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REVIEW 4 major objections 6 minor 54 references

Sparse Identification of Nonlinear Dynamics with Conformal Prediction

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that conformal prediction can be layered onto ensemble SINDy to produce forecast, feature-importance, and coefficient-uncertainty outputs whose coverage and ranking hold under process and non-Gaussian measurement noise.

desk verdict A useful exploratory paper that shows conformal prediction can be wired onto E-SINDy for forecasting and feature importance, but the coefficient-uncertainty claim rests on a heuristic the paper itself acknowledges lacks formal coverage guarantees. read the letter →

arxiv 2507.11739 v1 pith:QLM4DWBO submitted 2025-07-15 cs.LG cs.CEmath.DS

Urban Fasel This is my paper
classification cs.LGcs.CEmath.DS
keywords conformalpredictionsparseidentificationofnonlineardynamicsensembleSINDyuncertaintyquantificationtimeseriesforecastingfeatureimportancemodelselectiondynamicalsystems
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

The paper argues that conformal prediction — a framework that converts any predictive algorithm into one with user-specified coverage intervals under minimal distributional assumptions — can be layered onto Ensemble-SINDy to give honest uncertainty estimates for discovered dynamical models. It demonstrates three pairings: EnbPI and conformal PID control for forecast intervals, LOCO and LOCO-path for ranking library terms, and feature conformal prediction (feature-CP) for intervals on identified coefficients. On a stochastic predator–prey system and several chaotic benchmarks, the forecast intervals reach the requested coverage, the importance scores separate active from inactive library terms quickly, and the coefficient intervals contain the true values under process noise and gamma-distributed measurement noise where standard E-SINDy intervals miss. The payoff, if correct, is that practitioners can attach calibrated intervals and defensible feature rankings to SINDy models without assuming Gaussian or independent noise.

What carries the argument

The carrying mechanism is the non-conformity score, a scalar measuring how badly a candidate prediction or candidate coefficient vector deviates from the calibration data. For forecasting, EnbPI builds scores from out-of-sample residuals of bootstrap SINDy models and takes a quantile, while conformal PID control replaces that quantile with a P-I controller that widens or narrows the interval to keep long-run coverage at the target. For model selection, LOCO quantifies the excess prediction error when one library feature is removed, and LOCO-path sums the $\ell^1$ deviation between the full and leave-one-feature-out coefficient solutions across the regularization path. For coefficient uncertainty, feature-CP fits a surrogate SINDy model to each leave-one-out training set under a constraint that the surrogate reproduces the held-out derivative, and uses the $\ell^1$ distance $\|\tilde{\Xi}_i - \Xi_i\|_1$ between surrogate and ensemble coefficients as the score; the quantile of these scores becomes the interval. These scores let a user pick a coverage level and read off intervals without changing the underlying E-SINDy estimator.

What would settle it

Repeatedly simulate the stochastic predator–prey system over many noise realizations at increasing process-noise levels, compute the empirical finite-sample coverage of the 90% feature-CP intervals for the active coefficients, and compare it with the target. If coverage falls systematically below 90% as noise grows or as trajectory dependence strengthens, the coefficient-uncertainty claim would be falsified; a positive control would be the same experiment with an independent calibration set.

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Extended reading notes

Core claim

The paper's central claim is that conformal prediction integrated with ensemble SINDy (E-SINDy) can reliably achieve target coverage for time series forecasting, effectively quantify library feature importance, and produce more robust uncertainty intervals for model coefficients, even under non-Gaussian noise, compared with standard E-SINDy coefficient estimates. The three applications are worked out concretely: two online recalibration schemes for forecast intervals, two jackknife-based feature-importance schemes for model selection, and a surrogate-feature scheme that places a 90% interval on the model coefficients. The numerical evidence shows standard E-SINDy intervals are narrow but under-cover when measurement noise is gamma-distributed or when process noise is present, while feature-CP intervals remain conservative and contain the true coefficients. The authors present the results as demonstrations of practical recipes, and they state explicitly that formal finite-sample coverage guarantees for the SINDy-specific procedures remain future work.

