StateFlow: Dual-State Recurrent Modeling for Long-Horizon Time Series Forecasting

Long-horizon multivariate time series forecasting (LTSF) remains challenging due to non-stationarity, regime shifts, and error accumulation. The Variability-Aware Recursive Neural Network (VARNN) is designed to track such variability by maintaining a residual-memory state driven by one-step prediction errors. However, its original formulation is limited to one-step sequence regression and does not directly support multi-step forecasting. In this work, we extend VARNN to long-horizon forecasting and introduce StateFlow, a recurrent forecasting framework that uses VARNN as a dual-state recurrent backbone to capture two complementary signals from the lookback sequence: a hidden-state trajectory representing primary temporal dynamics, including trend, seasonality, level changes, and recurring patterns, and a residual-memory trajectory representing structured local prediction deviations, driven from a nonlinear recurrent transformation of errors between one-step base predictions and observed values. A chunk-based decoder separately summarizes these trajectories and maps them to the future horizon for direct multi-step forecasting. We further employ a two-stage optimization strategy that first trains the VARNN encoder through a one-step base prediction objective to optimize the internal representations over the lookback sequence, and then trains a horizon-specific decoder for direct multi-step forecasting. Experiments on standard LTSF benchmarks show that StateFlow achieves competitive performance against strong linear, recurrent, convolutional, and Transformer-based baselines while preserving linear recurrent encoding and a compact model design.

Diffusion-corrected Autoregressive Fourier Neural Operator for Droplet Evolution Prediction

Predicting droplet evolution in material jetting, or Inkjet Printing (IJP), is essential for maintaining printing quality. However, long-horizon forecasts remain challenging due to error accumulation and the complex coupling of process variables. In this work, we introduce the Diffusion-corrected Auto-Regressive Fourier Neural Operator (DiffARFNO), a two-stage framework that combines an autoregressive Fourier-MIONet with a conditional Denoising Diffusion Implicit Model (DDIM) corrector. Fourier-MIONet is trained as a coarse predictor and deployed autoregressively for long-horizon forecasting. In the second stage, a DDIM-based conditional corrector refines the coarse prediction within each sliding window through efficient iterative denoising. By combining coarse predictions from Fourier-MIONet with a DDIM corrector that restores fine details, DiffARFNO aims to provide high-fidelity predictions for long-horizon forecasts. Extensive experiments on droplet datasets from ANSYS Fluent demonstrate that DiffARFNO significantly outperforms existing state-of-the-art models.