Conformal Prediction & Ensemble-SINDy

• The paper demonstrates how conformal prediction calibrates ensemble forecasts to achieve robust uncertainty quantification in sparse nonlinear dynamics.
• It employs bootstrap-based ensemble techniques such as bagging and stability selection to enhance the reliability of SINDy model outputs.
• The framework delivers joint prediction intervals and feature importance measures, addressing challenges in time-series forecasting and coefficient uncertainty.

Searching arXiv for papers on conformal prediction with SINDy and related time-series conformal methods. {"query":"all:(\"Conformal Prediction\" AND SINDy) OR all:(Ensemble-SINDy conformal prediction) OR all:(\"Sparse Identification of Nonlinear Dynamics with Conformal Prediction\")","max_results":10,"sort_by":"submittedDate"} Conformal Prediction with Ensemble-SINDy denotes the use of conformal prediction to calibrate the outputs of an ensemble of sparse identification models for nonlinear dynamical systems. In this literature, the conformal layer is wrapped around a deterministic dynamical predictor through residual-based nonconformity scores on one-step transitions or trajectory segments, while the ensemble layer is realized through bootstrap, bagging, stability selection, or leave-one-out aggregation. The combined framework is used for three distinct targets: time-series prediction uncertainty, model structure or library feature importance, and uncertainty in identified coefficients (Fasel, 15 Jul 2025). Related work on PDE-driven dynamical systems provides the same basic residual-based construction in high-dimensional state space (Liang et al., 2024), and EnbPI supplies a time-series-specific ensemble wrapper that avoids data-splitting, is computationally efficient by avoiding retraining, and does not require data exchangeability (Xu et al., 2020).

1. Sparse identification and ensemble construction

Sparse Identification of Nonlinear Dynamics (SINDy) models a dynamical system

$$\frac{d}{dt}\mathbf{x}(t)=\mathbf{f}(\mathbf{x}(t)),$$

using a library of candidate nonlinear functions $\boldsymbol{\Theta}(\mathbf{X})$ and a sparse coefficient matrix $\boldsymbol{\Xi}$ such that

$$\dot{\mathbf{X}}=\boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}.$$

In the standard formulation, sparse regression is posed as

$$\boldsymbol{\Xi}=\arg\min_{\hat{\boldsymbol{\Xi}}}\Big\|\dot{\mathbf{X}}-\boldsymbol{\Theta}(\mathbf{X})\hat{\boldsymbol{\Xi}}\Big\|_2^2+\lambda\|\hat{\boldsymbol{\Xi}}\|_0,$$

and in practice sequential thresholded least squares is used: least squares is solved, coefficients smaller than $\lambda$ are zeroed, and least squares is refit on the remaining active terms (Fasel, 15 Jul 2025).

Ensemble-SINDy introduces bootstrap-based ensembles to improve robustness and quantify uncertainty. In the formulation summarized for E-SINDy, $B$ bootstrap samples are constructed by sampling rows with replacement from the combined matrix $[\dot{\mathbf{X}},\boldsymbol{\Theta}(\mathbf{X})]$, a SINDy model is fit on each bootstrap sample to obtain $\boldsymbol{\Xi}^{(b)}$, and coefficients are then aggregated, for example by row-wise median or mean, to produce an ensemble estimate $\bar{\boldsymbol{\Xi}}$. This yields empirical distributions for coefficients, inclusion probabilities for library terms, and a collection of ensemble forecasts obtained by integrating each ensemble member forward.

A closely related instantiation appears in the Adaptive Ensemble Sparse Identification framework for Li-ion battery modeling. There, the sparse model is learned for the voltage error dynamics,

$\boldsymbol{\Theta}(\mathbf{X})$0

with STRidge used for sparsity enforcement, 100 Moving Block Bootstrap resampled datasets used for base-model generation, and the top 10% models retained as the ensemble. Two aggregation strategies are used: bagging, with

$\boldsymbol{\Theta}(\mathbf{X})$1

and stability selection, based on inclusion probabilities

$\boldsymbol{\Theta}(\mathbf{X})$2

and subsequent refitting on the selected library (Silva et al., 1 Jul 2025).

