Pith. sign in

REVIEW 3 major objections 5 minor 56 references

A Deep State Space Model for Rainfall-Runoff Simulations

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that a state space model, S4D-FT, outperforms the LSTM benchmark for rainfall-runoff simulation across 531 US watersheds, challenging the long-standing LSTM dominance in hydrology.

desk verdict A useful first SSM benchmark for rainfall-runoff, but the manual tuning protocol and missing significance tests leave the 'outperforms LSTM' claim one step short of fully supported. read the letter →

arxiv 2501.14980 v1 pith:7P67JC3I submitted 2025-01-24 cs.LG cs.AIphysics.ao-ph

Yihan Wang , Lujun Zhang , Annan Yu , N. Benjamin Erichson , Tiantian Yang This is my paper · ORCID
classification cs.LGcs.AIphysics.ao-ph
keywords statespacemodelsS4D-FTrainfall-runoffsimulationLSTMbenchmarkCAMELShydrologicdeeplearningfrequencytuningstreamflowprediction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that a state space model architecture, the Frequency Tuned Diagonal State Space Sequence model (S4D-FT), can outperform the long-standing LSTM benchmark for rainfall-runoff simulation across the contiguous United States. If correct, it breaks the pattern in which LSTM has remained the best deep-learning model for this task despite newer architectures such as Transformers, and it gives hydrologists a new benchmark and a faster, parallelizable modeling alternative. The evidence is a CONUS-wide comparison on 531 CAMELS watersheds under the community's standard training and testing setup, where S4D-FT achieves higher median NSE (0.74 vs 0.72) and KGE (0.75 vs 0.74) than LSTM. The model wins in most regions but lags in high-flow, high-volume watersheds such as the Mid-Atlantic and Pacific Northwest.

What carries the argument

The load-bearing object is S4D-FT, a diagonal state space sequence model whose imaginary part of the state matrix A is rescaled at initialization by a frequency-tuning hyperparameter (set to cfi=cfr=10), correcting the spectral bias of the base S4D model. The model operates as a deep stack of continuous-time linear time-invariant systems, discretized with a trainable time step, trained in a sequence-to-one mode with a 365-day context. The comparison is carried inside the community's standard setup: 531 unimpaired CONUS watersheds, NLDAS meteorology plus 27 static attributes, training in 1999–2008 and testing in 1989–1999.

What would settle it

Retrain S4D-FT and LSTM under identical conditions with a fixed validation period (for example, the last year of the training interval) used for early stopping and hyperparameter selection, then compare NSE and KGE on the held-out test period; if S4D-FT's advantage disappears or reverses, the published comparison is not evidence of an architectural benefit.

Watch

Extended reading notes

Core claim

Using a standardized setup of 531 CAMELS watersheds, 32 input variables, an eight-member ensemble, and a 365-day look-back window, S4D-FT reaches the best median NSE, KGE, Pearson-r, and FHV among all compared models, including LSTM, MC-LSTM, Transformers, basic S4D, and Sac-SMA. Frequency tuning is what makes the difference: basic S4D lands near Transformer level and below LSTM, while S4D-FT surpasses LSTM. The regional analysis shows the advantage concentrates in snowmelt-driven and intermittent-streamflow watersheds, while LSTM retains an edge where daily mean and peak flows are large, which the authors trace to S4D-FT's weaker improvement on high-flow volume bias (FHV).

Load-bearing premise

The load-bearing premise is that S4D-FT's hand-tuned hyperparameters, selected by trial and error with no reported validation split, were not chosen using test-period performance; if that premise fails, the reported advantage over LSTM could be a tuning artifact rather than a property of the architecture.