Load-bearing premise

The load-bearing premise is that surrogate models forced to reproduce each held-out data point yield non-conformity scores that are valid for the finally selected SINDy coefficients, even though the same data were used to pick the thresholded library; the paper notes that valid post-selection intervals are challenging and gives no proof for this step.

Editorial extensions

If this is right

  • Forecast intervals can be recalibrated online from newly arriving measurements, so a SINDy forecaster can maintain stated coverage as the system drifts, without retraining the ensemble at every step.
  • Feature-importance scores stabilize quickly in the low-data regime, and LOCO-path removes the need to hand-tune the sparsity threshold, giving a principled way to choose which library terms belong in the model.
  • Coefficient uncertainty reported by standard E-SINDy can be overconfident under process and non-Gaussian measurement noise; feature-CP corrects this overconfidence, with the trade-off of wider intervals.
  • Because the conformal machinery makes no distributional assumption beyond exchangeability (or, for the time-series variants, mild asymptotic conditions), the same code path applies to Lorenz, hyper-Rössler, and quadrupole-boson systems without per-system tuning.

Reading between the lines

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

  • The feature-CP intervals are calibrated on scores computed after the thresholded library has already been selected from the same data, so the reported coverage does not yet account for post-selection bias; a hold-out model-selection set or a post-selection conformal correction would be needed to test whether the claimed coverage survives honest model selection.
  • The time-series coverage shown is marginal over the trajectory rather than conditional on the current state; for safety-critical monitoring the relevant guarantee would be per-timestep conditional coverage, which neither EnbPI nor conformal PID control claims.
  • A natural extension is per-coefficient calibration instead of a single $\ell^1$ score for the whole model, which could shrink the currently wide intervals, and combining feature-CP with an errors-in-variables treatment of library noise could remove the remaining bias in quadratic-term estimates.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The manuscript introduces three conformal-prediction applications built on Ensemble-SINDy: time-series forecasting intervals via EnbPI and CP-PID, library feature importance via LOCO and LOCO-path, and coefficient uncertainty via feature-CP. Experiments on a noisy predator-prey system and chaotic benchmarks show that the forecasting methods reach the nominal 90% coverage, that LOCO importance scores stabilize and separate active terms, and that feature-CP intervals are wider than standard E-SINDy intervals and contain the true coefficients under several non-Gaussian noise settings. The paper states explicitly that no finite-sample coverage guarantees are provided for the SINDy-specific conformal procedures.

Significance. The paper offers a practical, open-source integration of conformal prediction with SINDy and demonstrates that two established time-series conformal methods work with an E-SINDy forecaster. The LOCO and LOCO-path applications are useful and empirically supported. The main significance, however, is limited by the feature-CP method: its surrogate scores do not satisfy standard conformal exchangeability, and the paper's own text acknowledges the post-selection inference challenge. If the authors can either provide validity guarantees or explicitly reframe the coefficient intervals as a heuristic, the work would be an honest and useful engineering contribution.