Within this architecture, the ensemble is not the uncertainty quantifier by itself. It supplies a stabilized point predictor or a stabilized set of coefficient estimates; conformal prediction is then used to calibrate these outputs.

2. Core conformal construction on top of an ensemble predictor

The basic conformal construction in dynamical systems is residual-based. For a trained dynamics model $\boldsymbol{\Theta}(\mathbf{X})$3, the nonconformity score is

$\boldsymbol{\Theta}(\mathbf{X})$4

and, under exchangeability of samples $\boldsymbol{\Theta}(\mathbf{X})$5, the conformal prediction set satisfies

$\boldsymbol{\Theta}(\mathbf{X})$6

With a fixed prediction $\boldsymbol{\Theta}(\mathbf{X})$7, the set becomes an $\boldsymbol{\Theta}(\mathbf{X})$8 ball,

$\boldsymbol{\Theta}(\mathbf{X})$9

where $\boldsymbol{\Xi}$0 is obtained from calibration residuals (Liang et al., 2024).

For Ensemble-SINDy, the same template is used with the ensemble mean as the point predictor. In the synthesis accompanying the PDE work, the ensemble mean is written as

$\boldsymbol{\Xi}$1

and the conformal score is defined on the ensemble-mean residual,

$\boldsymbol{\Xi}$2

The resulting prediction set is

$\boldsymbol{\Xi}$3

For multi-step trajectory segments, the same construction is applied to stacked trajectories,

$\boldsymbol{\Xi}$4

so that conformal calibration is performed at the trajectory level rather than coordinate-by-coordinate (Liang et al., 2024).

Two features of this construction are central. First, the score is global: the entire state vector, or the entire stacked trajectory, is compressed into a single scalar via an $\boldsymbol{\Xi}$5 norm. Second, the prediction set is joint in the full output space. This yields joint coverage over all outputs represented in the score. If marginal coverage is required per coordinate, separate scores

$\boldsymbol{\Xi}$6

must be calibrated separately, but the joint guarantee is then lost (Liang et al., 2024).

The same logic appears in the direct E-SINDy paper for time-series forecasting, where conformal intervals are built around the ensemble mean forecast

$\boldsymbol{\Xi}$7

using a quantile $\boldsymbol{\Xi}$8 of calibration scores and the interval

$\boldsymbol{\Xi}$9

applied componentwise (Fasel, 15 Jul 2025).

3. Time-series formulations: EnbPI, CP-PID, and sequential calibration

Classical split conformal relies on exchangeability, which is violated in time series. EnbPI addresses this by combining bootstrap ensembles with leave-one-out aggregation. With training data $$\dot{\mathbf{X}}=\boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}.$$0, bootstrap samples $$\dot{\mathbf{X}}=\boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}.$$1, base learners $$\dot{\mathbf{X}}=\boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}.$$2, and an aggregation function $$\dot{\mathbf{X}}=\boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}.$$3, the leave-one-out ensemble predictor is

$$\dot{\mathbf{X}}=\boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}.$$4

and the leave-one-out residual is

$$\dot{\mathbf{X}}=\boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}.$$5

Prediction intervals are constructed from empirical residual quantiles,

$$\dot{\mathbf{X}}=\boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}.$$6

with $$\dot{\mathbf{X}}=\boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}.$$7 chosen to minimize interval width by line search over residual quantiles. EnbPI avoids data-splitting, is computationally efficient by avoiding retraining, and is scalable to sequentially producing prediction intervals (Xu et al., 2020).

The E-SINDy paper adapts EnbPI directly to ensemble sparse models. For each time $$\dot{\mathbf{X}}=\boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}.$$8 in a sliding calibration window, each ensemble model that did not use $$\dot{\mathbf{X}}=\boldsymbol{\Theta}(\mathbf{X})\boldsymbol{\Xi}.$$9 in its bootstrap sample yields a forecast residual

$$\boldsymbol{\Xi}=\arg\min_{\hat{\boldsymbol{\Xi}}}\Big\|\dot{\mathbf{X}}-\boldsymbol{\Theta}(\mathbf{X})\hat{\boldsymbol{\Xi}}\Big\|_2^2+\lambda\|\hat{\boldsymbol{\Xi}}\|_0,$$0