Editorial extensions

If this is right

  • S4D-FT becomes the new reference deep-learning model for CONUS-wide rainfall-runoff simulation, with median NSE 0.74 and KGE 0.75.
  • Hydrologists can expect SSM-based models to be competitive with LSTM on long-range hydrologic sequences while training and inferring in parallel.
  • In watersheds with frequent, prolonged high- and low-flow events and smaller flow magnitudes—snowmelt and intermittent regimes—S4D-FT is the safer choice.
  • In pluvio-nival watersheds with large flow volumes, LSTM remains competitive, so high-flow simulation is the remaining gap for SSMs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the trial-and-error hyperparameter selection should be checked with a proper validation split before treating the benchmark as settled; this is a direct test of the paper's conclusion.
  • Editorial extension: if the architectural advantage holds, other linear-time state space families such as Mamba or S5 would be natural candidates for the same hydrologic benchmark.
  • Editorial extension: combining S4D-FT with mass-conserving structure, along the lines the paper discusses for MC-LSTM, could address its weaker high-flow volume bias and improve KGE.
  • Editorial extension: the reported regional skill patterns suggest a practical model-selection rule—use S4D-FT in snowmelt and intermittent basins and LSTM in high-volume pluvio-nival basins—though the paper stops short of recommending it.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper introduces Frequency Tuned Diagonal State Space Sequence (S4D-FT), a state-space model, for daily rainfall-runoff simulation and benchmarks it across 531 CAMELS watersheds in the contiguous United States. Using a standard training period (1999–2008) and test period (1989–1999), the authors compare S4D-FT against an LSTM baseline, a basic S4D, and several previously published models (MC-LSTM, Transformers, Sac-SMA) using six statistical metrics. They report that S4D-FT achieves better median NSE and KGE than LSTM, present spatial skill-score maps showing regional variation, and offer an attribution analysis linking S4D-FT's relative performance to hydrologic signatures such as flow magnitude and event frequency. The paper concludes that S4D-FT outperforms LSTM and 'sets a new community-based standard' for CONUS-wide rainfall-runoff simulations with deep learning.

Significance. If the central claim holds, this would be the first demonstration that a state-space model outperforms the LSTM benchmark on the standard large-sample CAMELS rainfall-runoff task, a result of genuine interest to both hydrology and the broader deep-learning time-series community. The paper uses the standard CAMELS benchmark, an eight-member ensemble, public data, and a public code repository, which are strengths. The attribution analysis, while correlational, is a useful step toward understanding where a new architecture helps or hurts. However, the headline gains are small and per-watershed results are mixed, and the manual hyperparameter tuning protocol is not documented with a validation split; these issues currently prevent the strong 'new standard' conclusion from being fully supported.

major comments (3)
  1. [Section 2, Tables S2–S3] The central claim that S4D-FT outperforms LSTM is not fully supported because the S4D and S4D-FT hyperparameters were 'manually tunned through a trial-and-error process' without any reported validation split or selection criterion. In particular, the S4D-FT-specific frequency-tuning scales cfi=10 and cfr=10 may have been chosen based on the 1989–1999 test period, which is the same period used for the benchmark comparison. If so, the small observed gains in Table 1 (median NSE 0.74 vs. 0.72; KGE 0.75 vs. 0.74) could reflect test-set selection bias rather than an architectural advantage of state-space models. The authors should report the validation period used for hyperparameter selection, state the selection criterion, and ideally re-evaluate S4D-FT with fixed hyperparameters selected on a held-out validation split before assessing test performance.
  2. [Table 1 and Table S6] The conclusion that S4D-FT 'outperforms the LSTM model across diverse regions' is stronger than the reported evidence. The median gains are small (NSE +0.02, KGE +0.01), and Table S6 shows that S4D-FT improves NSE in only 68.7% and KGE in only 54.4% of watersheds, while FHV, FLV, and PBias are improved in roughly half the watersheds or fewer. No significance tests or confidence intervals are provided for the median differences or for the fraction of improved watersheds. The authors should add paired statistical tests (e.g., Wilcoxon signed-rank or bootstrap) and should either temper the 'new community-based standard' claim in Section 5 or define explicitly that it refers to median performance rather than consistently improved simulation quality.
  3. [Section 4, Figure 2] The attribution analysis is plausible but correlational: the conclusion that S4D-FT excels in snowmelt-driven and intermittent watersheds and underperforms in high-flow pluvio-nival watersheds is based on percentage differences and pairwise correlations for two broad groups, with only two example hydrographs. These regime classifications are not actually tested against the eight hydrologic signatures in a multivariate way, and no uncertainty is reported for the correlation coefficients. Since this attribution is a central part of the paper's interpretation (though secondary to the headline benchmark claim), the authors should either add a multivariate or conditional analysis (e.g., regression of skill scores on signatures) or soften the causal-sounding language.
minor comments (5)
  1. [Equation (1)] The dimensions of matrix D are incorrect: for u(t) in C^m and y(t) in C^p, D should be in C^{p x m}, not C^{m x p}.
  2. [Abstract and Section 1] The manuscript contains typos, including 'Sacramental Soil Moisture Accounting' (should be 'Sacramento') and 'manually tunned' (should be 'manually tuned').
  3. [Section 2.1 and Table S3] The tuning parameters cfi and cfr are not defined in the main text; readers have to infer their meaning from the supplementary table.
  4. [References] The reference 'Naiman et al.' appears without a year, and the Hochreiter 1997 citation is incomplete; these should be corrected.
  5. [Throughout] The metric names FLV and FL V are used inconsistently, and 'Supplementary' versus 'Appendix' labels should be harmonized.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: S4D-FT is benchmarked empirically against an external LSTM baseline on the public CAMELS dataset, and the self-cited prior work that introduces the architecture is not used as evidence for the performance claim.