major comments (4)
  1. [Sec. 3.3, Eqs. (15)–(18)] The feature-CP surrogate scores are not exchangeable with the final model's coefficients. The constrained least-squares problem uses a thresholded library selected from the same full dataset used to fit the ensemble and to choose the final model, and the score s_i is computed in-sample relative to the retained model. The quantile of these scores is not a valid conformal quantile for the selected coefficient vector, and no finite-sample or asymptotic argument is supplied. The paper itself concedes the difficulty in the paragraph following Eq. (18) and the Discussion states that formal guarantees are not provided. Nevertheless, the abstract and Section 5 claim that feature-CP 'produces more robust uncertainty intervals' and 'successfully covers the true coefficients across noise types.' This is load-bearing because the third central claim rests on this method. Either a validity proof should be supplied, or the claims should be explicitly labeled as heuristic and empirical, with a statement that coverage is not guaranteed.
  2. [Sec. 3.3, Eq. (18) and Fig. 5] The scalar L1 score s_i is used to compute a single quantile that is applied to all coefficients, and the text in Section 4.3 states that calibration is done 'on the full model and not on the individual model coefficients.' Consequently, the reported 90% intervals do not imply per-coefficient marginal coverage, and the simultaneous coverage of the true coefficient vector is not controlled. The experiments in Figure 5 and Appendix B display each coefficient's interval separately, which may mislead readers into thinking each has 90% coverage. Please report per-coefficient empirical coverage rates (and, if appropriate, simultaneous coverage) and state explicitly which property the intervals are designed to have.
  3. [Sec. 4] The paper does not report the sparsity threshold λ used for SINDy in any experiment, nor the values of the E-SINDy hyperparameters (ensemble size B, calibration window size l_r, prediction horizon, CP-PID gains η and r_t, and the noise levels). This is particularly important because λ directly controls model selection and hence the coefficient values, and because LOCO-path is introduced to bypass λ selection, but the comparison in Figure 4 does not state whether LOCO and LOCO-path use the same λ. Without these details the experimental claims are not reproducible from the text alone. Please add a table listing all hyperparameters per experiment, or provide a reproducible configuration file.
  4. [Sec. 4.1, Eq. (8)] The description of the EnbPI non-conformity scores is ambiguous. It says residuals are computed 'using only those models that were not trained on the corresponding state x_t.' Since E-SINDy uses bootstrap resampling, a given model is trained on a random subset of rows and may or may not include x_t; the paper does not explain how the 'not trained on' condition is determined or what fraction of the B models are used when all models have seen a particular point. Clarify the algorithmic implementation, or if all B models are used regardless, state that the out-of-sample property is approximate.
minor comments (6)
  1. [Sec. 2.1, Eq. (3)] The notation ||·||_0 is used for sparsity but is not defined; the standard definition (number of nonzero entries) should be given.
  2. [Sec. 4] The phrase 'a small library of candidate functions containing only quadratic terms' is inconsistent with the predator-prey dynamics, which include linear terms (x1 and x2) that are among the true coefficients reported in Figure 5. Clarify whether the library includes linear terms as well as quadratic ones.
  3. [Sec. 3.3] The parameter n is used both for the number of snapshots (Section 2.1) and for the number of states (in the definition of Ξ = [ξ1, ξ2] for n=2). Use different symbols to avoid confusion.
  4. [Sec. 2.2] The definition of k in 'k = r(n/2 + 1)(1−α)s' is nonstandard; provide a precise definition of the quantile (e.g., the smallest value such that at least (1−α) of the scores are below it).
  5. [Appendix B] The gamma noise parameters and the hyperparameters for WENDy and BSINDy are not reported; add a description of all baseline method settings.
  6. [References] Reference [16] is listed as an arXiv preprint only; if a journal version exists, cite it. Also, reference [24] is for SCP but the paper uses jackknife+ which is [30]; ensure citations are consistently matched.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: conformal intervals are calibrated on out-of-sample residuals and coverage is checked against synthetic ground truth, not enforced by construction.

full rationale

The paper's central claims are empirical demonstrations that conformal prediction methods combined with E-SINDy produce intervals with desired coverage and useful feature-importance measures. The time-series intervals in Sec. 3.1 are constructed from out-of-sample residuals: non-conformity scores are computed only from ensemble members not trained on the corresponding state, and the quantile is then applied to new predictions. The feature-CP coefficient intervals in Sec. 3.3 use surrogate models constrained to reproduce the held-out point, with scores s_i = ||Ξ̃_i - Ξ_i||_1; these scores are computed from held-out data and the resulting quantile is not fitted to make the reported coverage true. Coverage is evaluated against known synthetic ground-truth coefficients and trajectories, so the favorable coverage results are not guaranteed by the construction itself. The paper explicitly acknowledges the main limitation, that post-selection inference is challenging due to selection bias, and disclaims formal finite-sample guarantees; this is a validity gap, not circularity. Self-citations to E-SINDy [14] and related benchmark work [52] provide the base estimator and test systems, but the conformal guarantees come from cited external frameworks (EnbPI, CP-PID, feature-CP) and are not justified by the author's own prior results. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors' prior work is invoked, and no known result is repackaged as new organization. The derivation chain is therefore self-contained with respect to the paper's claimed contributions.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central demonstrations depend on standard conformal prediction guarantees, on the quality of the weak-form SINDy approximation, and on an unproven post-selection surrogate calibration. No parameters were fitted directly to the target coverage, but the sparsity threshold and PID gains are hand-chosen and unreported.

