✗	✗
POD-DeepONet ‡	✓	✗	✗	✗

TABLE III : Capability comparison of LAPIS-SHRED and related methods. ✓ = supported; ✗ = not supported.

Fig. 14 : Distribution of relative reconstruction errors on the fluid-around-an-obstacle benchmark (reproduced from Tomasetto et al. [ 22 ] , Figure 7b). SHRED-ROM outperforms all competing methods on the reconstruction task alone; the temporal inference setting addressed by LAPIS-SHRED introduces additional challenges that none of these methods are designed to handle.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Fig. 15 : LAPIS-SHRED backward reconstruction on 2D Kolmogorov flow: velocity field. Ground truth (top row), LAPIS-SHRED reconstruction from the terminal 10% of the trajectory (second row), and SHRED baseline using the full sensor time-series (third row), shown at selected time steps.

Fig. 16 : LAPIS-SHRED backward reconstruction on the 2D von Karman vortex street. Ground truth (top row), LAPIS-SHRED reconstruction from the terminal $10\%$ of the trajectory (second row), and SHRED baseline using the full sensor time-series (third row), shown at selected time steps.

Fig. 17 : NDSI data preview: spatiotemporal structure of the 2020-2025 snow melt seasons after SCAF trimming.

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Distributional Regression Uncertainty Quantification Uncertainty Calibration

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Reconstructing high-dimensional spatiotemporal fields from sparse sensor measurements is critical in a wide range of scientific applications. The SHallow REcurrent Decoder (SHRED) architecture is a recent state-of-the-art architecture that reconstructs high-quality spatial domain from hyper-sparse sensor measurement streams. An important limitation of SHRED is that in complex, data-scarce, high-frequency, or stochastic systems, portions of the spatiotemporal field must be modeled with valid uncertainty estimation. We introduce UQ-SHRED, a distributional learning framework for sparse sensing problems that provides uncertainty quantification through a neural network-based distributional regression called engression. UQ-SHRED models the uncertainty by learning the predictive distribution of the spatial state conditioned on the sensor history. By injecting stochastic noise into sensor inputs and training with an energy score loss, UQ-SHRED produces predictive distributions with minimal computational overhead, requiring only noise injection at the input and resampling through a single architecture without retraining or additional network structures. On complicated synthetic and real-life datasets including turbulent flow, atmospheric dynamics, neuroscience and astrophysics, UQ-SHRED provides a distributional approximation with well-calibrated confidence intervals. We further conduct ablation studies to understand how each model setting affects the quality of the UQ-SHRED performance, and its validity on uncertainty quantification over a set of different experimental setups.

Abstract

Spatiotemporal Forecasting

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