full rationale

The paper's central claim that S4D-FT outperforms LSTM rests on Table 1, which reports median NSE, KGE, Pearson-r, FHV, FLV, and PBias over 531 CAMELS watersheds for models trained on the standard 1999-2008 period and tested on 1989-1999. The LSTM baseline follows the published implementation of Kratzert et al. (2019a,b), and the authors state that their LSTM results align with benchmarks in Frame et al. (2022) and Liu et al. (2024). The S4D-FT architecture is adopted from Yu et al. (2025a), a prior paper co-authored by two of the present authors, but this citation only supplies the model definition and initialization scheme; it does not supply the hydrological performance numbers. The conclusion that S4D-FT is competitive or superior is derived from the authors' own experiments, not from the cited paper's conclusions. The manual trial-and-error tuning of cfi=10 and cfr=10 (Supplementary Table S3) is a hyperparameter-selection procedure; without evidence that the test period was used for selection, this is a potential bias concern, not a demonstrated circular reduction. The attribution analyses in Section 3.3 are post-hoc correlations between already-computed skill scores and watershed signatures; they do not define the skill scores in terms of the signatures, so they cannot be circular. No equation in the paper defines the predicted performance as equivalent to a fitted parameter or to a self-cited uniqueness theorem. The comparison is self-contained against external benchmarks, so no circularity is found.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central empirical claim rests on a hand-tuned model configuration, an assumed standard benchmark setup, and adopted baseline numbers from prior studies; no new physical entity is postulated.

free parameters (2)
  • Frequency tuning scale alpha (cfi, cfr) = 10 (both)
    Manual trial-and-error; separates S4D-FT from basic S4D; no validation protocol reported (Section 2, Table S3).
  • S4D/S4D-FT network hyperparameters (d_model, d_state, n_layer, dropout, learning rate) = 128, 128, 6, 0.12, 4e-4 (Table S3)
    Hand-selected; LSTM uses previously published hyperparameters, creating tuning asymmetry.
assumptions (4)
  • domain assumption The 531 CAMELS watersheds with NLDAS forcing and the 1989-1999 test period form a representative benchmark for CONUS rainfall-runoff simulation.
    Paper adopts the 'standard setup' from Frame et al. (2022) and Liu et al. (2024) without re-validating its representativeness.
  • domain assumption A 365-day look-back window and 32 input variables capture all relevant hydrologic memory and catchment controls.
    Used for all DL models (Section 2); no sensitivity analysis for window length is reported.
  • domain assumption The adopted metrics (NSE, KGE, FHV, FLV, PBias) and skill scores are sufficient to judge model superiority.
    No significance testing is reported; differences may be within noise for some metrics.
  • domain assumption Published results for Sac-SMA, MC-LSTM, and Transformers are directly comparable to this paper's experiments.
    Those values are adopted from Frame et al. (2022) and Liu et al. (2024); cross-study comparability is assumed.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Deep State Space Model for Rainfall-Runoff Simulations." pith.science (2026). https://pith.science/paper/7P67JC3I

@misc{pith2026250114980,
  author       = {Pith},
  title        = {Pith review of: A Deep State Space Model for Rainfall-Runoff Simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7P67JC3I}},
  note         = {Machine review of arXiv:2501.14980}
}
read the original abstract

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.

Figures

Figures reproduced from arXiv: 2501.14980 by the authors.