free parameters (5)
  • SINDy sparsity threshold lambda = not reported
    Controls model sparsity in all experiments; selection procedure is not stated, so results may depend on this hand-chosen value.
  • Prediction horizon = 2 time steps (main), 10 (appendix)
    Used for EnbPI and CP-PID forecasts; changes interval width and coverage behavior.
  • Ensemble size B = not reported
    Number of bootstrap E-SINDy models; affects residual aggregation and interval stability.
  • Calibration window size l_r = not reported
    Sliding window of residuals used to estimate quantiles in EnbPI.
  • PID gains eta and saturation r_t = not tuned
    CP-PID controller gains; the paper notes they were not tuned in appendix experiments.
assumptions (5)
  • standard math Split conformal prediction and jackknife+ have valid coverage under exchangeable data (Vovk et al. 2005; Barber et al. 2021).
    Background from Sec. 2.2; the paper uses these results to justify the conformal framework.
  • domain assumption The time series residuals used for conformal calibration are exchangeable or satisfy the bounded-score/long-run coverage conditions of EnbPI and CP-PID.
    Invoked in Sec. 3.1; the validity of the prediction intervals depends on these assumptions, which may be violated by thresholded sparse regression and non-stationary dynamics.
  • domain assumption The weak-form SINDy derivative estimates and the chosen library are sufficiently accurate that the discovered model is close to the true dynamics.
    Used throughout Sec. 4; if the library omits needed terms or derivative estimation is biased, coverage and importance scores can be unreliable.
  • ad hoc to paper Post-selection surrogate scores from constrained least-squares (Eq. 17) are informative for coefficient uncertainty even though the library was selected from the same data.
    Assumed in Sec. 3.3; the paper acknowledges selection bias is an open issue and provides no proof of valid coverage.
  • domain assumption Ground-truth coefficients in synthetic benchmarks are known, allowing coverage of coefficient intervals to be evaluated.
    Used in Sec. 4.3; this is a standard benchmarking assumption, but it means the reported coverage is not a property of real data.

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Cite this review

Pith. "Pith review of Sparse Identification of Nonlinear Dynamics with Conformal Prediction." pith.science (2026). https://pith.science/paper/QLM4DWBO

@misc{pith2026250711739,
  author       = {Pith},
  title        = {Pith review of: Sparse Identification of Nonlinear Dynamics with Conformal Prediction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLM4DWBO}},
  note         = {Machine review of arXiv:2507.11739}
}
read the original abstract

The Sparse Identification of Nonlinear Dynamics (SINDy) is a method for discovering nonlinear dynamical system models from data. Quantifying uncertainty in SINDy models is essential for assessing their reliability, particularly in safety-critical applications. While various uncertainty quantification methods exist for SINDy, including Bayesian and ensemble approaches, this work explores the integration of Conformal Prediction, a framework that can provide valid prediction intervals with coverage guarantees based on minimal assumptions like data exchangeability. We introduce three applications of conformal prediction with Ensemble-SINDy (E-SINDy): (1) quantifying uncertainty in time series prediction, (2) model selection based on library feature importance, and (3) quantifying the uncertainty of identified model coefficients using feature conformal prediction. We demonstrate the three applications on stochastic predator-prey dynamics and several chaotic dynamical systems. We show that conformal prediction methods integrated with E-SINDy can reliably achieve desired target coverage for time series forecasting, effectively quantify feature importance, and produce more robust uncertainty intervals for model coefficients, even under non-Gaussian noise, compared to standard E-SINDy coefficient estimates.

