Research map / Time series / Causal and generative time series
Shifting Time: Time-series Forecasting with Khatri-Rao Neural Operators
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes learning a continuous time-shift operator for temporal and spatio-temporal forecasting, parametrized by Khatri-Rao neural operators (KRNOs) with near-linear scaling in quadrature nodes.
Problem
Time-series and spatio-temporal forecasting with autoregressive models that require regularly sampled observations; irregular sampling and super-resolution forecasting are hard for discrete-lag and existing neural operator methods.
Approach
Cast forecasting as learning a continuous time-shift operator mapping a past history window to a future window, and parametrize it with a neural operator whose kernel is a Khatri-Rao (element-wise product) decomposition of component kernels, enabling exact non-stationary kernel evaluation at about O(N^(1+1/d)) cost.
Key results
- KRNO achieves 0.00145 relative L2 error on shallow water density versus 0.00211 for FNO-3D, using 146,159 parameters versus 2,462,895 for FNO-3D (about 6%).
- On MuJoCo regular sampling, KRNO reaches test MSE 0.007 versus 0.012-0.028 for the best prior Neural SDE/NCDE baselines in Table 5.
- On irregular time-series, KRNO attains the best MSE on MIMIC (1.57e-2) and USHCN (4.95e-1), while T-PatchGNN is best on Human Activity MSE.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use KRNO instead of FNO-3D | Shallow-water forecasting · Shallow water | L2 relative error: 0.00211 -> 0.00145 (ρ); 0.02606 -> 0.01497 (u); 0.02637 -> 0.01459 (v) | Helped |
| Use KRNO instead of T-PatchGNN | Irregular time-series forecasting · USHCN | MSE×10−1: 5.00 ± 0.04 -> 4.95 ± 0.08 | Helped |
| Use KRNO instead of T-PatchGNN | Irregular time-series forecasting · MIMIC | MAE×10−2: 7.22 ± 0.09 -> 7.43 ± 0.06 | Hurt |
| Use KRNO instead of T-PatchGNN | Irregular time-series forecasting · Human Activity | MSE×10−3: 2.66 ± 0.03 -> 2.85 ± 0.03 | Hurt |
| Use KRNO instead of Neural LNSDE | MuJoCo forecasting · MuJoCo, regular | Test MSE: 0.013 ± 0.000 -> 0.007 ± 0.002 | Helped |
| Use KRNO instead of LEAP | MuJoCo forecasting · MuJoCo, 30% dropped | Test MSE: 0.022 ± 0.001 -> 0.008 ± 0.002 | Helped |
| Use KRNO instead of LEAP | MuJoCo forecasting · MuJoCo, 50% dropped | Test MSE: 0.022 ± 0.002 -> 0.0114 ± 0.004 | Helped |
| Use KRNO instead of LEAP | MuJoCo forecasting · MuJoCo, 70% dropped | Test MSE: 0.022 ± 0.001 -> 0.0115 ± 0.002 | Helped |
| Use KRNO instead of Nbeats-I+G | M4 forecasting · M4-Quarterly | sMAPE: 9.212 -> 10.503 | Hurt |
| Use KRNO instead of KNF | M4 forecasting · M4-Weekly | sMAPE: 7.254† -> 6.934 | Helped |
| Use KRNO instead of KNF | M4 forecasting · M4-Daily | sMAPE: 2.990† -> 3.086 | Hurt |
| Use KRNO instead of KNF | Multivariate forecasting · Crypto | Weighted RMSE 10−3: 6.91±0.01 -> 6.95±0.16 | Hurt |
Surprising findings
- On M4 Quarterly, KRNO's sMAPE (10.503) is worse than several baselines, while it is best on Weekly among listed methods (6.934).
Limitations
- Theoretical time-shift operator existence relies on Hilbert-Schmidt and Lipschitz assumptions that may not hold for general nonlinear dynamics
- Time window hyperparameters tp and tf must be tuned via cross-validation or hypergradient methods
- Results on M4 Quarterly and Crypto/Basketball are not uniformly best among baselines
Experimental setup
Datasets: Shallow water (Kissas et al. 2022) Climate modeling (Kissas et al. 2022) Darcy-flow (Lu et al. 2022) Hyper-elasticity (Li et al. 2023) MuJoCo Hopper MIMIC-III USHCN Human Activity Darts M4 Crypto NBA player trajectories
Benchmarks: Darcy-flow Hyper-elasticity Shallow water Climate modeling MuJoCo MIMIC-III USHCN Human Activity Darts M4 Crypto NBA Player Trajectory
Metrics: L2 relative error MSE MAE NMAE sMAPE weighted RMSE RMSE
Baselines: FNO-3D LOCA T-PatchGNN CRU Neural Flow Latent-ODE N-BEATS KNF LLMTIME Neural SDE
Builds on: Fourier Neural Operator DeepONet Neural ODE Gaussian process product kernels RevIN
Closest papers by meaning
- SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting (ICML 2025 · similarity 0.85)
- OLinear: A Linear Model for Time Series Forecasting in Orthogonally Transformed Domain (NeurIPS 2025 · similarity 0.85)
- Learning Pattern-Specific Experts for Time Series Forecasting Under Patch-level Distribution Shift (NeurIPS 2025 · similarity 0.85)
- Learning with Calibration: Exploring Test-Time Computing of Spatio-Temporal Forecasting (NeurIPS 2025 · similarity 0.85)
- TimeEmb: A Lightweight Static-Dynamic Disentanglement Framework for Time Series Forecasting (NeurIPS 2025 · similarity 0.85)
- PAMod: Modeling Cyclical Shifts via Phase-Amplitude Modulation for Non-stationary Time Series Forecasting (NeurIPS 2026 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.