Figure 1
Figure 1. Simulation performance of S4D-FT relative to the LSTM model across study wa [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Analysis of S4D-FT’s performance relative to LSTM considering multiple evaluation [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 44 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

  2. [2]

    The camels data set: catchment attributes and meteorology for large-sample studies

    Nans Addor, Andrew J Newman, Naoki Mizukami, and Martyn P Clark. The camels data set: catchment attributes and meteorology for large-sample studies. Hydrology and Earth System Sciences, 21 0 (10): 0 5293--5313, 2017

    work page 2017

  3. [3]

    Spectral State Space Models

    Naman Agarwal, Daniel Suo, Xinyi Chen, and Elad Hazan. Spectral state space models. arXiv preprint arXiv:2312.06837, 2023

    work page Pith review arXiv 2023

  4. [4]

    Sacramento soil moisture accounting model (sac-sma)

    MG Anderson and JJ McDonnell. Sacramento soil moisture accounting model (sac-sma). Encyclopedia of Hydrological Sciences, 2005

    work page 2005

  5. [5]

    Basri, D

    R. Basri, D. Jacobs, Y. Kasten, and S. Kritchman. The convergence rate of neural networks for learned functions of different frequencies. Adv. Neur. Info. Proc. Syst., 32, 2019

    work page 2019

  6. [6]

    Changing ideas in hydrology—the case of physically-based models

    Keith Beven. Changing ideas in hydrology—the case of physically-based models. Journal of hydrology, 105 0 (1-2): 0 157--172, 1989

    work page 1989

  7. [7]

    A discussion of distributed hydrological modelling

    Keith J Beven. A discussion of distributed hydrological modelling. In Distributed hydrological modelling, pp.\ 255--278. Springer, 1996

    work page 1996

  8. [8]

    Rainfall-runoff modelling: the primer

    Keith J Beven. Rainfall-runoff modelling: the primer. John Wiley & Sons, 2012

    work page 2012

Show all 56 references
  1. [9]

    A physically based, variable contributing area model of basin hydrology/un mod \`e le \`a base physique de zone d'appel variable de l'hydrologie du bassin versant

    Keith J Beven and Michael J Kirkby. A physically based, variable contributing area model of basin hydrology/un mod \`e le \`a base physique de zone d'appel variable de l'hydrologie du bassin versant. Hydrological sciences journal, 24 0 (1): 0 43--69, 1979

    1979

  2. [10]

    Future streamflow regime changes in the united states: assessment using functional classification

    Manuela I Brunner, Lieke A Melsen, Andrew J Newman, Andrew W Wood, and Martyn P Clark. Future streamflow regime changes in the united states: assessment using functional classification. Hydrology and Earth System Sciences, 24 0 (8): 0 3951--3966, 2020

    2020

  3. [11]

    The evolution of process-based hydrologic models: historical challenges and the collective quest for physical realism

    Martyn P Clark, Marc FP Bierkens, Luis Samaniego, Ross A Woods, Remko Uijlenhoet, Katrina E Bennett, Valentijn Pauwels, Xitian Cai, Andrew W Wood, and Christa D Peters-Lidard. The evolution of process-based hydrologic models: historical challenges and the collective quest for ...

    2017

  4. [12]

    Lipschitz recurrent neural networks

    N Benjamin Erichson, Omri Azencot, Alejandro Queiruga, Liam Hodgkinson, and Michael W Mahoney. Lipschitz recurrent neural networks. In International Conference on Learning Representations, 2021

    2021

  5. [13]

    Gated recurrent neural networks with weighted time-delay feedback

    N Benjamin Erichson, Soon Hoe Lim, and Michael W Mahoney. Gated recurrent neural networks with weighted time-delay feedback. arXiv preprint arXiv:2212.00228, 2025

    2025 arXiv

  6. [14]

    Differentiable, learnable, regionalized process-based models with multiphysical outputs can approach state-of-the-art hydrologic prediction accuracy

    Dapeng Feng, Jiangtao Liu, Kathryn Lawson, and Chaopeng Shen. Differentiable, learnable, regionalized process-based models with multiphysical outputs can approach state-of-the-art hydrologic prediction accuracy. Water Resources Research, 58 0 (10): 0 e2022WR032404, 2022

    2022

  7. [15]

    Deep learning rainfall--runoff predictions of extreme events

    Jonathan M Frame, Frederik Kratzert, Daniel Klotz, Martin Gauch, Guy Shalev, Oren Gilon, Logan M Qualls, Hoshin V Gupta, and Grey S Nearing. Deep learning rainfall--runoff predictions of extreme events. Hydrology and Earth System Sciences, 26 0 (13): 0 3377--3392, 2022

    2022

  8. [16]