Figures

Figures reproduced from arXiv: 2507.11739 by the authors.

Figure 1
Figure 1. Schematic of the E-SINDy with conformal prediction framework for (a) uncertainty [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. E-SINDy with conformal prediction for time series prediction using EnbPI and CP-PID. [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Coverage and width over target coverage for the predator-prey dynamics using EnbPI [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: E-SINDy inclusion probability, LOCO, and LOCO-path results showing a measure of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Feature-CP results evaluating SINDy conformal prediction on model coefficients, for a [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

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Works this paper leans on

54 extracted references · 44 canonical work pages

  1. [1]

    Discovering governing equations from data by sparse identification of nonlinear dynamical systems

    Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the national academy of sciences, 2016

    work page 2016

  2. [2]

    Sparse identification of non- linear dynamics with low-dimensionalized flow representations

    Kai Fukami, Takaaki Murata, Kai Zhang, and Koji Fukagata. Sparse identification of non- linear dynamics with low-dimensionalized flow representations. Journal of Fluid Mechanics , 2021

    work page 2021

  3. [3]

    Hypergraph reconstruction from dynamics

    Robin Delabays, Giulia De Pasquale, Florian D ¨orfler, and Yuanzhao Zhang. Hypergraph reconstruction from dynamics. Nature Communications, 2025

    work page 2025

  4. [4]

    Algorithmic discovery of dynamic models from in- fectious disease data

    Jonathan Horrocks and Chris T Bauch. Algorithmic discovery of dynamic models from in- fectious disease data. Scientific reports, 2020

    work page 2020

  5. [5]

    Safe physics-informed machine learning for dynamics and control

    Jan Drgona, Truong X Nghiem, Thomas Beckers, Mahyar Fazlyab, Enrique Mallada, Colin Jones, Draguna Vrabie, Steven L Brunton, and Rolf Findeisen. Safe physics-informed machine learning for dynamics and control. arXiv:2504.12952, 2025

    arXiv 2025

  6. [6]

    Robust data-driven discovery of governing physical laws with error bars

    Sheng Zhang and Guang Lin. Robust data-driven discovery of governing physical laws with error bars. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 2018

    work page 2018

  7. [7]

    Bayesian differential programming for robust systems identification under uncertainty

    Yibo Yang, Mohamed Aziz Bhouri, and Paris Perdikaris. Bayesian differential programming for robust systems identification under uncertainty. Proceedings of the Royal Society A, 2020

    work page 2020

  8. [8]

    Bayesian system ID: optimal management of parameter, model, and measurement uncertainty

    Nicholas Galioto and Alex Arkady Gorodetsky. Bayesian system ID: optimal management of parameter, model, and measurement uncertainty. Nonlinear Dynamics, 2020

    work page 2020

Show all 54 references
  1. [9]

    Sparsifying priors for bayesian uncertainty quantification in model discovery

    Seth M Hirsh, David A Barajas-Solano, and J Nathan Kutz. Sparsifying priors for bayesian uncertainty quantification in model discovery. Royal Society open science, 2022

    2022

  2. [10]

    A bayesian approach for data- driven dynamic equation discovery.Journal of Agricultural, Biological and Environmental Statis- tics, 2022

    Joshua S North, Christopher K Wikle, and Erin M Schliep. A bayesian approach for data- driven dynamic equation discovery.Journal of Agricultural, Biological and Environmental Statis- tics, 2022

    2022

  3. [11]

    17 Figure B.1: E-SINDy with feature-CP (blue) compared with WENDy (red) and BSINDy (green)

    Robert K Niven, Laurent Cordier, Ali Mohammad-Djafari, Markus Abel, and Markus Quade. 17 Figure B.1: E-SINDy with feature-CP (blue) compared with WENDy (red) and BSINDy (green). Results shown for a fixed time series length and varying noise levels. Each noise level is run on 1...

