# A Deep State Space Model for Rainfall-Runoff Simulations

arXiv: https://arxiv.org/abs/2501.14980 · HTML: https://mlsanity.com/paper/2501.14980/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 69 references that are in the map.
- First posted on arXiv 24 Jan 2025.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Language Modeling with Gated Convolutional Networks](https://mlsanity.com/paper/1612.08083/) (2016)
- [Combining Recurrent, Convolutional, and Continuous-time Models with Linear State-Space Layers](https://mlsanity.com/paper/2110.13985/) (2021)
- [On the Parameterization and Initialization of Diagonal State Space Models](https://mlsanity.com/paper/2206.11893/) (2022)
- [Liquid Structural State-Space Models](https://mlsanity.com/paper/2209.12951/) (2022)
- [The Convergence Rate of Neural Networks for Learned Functions of Different Frequencies](https://mlsanity.com/paper/1906.00425/) (2019)
- [Lipschitz Recurrent Neural Networks](https://mlsanity.com/paper/2006.12070/) (2020)
- [Towards Learning Universal, Regional, and Local Hydrological Behaviors via Machine-Learning Applied to Large-Sample Datasets](https://mlsanity.com/paper/1907.08456/) (2019)
- [Mamba-360: Survey of State Space Models as Transformer Alternative for Long Sequence Modelling: Methods, Applications, and Challenges](https://mlsanity.com/paper/2404.16112/) (2024)
- [UnICORNN: A recurrent model for learning very long time dependencies](https://mlsanity.com/paper/2103.05487/) (2021)
- [Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs](https://mlsanity.com/paper/2310.02619/) (2023)
- [Long Expressive Memory for Sequence Modeling](https://mlsanity.com/paper/2110.04744/) (2021)
- [Deep Latent State Space Models for Time-Series Generation](https://mlsanity.com/paper/2212.12749/) (2022)
- [StableSSM: Alleviating the Curse of Memory in State-space Models through Stable Reparameterization](https://mlsanity.com/paper/2311.14495/) (2023)
- [From calibration to parameter learning: Harnessing the scaling effects of big data in geoscientific modeling](https://mlsanity.com/paper/2007.15751/) (2020)
- [Differentiable, learnable, regionalized process-based models with physical outputs can approach state-of-the-art hydrologic prediction accuracy](https://mlsanity.com/paper/2203.14827/) (2022)
- [Spectral State Space Models](https://mlsanity.com/paper/2312.06837/) (2023)
- [Robustifying State-space Models for Long Sequences via Approximate Diagonalization](https://mlsanity.com/paper/2310.01698/) (2023)
- [State-Free Inference of State-Space Models: The Transfer Function Approach](https://mlsanity.com/paper/2405.06147/) (2024)
- [MC-LSTM: Mass-Conserving LSTM](https://mlsanity.com/paper/2101.05186/) (2021)
- [Tuning Frequency Bias of State Space Models](https://mlsanity.com/paper/2410.02035/) (2024)
- [HOPE for a Robust Parameterization of Long-memory State Space Models](https://mlsanity.com/paper/2405.13975/) (2024)
- [Elucidating the Design Choice of Probability Paths in Flow Matching for Forecasting](https://mlsanity.com/paper/2410.03229/) (2024)
- [Gated Recurrent Neural Networks with Weighted Time-Delay Feedback](https://mlsanity.com/paper/2212.00228/) (2022)

## Built on by (most cited first, 3 of 3)

- [Block-Biased Mamba for Long-Range Sequence Processing](https://mlsanity.com/paper/2505.09022/) (2025)
- [HydroDiffusion: Diffusion-Based Probabilistic Streamflow Forecasting with a State Space Backbone](https://mlsanity.com/paper/2512.12183/) (2025)
- [Continuity Laws for Sequential Models](https://mlsanity.com/paper/2605.08539/) (2026)

## Abstract (arXiv)

The classical way of studying the rainfall-runoff processes in the water cycle relies on conceptual or physically-based hydrologic models. Deep learning (DL) has recently emerged as an alternative and blossomed in hydrology community for rainfall-runoff simulations. However, the decades-old Long Short-Term Memory (LSTM) network remains the benchmark for this task, outperforming newer architectures like Transformers. In this work, we propose a State Space Model (SSM), specifically the Frequency Tuned Diagonal State Space Sequence (S4D-FT) model, for rainfall-runoff simulations. The proposed S4D-FT is benchmarked against the established LSTM and a physically-based Sacramento Soil Moisture Accounting model across 531 watersheds in the contiguous United States (CONUS). Results show that S4D-FT is able to outperform the LSTM model across diverse regions. Our pioneering introduction of the S4D-FT for rainfall-runoff simulations challenges the dominance of LSTM in the hydrology community and expands the arsenal of DL tools available for hydrological modeling.