    On strictly enforced mass conservation constraints for modelling the rainfall-runoff process

    Jonathan M Frame, Frederik Kratzert, Hoshin V Gupta, Paul Ullrich, and Grey S Nearing. On strictly enforced mass conservation constraints for modelling the rainfall-runoff process. Hydrological Processes, 37 0 (3): 0 e14847, 2023

    2023

  9. [17]

    Mamba: Linear-time sequence modeling with selective state spaces

    Albert Gu and Tri Dao. Mamba: Linear-time sequence modeling with selective state spaces. arXiv preprint arXiv:2312.00752, 2023

    2023 arXiv

  10. [18]

    Efficiently modeling long sequences with structured state spaces

    Albert Gu, Karan Goel, and Christopher R \'e . Efficiently modeling long sequences with structured state spaces. arXiv preprint arXiv:2111.00396, 2021 a

    2021 arXiv

  11. [19]

    Combining recurrent, convolutional, and continuous-time models with linear state space layers

    Albert Gu, Isys Johnson, Karan Goel, Khaled Saab, Tri Dao, Atri Rudra, and Christopher R \'e . Combining recurrent, convolutional, and continuous-time models with linear state space layers. Advances in neural information processing systems, 34: 0 572--585, 2021 b

    2021

  12. [20]

    On the parameterization and initialization of diagonal state space models

    Albert Gu, Karan Goel, Ankit Gupta, and Christopher R \'e . On the parameterization and initialization of diagonal state space models. Advances in Neural Information Processing Systems, 35: 0 35971--35983, 2022

    2022

  13. [21]

    Decomposition of the mean squared error and nse performance criteria: Implications for improving hydrological modelling

    Hoshin V Gupta, Harald Kling, Koray K Yilmaz, and Guillermo F Martinez. Decomposition of the mean squared error and nse performance criteria: Implications for improving hydrological modelling. Journal of hydrology, 377 0 (1-2): 0 80--91, 2009

    2009

  14. [22]

    Liquid structural state-space models

    Ramin Hasani, Mathias Lechner, Tsun-Hsuan Wang, Makram Chahine, Alexander Amini, and Daniela Rus. Liquid structural state-space models. International Conference on Learning Representations, 2023

    2023

  15. [23]

    Long short-term memory

    S Hochreiter. Long short-term memory. Neural Computation MIT-Press, 1997

    1997

  16. [24]

    Mc-lstm: Mass-conserving lstm

    Pieter-Jan Hoedt, Frederik Kratzert, Daniel Klotz, Christina Halmich, Markus Holzleitner, Grey S Nearing, Sepp Hochreiter, and G \"u nter Klambauer. Mc-lstm: Mass-conserving lstm. In International conference on machine learning, pp.\ 4275--4286. PMLR, 2021

    2021

  17. [25]

    Physics-informed neural network for diffusive wave model

    Qingzhi Hou, Yixin Li, Vijay P Singh, and Zewei Sun. Physics-informed neural network for diffusive wave model. Journal of Hydrology, 637: 0 131261, 2024

    2024

  18. [26]

    Groundwater inverse modeling: Physics-informed neural network with disentangled constraints and errors

    Yuzhe Ji, Yuanyuan Zha, Tian-Chyi J Yeh, Liangsheng Shi, and Yanling Wang. Groundwater inverse modeling: Physics-informed neural network with disentangled constraints and errors. Journal of Hydrology, 640: 0 131703, 2024

    2024

  19. [27]

    A review of rainfall-runoff modeling for stormwater management

    H Vernon Knapp, Ali Durguno g lu, and Terry W Ortel. A review of rainfall-runoff modeling for stormwater management. ISWS Contract Report CR 516, 1991

    1991

  20. [28]

    Toward improved predictions in ungauged basins: Exploiting the power of machine learning

    Frederik Kratzert, Daniel Klotz, Mathew Herrnegger, Alden K Sampson, Sepp Hochreiter, and Grey S Nearing. Toward improved predictions in ungauged basins: Exploiting the power of machine learning. Water Resources Research, 55 0 (12): 0 11344--11354, 2019 a

    2019

  21. [29]

    Towards learning universal, regional, and local hydrological behaviors via machine learning applied to large-sample datasets

    Frederik Kratzert, Daniel Klotz, Guy Shalev, G \"u nter Klambauer, Sepp Hochreiter, and Grey Nearing. Towards learning universal, regional, and local hydrological behaviors via machine learning applied to large-sample datasets. Hydrology and Earth System Sciences, 23 0 (12): 0...

