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FlowMixer: A Depth-Agnostic Neural Architecture for Interpretable Spatiotemporal Forecasting
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Introduces FlowMixer, a single-layer, shape-preserving mixing architecture with non-negative time/feature mixing matrices that yields Kronecker-Koopman eigenmodes and competitive long-horizon forecasting, chaos prediction (with SOBR), and 2D turbulent flow prediction.
Problem
Existing forecasting architectures require extensive depth/architecture search, and their Koopman-based eigenmodes are embedded within networks and hard to interpret; a unified model across time-series forecasting, chaotic dynamics and turbulent flows is lacking.
Approach
A single reversible-mapping mixing layer F(X)=phi^-1(W_t phi(X) W_f^T) with non-negative time mixing (alpha I + W∘W) and stochastic static-attention feature mixing; its square matrices give a semi-group property (depth eliminated) and a Kronecker product eigen-decomposition yielding directly extractable space-time eigenmodes and algebraic horizon changes.
Key results
- Across seven benchmark datasets, FlowMixer achieves best MSE on most ETT horizons and Weather (e.g., ETTh1 h=96 MSE 0.355 vs Chimera 0.366), with the paper reporting Wilcoxon significance (p<0.05) against baselines on MSE.
- On chaotic attractors, FlowMixer with SOBR forecasts 1024 steps (about 9-10 Lyapunov times for Lorenz) by compounding 32-step predictions, visually matching ground truth while RC and N-BEATS show specific failure modes.
- On 2D flow past a cylinder and NACA airfoil, FlowMixer reports MSE around 3.3e-3 (cylinder) and 2e-3 to 8e-3 (airfoil), roughly 4-5% relative l2 error.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| remove RevIN (ablation) | FlowMixer · ETTh1 (720) | MSE: 0.620 -> 0.465 | Helped |
| remove feature mixing (ablation) | FlowMixer · ETTh1 (720) | MSE: 0.476 -> 0.465 | Helped |
| remove time mixing (ablation) | FlowMixer · ETTh1 (720) | MSE: 0.634 -> 0.465 | Helped |
| remove the positive time-mixing constraint (ablation) | FlowMixer · ETTh1 (720) | MSE: 0.482 -> 0.465 | Helped |
| use static attention instead of generic feature mixing (ablation) | FlowMixer · ETTh1 (720) | MSE: 0.519 -> 0.465 | Helped |
| remove adaptive skip connections (ablation) | FlowMixer · ETTh1 (720) | MSE: 0.497 -> 0.465 | Helped |
| use standard full feature mixing instead of channel-independent mixing (ablation) | FlowMixer · ETTh1 (720) | MSE: 0.491 -> 0.465 | Helped |
Surprising findings
- Channel-independent variant performs worse than the standard FlowMixer on ETTh1, and parameter count grows 489x on Traffic.
- The semi-group property is lost when SOBR lifting is used, so horizon algebra holds only in the lifted space.
Limitations
- Cubic computational complexity in sequence length.
- No explicit physical conservation guarantees.
- Semi-group/horizon algebra does not hold under SOBR lifting; per-horizon models are trained independently for the forecasting comparison.
Experimental setup
Datasets: ETTh1 ETTh2 ETTm1 ETTm2 Weather Electricity Traffic Lorenz Rössler Aizawa 2D cylinder flow (Re=150) NACA airfoil flow (Re=1000)
Benchmarks: ETTh1 ETTh2 ETTm1 ETTm2 Weather Electricity Traffic Lorenz Rössler Aizawa 2D cylinder flow NACA airfoil flow
Metrics: MSE MAE Wilcoxon signed-rank p-value relative l2 error
Baselines: TimeMixer++ Chimera iTransformer Ada-MSHyper CycleNet TSMixer N-HITS FEDformer Autoformer DLinear KooPA KNF
Builds on: TSMixer RevIN Dynamic Mode Decomposition reservoir computing N-BEATS
Closest papers by meaning
- Learning Spatiotemporal Dynamical Systems from Point Process Observations (ICLR 2025 · similarity 0.84)
- Space-Time Continuous PDE Forecasting using Equivariant Neural Fields (NeurIPS 2024 · similarity 0.84)
- TimeMixer: Decomposable Multiscale Mixing for Time Series Forecasting (ICLR 2024 · similarity 0.84)
- Graph-based Forecasting with Missing Data through Spatiotemporal Downsampling (ICML 2024 · similarity 0.84)
- Earthfarsser: Versatile Spatio-Temporal Dynamical Systems Modeling in One Model (AAAI 2024 · similarity 0.84)
- Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators (NeurIPS 2026 · similarity 0.84)
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