where $$\boldsymbol{\Xi}=\arg\min_{\hat{\boldsymbol{\Xi}}}\Big\|\dot{\mathbf{X}}-\boldsymbol{\Theta}(\mathbf{X})\hat{\boldsymbol{\Xi}}\Big\|_2^2+\lambda\|\hat{\boldsymbol{\Xi}}\|_0,$$1 is the observed future trajectory over the prediction horizon $$\boldsymbol{\Xi}=\arg\min_{\hat{\boldsymbol{\Xi}}}\Big\|\dot{\mathbf{X}}-\boldsymbol{\Theta}(\mathbf{X})\hat{\boldsymbol{\Xi}}\Big\|_2^2+\lambda\|\hat{\boldsymbol{\Xi}}\|_0,$$2. These are averaged to obtain

$$\boldsymbol{\Xi}=\arg\min_{\hat{\boldsymbol{\Xi}}}\Big\|\dot{\mathbf{X}}-\boldsymbol{\Theta}(\mathbf{X})\hat{\boldsymbol{\Xi}}\Big\|_2^2+\lambda\|\hat{\boldsymbol{\Xi}}\|_0,$$3

and the $$\boldsymbol{\Xi}=\arg\min_{\hat{\boldsymbol{\Xi}}}\Big\|\dot{\mathbf{X}}-\boldsymbol{\Theta}(\mathbf{X})\hat{\boldsymbol{\Xi}}\Big\|_2^2+\lambda\|\hat{\boldsymbol{\Xi}}\|_0,$$4-quantile of the windowed scores is used to form the interval around the ensemble mean forecast. In reported experiments, this is applied on the stochastic predator-prey system and on longer-horizon chaotic systems including Lorenz, hyper-Rössler, and a nuclear quadrupole system (Fasel, 15 Jul 2025).

A second sequential method in the same paper is Conformal PID control. Here the interval width is controlled by an adaptive quantile parameter $$\boldsymbol{\Xi}=\arg\min_{\hat{\boldsymbol{\Xi}}}\Big\|\dot{\mathbf{X}}-\boldsymbol{\Theta}(\mathbf{X})\hat{\boldsymbol{\Xi}}\Big\|_2^2+\lambda\|\hat{\boldsymbol{\Xi}}\|_0,$$5 and the miscoverage indicator

$$\boldsymbol{\Xi}=\arg\min_{\hat{\boldsymbol{\Xi}}}\Big\|\dot{\mathbf{X}}-\boldsymbol{\Theta}(\mathbf{X})\hat{\boldsymbol{\Xi}}\Big\|_2^2+\lambda\|\hat{\boldsymbol{\Xi}}\|_0,$$6

The update is

$$\boldsymbol{\Xi}=\arg\min_{\hat{\boldsymbol{\Xi}}}\Big\|\dot{\mathbf{X}}-\boldsymbol{\Theta}(\mathbf{X})\hat{\boldsymbol{\Xi}}\Big\|_2^2+\lambda\|\hat{\boldsymbol{\Xi}}\|_0,$$7

and the interval is

$$\boldsymbol{\Xi}=\arg\min_{\hat{\boldsymbol{\Xi}}}\Big\|\dot{\mathbf{X}}-\boldsymbol{\Theta}(\mathbf{X})\hat{\boldsymbol{\Xi}}\Big\|_2^2+\lambda\|\hat{\boldsymbol{\Xi}}\|_0,$$8

Under bounded nonconformity scores, the long-run average miscoverage converges to the target,

$$\boldsymbol{\Xi}=\arg\min_{\hat{\boldsymbol{\Xi}}}\Big\|\dot{\mathbf{X}}-\boldsymbol{\Theta}(\mathbf{X})\hat{\boldsymbol{\Xi}}\Big\|_2^2+\lambda\|\hat{\boldsymbol{\Xi}}\|_0,$$9

(Fasel, 15 Jul 2025).