    2019

  22. [30]

    Hydrological concept formation inside long short-term memory (lstm) networks

    Thomas Lees, Steven Reece, Frederik Kratzert, Daniel Klotz, Martin Gauch, Jens De Bruijn, Reetik Kumar Sahu, Peter Greve, Louise Slater, and Simon Dadson. Hydrological concept formation inside long short-term memory (lstm) networks. Hydrology and Earth System Sciences Discussi...

    2021

  23. [31]

    Elucidating the design choice of probability paths in flow matching for forecasting

    Soon Hoe Lim, Yijin Wang, Annan Yu, Emma Hart, Michael W Mahoney, Xiaoye S Li, and N Benjamin Erichson. Elucidating the design choice of probability paths in flow matching for forecasting. arXiv preprint arXiv:2410.03229, 2024

    arXiv

  24. [32]

    Probing the limit of hydrologic predictability with the transformer network

    Jiangtao Liu, Yuchen Bian, Kathryn Lawson, and Chaopeng Shen. Probing the limit of hydrologic predictability with the transformer network. Journal of Hydrology, 637: 0 131389, 2024

    2024

  25. [33]

    General review of rainfall-runoff modeling: model calibration, data assimilation, and uncertainty analysis

    Hamid Moradkhani and Soroosh Sorooshian. General review of rainfall-runoff modeling: model calibration, data assimilation, and uncertainty analysis. Springer, 2008

    2008

  26. [34]

    Generative modeling of regular and irregular time series data via koopman vaes

    Ilan Naiman, N Benjamin Erichson, Pu Ren, Michael W Mahoney, and Omri Azencot. Generative modeling of regular and irregular time series data via koopman vaes. In The Twelfth International Conference on Learning Representations

  27. [35]

    River flow forecasting through conceptual models part i—a discussion of principles

    J Eamonn Nash and Jonh V Sutcliffe. River flow forecasting through conceptual models part i—a discussion of principles. Journal of hydrology, 10 0 (3): 0 282--290, 1970

    1970

  28. [36]

    Andrew J Newman, Martyn P Clark, Kevin Sampson, Andrew Wood, Lauren E Hay, Andy Bock, Roland J Viger, David Blodgett, L Brekke, JR Arnold, et al. Development of a large-sample watershed-scale hydrometeorological data set for the contiguous usa: data set characteristics and ass...

    2015

  29. [37]

    State-free inference of state-space models: The transfer function approach

    Rom N Parnichkun, Stefano Massaroli, Alessandro Moro, Jimmy TH Smith, Ramin Hasani, Mathias Lechner, Qi An, Christopher R \'e , Hajime Asama, Stefano Ermon, et al. State-free inference of state-space models: The transfer function approach. arXiv preprint arXiv:2405.06147, 2024

    2024 arXiv

  30. [38]

    Mamba-360: Survey of state space models as transformer alternative for long sequence modelling: Methods, applications, and challenges

    Badri Narayana Patro and Vijay Srinivas Agneeswaran. Mamba-360: Survey of state space models as transformer alternative for long sequence modelling: Methods, applications, and challenges. arXiv preprint arXiv:2404.16112, 2024

    2024 arXiv

  31. [39]

    Evaluation of random forests for short-term daily streamflow forecasting in rainfall-and snowmelt-driven watersheds

    Leo Triet Pham, Lifeng Luo, and Andrew Finley. Evaluation of random forests for short-term daily streamflow forecasting in rainfall-and snowmelt-driven watersheds. Hydrology and Earth System Sciences, 25 0 (6): 0 2997--3015, 2021

    2021

  32. [40]

    Long expressive memory for sequence modeling

    T Konstantin Rusch, Siddhartha Mishra, N Benjamin Erichson, and Michael W Mahoney. Long expressive memory for sequence modeling. In International Conference on Learning Representations, 2022

    2022

  33. [41]

    Differentiable modelling to unify machine learning and physical models for geosciences

    Chaopeng Shen, Alison P Appling, Pierre Gentine, Toshiyuki Bandai, Hoshin Gupta, Alexandre Tartakovsky, Marco Baity-Jesi, Fabrizio Fenicia, Daniel Kifer, Li Li, et al. Differentiable modelling to unify machine learning and physical models for geosciences. Nature Reviews Earth ...