    2024

  4. [12]

    Rapid bayesian identification of sparse non- linear dynamics from scarce and noisy data

    Lloyd Fung, Urban Fasel, and Matthew Juniper. Rapid bayesian identification of sparse non- linear dynamics from scarce and noisy data. In Proceedings A. The Royal Society, 2025

    2025

  5. [13]

    Statistical mechanics of dynamical system identification

    Andrei A Klishin, Joseph Bakarji, J Nathan Kutz, and Krithika Manohar. Statistical mechanics of dynamical system identification. arXiv:2403.01723, 2024. 18

    2024

  6. [14]

    Nathan Kutz, Bingni W

    Urban Fasel, J. Nathan Kutz, Bingni W. Brunton, and Steven L. Brunton. Ensemble-sindy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control. Proceedings of the Royal Society A, 2022

    2022

  7. [15]

    Sta- bility selection enables robust learning of differential equations from limited noisy data

    Suryanarayana Maddu, Bevan L Cheeseman, Ivo F Sbalzarini, and Christian L M ¨uller. Sta- bility selection enables robust learning of differential equations from limited noisy data. Pro- ceedings of the Royal Society A, 2022

    2022

  8. [16]

    Convergence of uncertainty esti- mates in ensemble and bayesian sparse model discovery

    L Gao, Urban Fasel, Steven L Brunton, and J Nathan Kutz. Convergence of uncertainty esti- mates in ensemble and bayesian sparse model discovery. arXiv:2301.12649, 2023

    2023 arXiv

  9. [17]

    Ekf– sindy: Empowering the extended kalman filter with sparse identification of nonlinear dy- namics

    Luca Rosafalco, Paolo Conti, Andrea Manzoni, Stefano Mariani, and Attilio Frangi. Ekf– sindy: Empowering the extended kalman filter with sparse identification of nonlinear dy- namics. Computer Methods in Applied Mechanics and Engineering, 2024

    2024

  10. [18]

    Online learning in bifurcating dynamic systems via sindy and kalman filtering

    Luca Rosafalco, Paolo Conti, Andrea Manzoni, Stefano Mariani, and Attilio Frangi. Online learning in bifurcating dynamic systems via sindy and kalman filtering. Nonlinear Dynamics, 2025

    2025

  11. [19]

    Veni, vindy, vici: a variational reduced-order modeling framework with uncertainty quantification

    Paolo Conti, Jonas Kneifl, Andrea Manzoni, Attilio Frangi, J ¨org Fehr, Steven L Brunton, and J Nathan Kutz. Veni, vindy, vici: a variational reduced-order modeling framework with uncertainty quantification. arXiv:2405.20905, 2024

    2024

  12. [20]

    Algorithmic learning in a random world

    Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic learning in a random world. Springer, 2005

    2005

  13. [21]

    Theoretical foundations of conformal prediction

    Anastasios N Angelopoulos, Rina Foygel Barber, and Stephen Bates. Theoretical foundations of conformal prediction. arXiv:2411.11824, 2024

    2024 arXiv

  14. [22]

    Conformal prediction interval for dynamic time-series

    Chen Xu and Yao Xie. Conformal prediction interval for dynamic time-series. In International Conference on Machine Learning. PMLR, 2021

    2021

  15. [23]

    Conformal PID control for time series prediction

    Anastasios Angelopoulos, Emmanuel Candes, and Ryan J Tibshirani. Conformal PID control for time series prediction. Advances in neural information processing systems, 2023

    2023

  16. [24]

    Distribution-free predictive inference for regression

    Jing Lei, Max G’Sell, Alessandro Rinaldo, Ryan J Tibshirani, and Larry Wasserman. Distribution-free predictive inference for regression. Journal of the American Statistical As- sociation, 2018

    2018

  17. [25]