These time-series variants shift the role of conformal prediction. Instead of providing exact finite-sample marginal validity under exchangeability, they provide approximate marginal validity, long-run validity, or explicit coverage-gap bounds under assumptions on residual dependence and estimator quality. This is the principal route by which conformal prediction becomes usable for iterated dynamical forecasts.

4. Forecast uncertainty, feature importance, and coefficient uncertainty

The direct E-SINDy literature distinguishes three separate conformal targets: state prediction, library feature importance, and coefficient uncertainty (Fasel, 15 Jul 2025).

Target	Mechanism	Representative score
Time-series prediction	EnbPI or CP-PID on ensemble forecasts	$\lambda$0
Feature importance	LOCO or LOCO-path on SINDy library terms	$\lambda$1 or $\lambda$2
Coefficient uncertainty	feature-CP in coefficient space	$\lambda$3

For feature importance, Leave-One-Covariate-Out (LOCO) compares the full SINDy model with a reduced model omitting feature $\lambda$4. The excess prediction error is

$\lambda$5

If dropping feature $\lambda$6 increases prediction error substantially, $\lambda$7 is large. LOCO-path extends this across the regularization path: $\lambda$8 Both statistics are used as feature-importance measures rather than as forecast intervals.

For coefficient uncertainty, feature conformal prediction constructs surrogate constrained SINDy fits. For each held-out index $\lambda$9, a constrained least-squares problem is solved so that the excluded sample is fit exactly, producing $B$0. The coefficient-space nonconformity score is then

$B$1

A conformal quantile of these scores is used to calibrate coefficient intervals around the ensemble coefficient estimates. In the reported experiments, the conformalized coefficient intervals are wider than standard E-SINDy bootstrap intervals but more robust under Gaussian measurement noise, Gamma measurement noise, and Gaussian process noise (Fasel, 15 Jul 2025).

This tripartite use of conformal prediction is distinctive. In Ensemble-SINDy, conformalization is not restricted to the output space of the learned dynamics. It is also used to assess the relevance of candidate library terms and the uncertainty of identified sparse coefficients.

5. Empirical behavior in dynamical systems and hybrid models

On stochastic predator-prey dynamics, E-SINDy combined with EnbPI and CP-PID achieves empirical coverage close to nominal for time-series prediction. The experiments use a short horizon with $B$2, a calibration window of 50 steps, and prediction horizon $B$3; appendix experiments extend to horizon 10 on Lorenz, hyper-Rössler, and a nuclear quadrupole system. In these studies, interval width increases with noise intensity and with higher target coverage, EnbPI intervals are relatively stable, and CP-PID reacts more aggressively, with initially wider intervals that later shrink as the ensemble predictor improves (Fasel, 15 Jul 2025).

A physically grounded battery example gives a concrete hybrid realization. The Extended Single Particle Model is augmented by Adaptive Ensemble Sparse Identification, and conformal prediction is then applied to the residual voltage-error dynamics. With $B$4, Sequential Predictive Conformal Inference yields coverage $B$5 and average interval width $B$6 for the bagging ensemble, and coverage $B$7 and average interval width $B$8 for the stability-selection ensemble. On unseen data, the hybrid model reports mean squared error reductions of up to 46%, with test-set MSER $B$9 for ESPM + AESI I, MSER $[\dot{\mathbf{X}},\boldsymbol{\Theta}(\mathbf{X})]$0 for ESPM + AESI II, and MSER $[\dot{\mathbf{X}},\boldsymbol{\Theta}(\mathbf{X})]$1 for ESPM + tuned single SINDy-C model (Silva et al., 1 Jul 2025).