    2023

  34. [42]

    Smith, Andrew Warrington, and Scott Linderman

    Jimmy T.H. Smith, Andrew Warrington, and Scott Linderman. Simplified state space layers for sequence modeling. In The Eleventh International Conference on Learning Representations, 2023

    2023

  35. [43]

    From calibration to parameter learning: Harnessing the scaling effects of big data in geoscientific modeling

    Wen-Ping Tsai, Dapeng Feng, Ming Pan, Hylke Beck, Kathryn Lawson, Yuan Yang, Jiangtao Liu, and Chaopeng Shen. From calibration to parameter learning: Harnessing the scaling effects of big data in geoscientific modeling. Nature communications, 12 0 (1): 0 5988, 2021

    2021

  36. [44]

    Attention is all you need

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in Neural Information Processing Systems, 2017

    2017

  37. [45]

    Stablessm: Alleviating the curse of memory in state-space models through stable reparameterization

    Shida Wang and Qianxiao Li. Stablessm: Alleviating the curse of memory in state-space models through stable reparameterization. arXiv preprint arXiv:2311.14495, 2023

    2023 arXiv

  38. [46]

    Continental-scale water and energy flux analysis and validation for the north american land data assimilation system project phase 2 (nldas-2): 1

    Youlong Xia, Kenneth Mitchell, Michael Ek, Justin Sheffield, Brian Cosgrove, Eric Wood, Lifeng Luo, Charles Alonge, Helin Wei, Jesse Meng, et al. Continental-scale water and energy flux analysis and validation for the north american land data assimilation system project phase ...

    2012

  39. [47]

    Classification of watersheds in the conterminous united states using shape-based time-series clustering and random forests

    Mingyue Yang and Francisco Olivera. Classification of watersheds in the conterminous united states using shape-based time-series clustering and random forests. Journal of Hydrology, 620: 0 129409, 2023

    2023

  40. [48]

    A process-based diagnostic approach to model evaluation: Application to the nws distributed hydrologic model

    Koray K Yilmaz, Hoshin V Gupta, and Thorsten Wagener. A process-based diagnostic approach to model evaluation: Application to the nws distributed hydrologic model. Water resources research, 44 0 (9), 2008

    2008

  41. [49]

    Tuning frequency bias in neural network training with nonuniform data

    Annan Yu, Yunan Yang, and Alex Townsend. Tuning frequency bias in neural network training with nonuniform data. International Conference on Learning Representations, 2023

    2023

  42. [50]

    Mahoney, and N

    Annan Yu, Arnur Nigmetov, Dmitriy Morozov, Michael W. Mahoney, and N. Benjamin Erichson. Robustifying state-space models for long sequences via approximate diagonalization. In The Twelfth International Conference on Learning Representations, 2024

    2024

  43. [51]

    Tuning frequency bias of state space models

    Annan Yu, Dongwei Lyu, Soon Hoe Lim, Michael W Mahoney, and N Benjamin Erichson. Tuning frequency bias of state space models. The Thirteenth International Conference on Learning Representations, 2025 a

    2025

  44. [52]

    Hope for a robust parameterization of long-memory state space models

    Annan Yu, Michael W Mahoney, and N Benjamin Erichson. Hope for a robust parameterization of long-memory state space models. The Thirteenth International Conference on Learning Representations, 2025 b

    2025

  45. [53]

    Deep latent state space models for time-series generation

    Linqi Zhou, Michael Poli, Winnie Xu, Stefano Massaroli, and Stefano Ermon. Deep latent state space models for time-series generation. In International Conference on Machine Learning, pp.\ 42625--42643. PMLR, 2023

    2023

  46. [54]

    @esa (Ref

    \@ifxundefined[1] #1\@undefined \@firstoftwo \@secondoftwo \@ifnum[1] #1 \@firstoftwo \@secondoftwo \@ifx[1] #1 \@firstoftwo \@secondoftwo [2] @ #1 \@temptokena #2 #1 @ \@temptokena \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should ...

  47. [55]

    \@lbibitem[] @bibitem@first@sw\@secondoftwo \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 @tmp #1 NAT@b@open@#2 NAT@b@shut@#2 \@ifnum @merge>\@ne @bibitem@firs...

  48. [56]

    @open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifxundefined @sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifxundefined @heading @heading NAT@ctr thebibliography [1] @ \@biblabel @NAT@ctr \@bibset...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.