    Inference for sparse linear regression based on the leave-one-covariate-out solution path

    Xiangyang Cao, Karl Gregory, and Dewei Wang. Inference for sparse linear regression based on the leave-one-covariate-out solution path. Communications in Statistics-Theory and Methods, 2023

    2023

  18. [26]

    Predic- tive inference with feature conformal prediction

    Jiaye Teng, Chuan Wen, Dinghuai Zhang, Yoshua Bengio, Yang Gao, and Yang Yuan. Predic- tive inference with feature conformal prediction. arXiv:2210.00173, 2022

    2022 arXiv

  19. [27]

    Conformalized time series with semantic features

    Baiting Chen, Zhimei Ren, and Lu Cheng. Conformalized time series with semantic features. Advances in Neural Information Processing Systems, 2024

    2024

  20. [28]

    Weak sindy: Galerkin-based data-driven model selection

    Daniel A Messenger and David M Bortz. Weak sindy: Galerkin-based data-driven model selection. Multiscale Modeling & Simulation, 2021

    2021

  21. [29]

    Using noisy or incomplete data to discover models of spatiotemporal dynamics

    Patrick AK Reinbold, Daniel R Gurevich, and Roman O Grigoriev. Using noisy or incomplete data to discover models of spatiotemporal dynamics. Physical Review E, 2020

    2020

  22. [30]

    Predictive inference with the jackknife+

    Rina Foygel Barber, Emmanuel J Candes, Aaditya Ramdas, and Ryan J Tibshirani. Predictive inference with the jackknife+. The Annals of Statistics, 2021

    2021

  23. [31]

    Predictive inference is free with the jackknife+-after- bootstrap

    Byol Kim, Chen Xu, and Rina Barber. Predictive inference is free with the jackknife+-after- bootstrap. Advances in Neural Information Processing Systems, 2020

    2020

  24. [32]

    Adaptive conformal inference under distribution shift

    Isaac Gibbs and Emmanuel Candes. Adaptive conformal inference under distribution shift. Advances in Neural Information Processing Systems, 2021

    2021

  25. [33]

    Adap- tive conformal predictions for time series

    Margaux Zaffran, Olivier F ´eron, Yannig Goude, Julie Josse, and Aymeric Dieuleveut. Adap- tive conformal predictions for time series. In International Conference on Machine Learning . PMLR, 2022

    2022

  26. [34]

    Conformal prediction for 19 time series with modern hopfield networks.Advances in Neural Information Processing Systems, 2023

    Andreas Auer, Martin Gauch, Daniel Klotz, and Sepp Hochreiter. Conformal prediction for 19 time series with modern hopfield networks.Advances in Neural Information Processing Systems, 2023

    2023

  27. [35]

    Smoothing and differentiation of data by simplified least squares procedures

    Abraham Savitzky and Marcel JE Golay. Smoothing and differentiation of data by simplified least squares procedures. Analytical chemistry, 1964

    1964

  28. [36]

    Smoothing and parameter estimation by soft-adherence to governing equations

    Samuel H Rudy, Steven L Brunton, and J Nathan Kutz. Smoothing and parameter estimation by soft-adherence to governing equations. Journal of Computational Physics, 2019

    2019

  29. [37]

    Data assimilation: methods, algorithms, and ap- plications

    Mark Asch, Marc Bocquet, and Ma ¨elle Nodet. Data assimilation: methods, algorithms, and ap- plications. SIAM, 2016

    2016

  30. [38]

    Random forests

    Leo Breiman. Random forests. Machine learning, 2001

    2001

  31. [39]

    Conditional variable importance for random forests

    Carolin Strobl, Anne-Laure Boulesteix, Thomas Kneib, Thomas Augustin, and Achim Zeileis. Conditional variable importance for random forests. BMC bioinformatics, 2008

    2008

  32. [40]

    Regression shrinkage and selection via the lasso

    Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Sta- tistical Society Series B: Statistical Methodology, 1996

