# FlowMixer: A Depth-Agnostic Neural Architecture for Interpretable Spatiotemporal Forecasting

arXiv: https://arxiv.org/abs/2505.16786 · HTML: https://mlsanity.com/paper/2505.16786/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 69 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 30 of 33)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [A Time Series is Worth 64 Words: Long-term Forecasting with Transformers](https://mlsanity.com/paper/2211.14730/) (2022)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [Are Transformers Effective for Time Series Forecasting?](https://mlsanity.com/paper/2205.13504/) (2022)
- [FEDformer: Frequency Enhanced Decomposed Transformer for Long-term Series Forecasting](https://mlsanity.com/paper/2201.12740/) (2022)
- [Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks](https://mlsanity.com/paper/1703.07015/) (2017)
- [N-BEATS: Neural basis expansion analysis for interpretable time series forecasting](https://mlsanity.com/paper/1905.10437/) (2019)
- [Time-LLM: Time Series Forecasting by Reprogramming Large Language Models](https://mlsanity.com/paper/2310.01728/) (2023)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [TSMixer: An All-MLP Architecture for Time Series Forecasting](https://mlsanity.com/paper/2303.06053/) (2023)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Next Generation Reservoir Computing](https://mlsanity.com/paper/2106.07688/) (2021)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [TimeMixer++: A General Time Series Pattern Machine for Universal Predictive Analysis](https://mlsanity.com/paper/2410.16032/) (2024)
- [Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors](https://mlsanity.com/paper/2305.18803/) (2023)
- [CycleNet: Enhancing Time Series Forecasting through Modeling Periodic Patterns](https://mlsanity.com/paper/2409.18479/) (2024)
- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [Hybrid Forecasting of Chaotic Processes: Using Machine Learning in Conjunction with a Knowledge-Based Model](https://mlsanity.com/paper/1803.04779/) (2018)
- [Reservoir computing approaches for representation and classification of multivariate time series](https://mlsanity.com/paper/1803.07870/) (2018)
- [Zero-shot forecasting of chaotic systems](https://mlsanity.com/paper/2409.15771/) (2024)
- [Model scale versus domain knowledge in statistical forecasting of chaotic systems](https://mlsanity.com/paper/2303.08011/) (2023)
- [WPMixer: Efficient Multi-Resolution Mixing for Long-Term Time Series Forecasting](https://mlsanity.com/paper/2412.17176/) (2024)
- [Ada-MSHyper: Adaptive Multi-Scale Hypergraph Transformer for Time Series Forecasting](https://mlsanity.com/paper/2410.23992/) (2024)

## Built on by (most cited first, 1 of 1)

- [Fractional is Better: Learnable Derivative Orders in Neural Operator Learning](https://mlsanity.com/paper/2610.20363/) (2026)

## Abstract (arXiv)

We introduce FlowMixer, a single-layer neural architecture that leverages constrained matrix operations to model structured spatiotemporal patterns with enhanced interpretability. FlowMixer incorporates non-negative matrix mixing layers within a reversible mapping framework - applying transforms before mixing and their inverses afterward. This shape-preserving design enables a Kronecker-Koopman eigenmodes framework that bridges statistical learning with dynamical systems theory, providing interpretable spatiotemporal patterns and facilitating direct algebraic manipulation of prediction horizons without retraining. The architecture's semi-group property enables this single layer to mathematically represent any depth through composition, eliminating depth search entirely. Extensive experiments across diverse domains demonstrate FlowMixer's long-horizon forecasting capabilities while effectively modeling physical phenomena such as chaotic attractors and turbulent flows. Our results achieve performance matching state-of-the-art methods while offering superior interpretability through directly extractable eigenmodes. This work suggests that architectural constraints can simultaneously maintain competitive performance and enhance mathematical interpretability in neural forecasting systems.