Broader dynamical-systems evidence comes from operator-learning neural PDE solvers. In the Navier–Stokes turbulence benchmark, conformal prediction is compared with snapshot ensembles and Monte Carlo Dropout on FNO, UNO, and TFNO. CP generally has the lowest MAE and RMSE across operators, larger sharpness values indicating looser intervals, substantially smaller Miscalibration Area, and near-zero Recalibration Area after isotonic regression. For FNO, the reported values are CP MA $[\dot{\mathbf{X}},\boldsymbol{\Theta}(\mathbf{X})]$2, RA $[\dot{\mathbf{X}},\boldsymbol{\Theta}(\mathbf{X})]$3, versus Ensemble MA $[\dot{\mathbf{X}},\boldsymbol{\Theta}(\mathbf{X})]$4, RA $[\dot{\mathbf{X}},\boldsymbol{\Theta}(\mathbf{X})]$5, and Dropout MA $[\dot{\mathbf{X}},\boldsymbol{\Theta}(\mathbf{X})]$6, RA $[\dot{\mathbf{X}},\boldsymbol{\Theta}(\mathbf{X})]$7. For TFNO, CP MA $[\dot{\mathbf{X}},\boldsymbol{\Theta}(\mathbf{X})]$8 and RA is approximately zero, versus Ensemble MA $[\dot{\mathbf{X}},\boldsymbol{\Theta}(\mathbf{X})]$9 and Dropout MA $\boldsymbol{\Xi}^{(b)}$0. The same study reports that CP remains effective after rotation symmetry tests when the calibration set is rotated together with the test set (Liang et al., 2024).

These results do not establish a single canonical Ensemble-SINDy pipeline, but they do show a recurring pattern: ensemble averaging or sparse ensembling stabilizes the point predictor, and conformal calibration supplies the interval or set with the coverage guarantee.

6. Guarantees, related aggregation schemes, and limitations

The strongest finite-sample guarantee comes from standard split conformal prediction under exchangeability: $\boldsymbol{\Xi}^{(b)}$1 For dynamical systems, this is typically invoked by treating transition pairs or trajectory segments as exchangeable samples. The PDE literature explicitly relies on this transition-level interpretation, and the direct Ensemble-SINDy literature uses approximate or long-run analogues when temporal dependence is unavoidable (Liang et al., 2024).

Time-series-specific guarantees are weaker but operationally more relevant. EnbPI establishes explicit bounds on conditional and marginal coverage gaps of estimated prediction intervals and similar bounds on the size of set differences between oracle and estimated prediction intervals; these bounds asymptotically converge to zero under additional assumptions, including estimator quality and i.i.d., linear-process, or strongly mixing errors (Xu et al., 2020). CP-PID provides long-run validity rather than exact finite-sample marginal validity (Fasel, 15 Jul 2025).

Several related aggregation frameworks extend the conformal-ensemble idea beyond simple bootstrap averaging. Conformal Online Model Aggregation conformalizes each expert separately and then combines the resulting prediction sets through weighted majority vote,

$\boldsymbol{\Xi}^{(b)}$2

with coverage

$\boldsymbol{\Xi}^{(b)}$3

under the stated assumptions, and with online weights updated by exponential weights or AdaHedge (Gasparin et al., 2024). Multi-model Ensemble Conformal Prediction in Dynamic Environments instead performs online model selection among multiple conformal models and proves strongly adaptive regret over all intervals while maintaining valid coverage in the adaptive-coverage sense (Hajihashemi et al., 2024). Stacked conformal prediction conformalizes a stacked ensemble of predictive models at the meta-learner level and shows approximate marginal validity without requiring the use of a separate calibration sample (F, 18 May 2025).

Several limitations recur across this literature. Exchangeability is exact only in the classical conformal setting; time-series methods replace it with approximate validity, long-run validity, or adaptive coverage error. Joint $\boldsymbol{\Xi}^{(b)}$4-ball calibration provides joint coverage over the full state or trajectory, not marginal coverage per component. A single conformal radius calibrated on short horizons can become conservative or miscalibrated at long horizons. Ensemble variance alone is not a coverage guarantee: in the PDE benchmark, ensembles and MC Dropout do not use calibration data, and their coverage can be inconsistent (Liang et al., 2024). Feature-level conformal procedures such as LOCO, LOCO-path, and feature-CP are computationally expensive because they require repeated SINDy refits or constrained surrogate solves (Fasel, 15 Jul 2025).

Taken together, the literature supports a precise interpretation of Conformal Prediction with Ensemble-SINDy: an uncertainty-quantified sparse-modeling framework in which an ensemble of SINDy models supplies stabilized forecasts or coefficients, and conformal prediction calibrates those outputs into prediction intervals, feature-importance diagnostics, or coefficient uncertainty sets under assumptions appropriate to the data regime.