    1996

  33. [41]

    Model selection for dynamical systems via sparse regression and information criteria

    Niall M Mangan, J Nathan Kutz, Steven L Brunton, and Joshua L Proctor. Model selection for dynamical systems via sparse regression and information criteria. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2017

    2017

  34. [42]

    Direct estimation of parameters in ode models using wendy: Weak-form estimation of nonlinear dynamics

    David M Bortz, Daniel A Messenger, and Vanja Dukic. Direct estimation of parameters in ode models using wendy: Weak-form estimation of nonlinear dynamics. Bulletin of Mathematical Biology, 2023

    2023

  35. [43]

    Constrained sparse galerkin regression

    Jean-Christophe Loiseau and Steven L Brunton. Constrained sparse galerkin regression. Jour- nal of Fluid Mechanics, 2018

    2018

  36. [44]

    Multi-objective sindy for parameterized model dis- covery from single transient trajectory data

    Javier Lemus and Benjamin Herrmann. Multi-objective sindy for parameterized model dis- covery from single transient trajectory data. Nonlinear Dynamics, 2025

    2025

  37. [45]

    Valid post- selection inference

    Richard Berk, Lawrence Brown, Andreas Buja, Kai Zhang, and Linda Zhao. Valid post- selection inference. The Annals of Statistics, 2013

    2013

  38. [46]

    Confidence intervals for low dimensional parame- ters in high dimensional linear models

    Cun-Hui Zhang and Stephanie S Zhang. Confidence intervals for low dimensional parame- ters in high dimensional linear models. Journal of the Royal Statistical Society Series B: Statistical Methodology, 2014

    2014

  39. [47]

    Double/debiased machine learning for treatment and structural parameters, 2018

    Victor Chernozhukov, Denis Chetverikov, Mert Demirer, Esther Duflo, Christian Hansen, Whitney Newey, and James Robins. Double/debiased machine learning for treatment and structural parameters, 2018

    2018

  40. [48]

    Conformal prediction after efficiency- oriented model selection

    Ruiting Liang, Wanrong Zhu, and Rina Foygel Barber. Conformal prediction after efficiency- oriented model selection. arXiv:2408.07066, 2024

    2024 arXiv

  41. [49]

    Automatic differentiation to simultaneously identify nonlinear dynamics and extract noise probability distributions from data

    Kadierdan Kaheman, Steven L Brunton, and J Nathan Kutz. Automatic differentiation to simultaneously identify nonlinear dynamics and extract noise probability distributions from data. Machine Learning: Science and Technology, 2022

    2022

  42. [50]

    Time series analysis by state space methods

    James Durbin and Siem Jan Koopman. Time series analysis by state space methods . Oxford University Press (UK), 2012

    2012

  43. [51]

    Chaos as an interpretable benchmark for forecasting and data-driven mod- elling

    William Gilpin. Chaos as an interpretable benchmark for forecasting and data-driven mod- elling. Advances in Neural Information Processing Systems (NeurIPS), arXiv:2110.05266, 2021

    2021 arXiv

  44. [52]

    Benchmarking sparse system identification with low-dimensional chaos.Nonlinear Dynamics, 2023

    Alan A Kaptanoglu, Lanyue Zhang, Zachary G Nicolaou, Urban Fasel, and Steven L Brunton. Benchmarking sparse system identification with low-dimensional chaos.Nonlinear Dynamics, 2023

    2023

  45. [53]

    Improved online conformal pre- diction via strongly adaptive online learning

    Aadyot Bhatnagar, Huan Wang, Caiming Xiong, and Yu Bai. Improved online conformal pre- diction via strongly adaptive online learning. In International Conference on Machine Learning. PMLR, 2023

    2023

  46. [54]

    Conformal inference for online prediction with arbi- trary distribution shifts

    Isaac Gibbs and Emmanuel J Cand `es. Conformal inference for online prediction with arbi- trary distribution shifts. Journal of Machine Learning Research, 2024. 20

    2024

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.