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SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting

ICML 2025 · Time series › Causal and generative time series · approach: Dynamical system reconstruction · 0 citations (OpenAlex)

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Proposes SKOLR, a Koopman-theory time-series forecaster that equates a structured (block-diagonal) finite-dimensional Koopman operator on time-delay states with a parallel linear RNN stack, combined with learnable FFT frequency-branch decomposition and MLP encoder/decoder.

Problem

Finite-dimensional approximation of the infinite-dimensional Koopman operator for forecasting time series from partially observed nonlinear dynamics, with the aim of efficient and accurate forecasting.

Approach

Using a time-delay extended state, the EDMD-style Koopman matrix with enforced block structure is shown to equal a linear RNN state update; SKOLR splits the signal into learnable frequency-gated branches, each with its own linear RNN, and uses MLP measurement functions and decoders.

Key results

• SKOLR attains the best average MSE rank on long-term forecasting benchmarks, e.g. ECL T=48 MSE 0.137 and Weather T=192 MSE 0.193 (matching Koopa).
• On M4 average, SKOLR reports sMAPE 11.704, MASE 1.572, OWA 0.843, the best among compared models.
• On ETTm1 with a fixed dimension of 512, a (32, 16) branch configuration with 0.25M params matches the 1-branch model with 1.71M params (about 85% reduction).

Measured effects reported in the paper

Surprising findings

• Learnable frequency decomposition underperforms uniform masking on ETTh1 at horizon 48, which the authors attribute to overfitting on small data.
• Converting the LRU architecture to forecasting performs poorly relative to SKOLR, per the authors' comparison.

Limitations

• Gains on some datasets are small or absent (e.g., ETTh1 at T=48 and ETTm2 at T=144 where the learnable decomposition underperforms or ties baselines).
• Reported results are averages of 3 trials with hyperparameters chosen by grid search, leaving the stated results sensitive to tuning.
• Koopman connection is derived for a linear RNN with a time-delay state; the equivalence is for a structured approximation rather than the full Koopman operator.

Experimental setup

Datasets: Weather Traffic Electricity ILI ETTh1 ETTh2 ETTm1 ETTm2 M4

Benchmarks: Weather Traffic Electricity ILI ETTh1 ETTh2 ETTm1 ETTm2 M4 Pendulum Duffing Lotka-Volterra

Metrics: MSE MAE sMAPE MASE OWA

Baselines: Koopa iTransformer PatchTST TimesNet MICN DLinear KNF Autoformer KooPA N-HiTS N-BEATS LRU

Builds on: Extended Dynamic Mode Decomposition Koopa KNF multiresolution DMD RevIN

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