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Conditional Diffusion Model with Nonlinear Data Transformation for Time Series Forecasting

ICML 2025 · Time series › Causal and generative time series · approach: Diffusion models for time series

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Proposes CN-Diff, a conditional diffusion model for time series forecasting that adds a learnable nonlinear time-dependent data transformation and a learnable condition to the forward process, trained with a non-Markovian variational bound.

Problem

Standard conditional diffusion models for time series use a fixed linear Gaussian forward process and an isotropic Gaussian prior, which limits their ability to capture complex temporal patterns and widens the gap between the NLL and its variational bound.

Approach

Make the forward process learnable by applying a nonlinear time-dependent transformation T_phi to the data and a learnable condition c, so the prior becomes N(c, I) and the reverse process is non-Markovian; derive a new variational loss with prior, reconstruction, and diffusion terms.

Key results

• Multivariate average rank of 1.7 across nine datasets, with best MSE on 6 of 9 (e.g., NorPool 0.531 vs mr-Diff 0.645; ETTh1 0.405).
• Univariate average rank 2.0, best on four of nine datasets (e.g., ETTh1 0.066, ETTm1 0.038).
• Ablation: nonlinear transformation in both dimensions improves Electricity MSE from 0.536 to 0.145 and Caiso from 0.242 to 0.094 relative to no transformation.

Measured effects reported in the paper

Surprising findings

• Multivariate CN-Diff ranks 4th on Wind, where mr-Diff is best, which the authors attribute to multi-scale patterns in volatile wind data.

Limitations

• Wind dataset performance is weaker than mr-Diff, and multi-resolution extension is left for future work.
• Baseline results are copied from mr-Diff paper (Shen et al., 2024) rather than re-run under identical settings.
• Ablations are reported on a subset of datasets (ETTh1, Electricity, Traffic, Caiso, Wind).

Experimental setup

Datasets: NorPool Caiso Traffic Electricity Weather Exchange ETTh1 ETTm1 Wind

Benchmarks: NorPool Caiso Traffic Electricity Weather Exchange ETTh1 ETTm1 Wind

Metrics: MSE MAE

Baselines: mr-Diff TimeDiff TimeGrad CSDI SSSD D3VAE CPF PSA-GAN N-Hits FiLM DLinear NLinear

Builds on: DDPM CSDI TimeDiff NDM PriorGrad

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