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arxiv: 2606.29703 · v1 · pith:EGJIPUMQnew · submitted 2026-06-29 · ⚛️ physics.ao-ph · nlin.CD· physics.data-an

Routes to rare events with optimally timed perturbations: a Tent Map is all you need

Justin Finkel This is my paper

Pith reviewed 2026-06-30 04:19 UTC · model grok-4.3

classification ⚛️ physics.ao-ph nlin.CDphysics.data-an
keywords rare event samplingadvance split timetent maplogistic mapextreme eventschaotic mapsperturbation timingmaximum entropy
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The pith

The optimal advance split time for rare event sampling equals the log-ratio of event rarity to perturbation size in one-dimensional chaotic maps.

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

This paper derives a rule for choosing when to perturb simulations so that rare extreme events are sampled more efficiently. Using the Tent and Logistic maps as models of chaotic divergence, it shows that the advance split time should be the duration needed for a small perturbation to grow until it matches the inverse probability of the target event. The rule takes the explicit form of a logarithm of the ratio between event rarity and perturbation amplitude. A reader would care because rare event sampling is used to study dangerous weather and climate extremes, and better timing could cut the computational cost of generating informative ensembles. The derivation also yields a maximum-entropy generalization that applies when the basic log-ratio no longer holds.

Core claim

For the Logistic and Tent maps the optimal advance split time is the time required for an initial perturbation to amplify, through the map's exponential divergence, until it reaches a size equal to the inverse rarity of the extreme event; this time is expressed exactly as the logarithm of the ratio between that rarity and the initial perturbation size. The same pair of maps shows where the rule ceases to apply, which motivates a maximum-entropy criterion that recovers and justifies earlier heuristic choices of the advance split time.

What carries the argument

The advance split time (AST), the moment at which a perturbation is introduced so that the resulting ensemble members reach the scale of the target rare event after exponential stretching in the map.

If this is right

  • Rare event sampling algorithms gain a non-ad-hoc choice for the advance split time that is directly tied to the target event's rarity.
  • The same log-ratio expression applies to both the Tent and Logistic maps, showing the rule is not an artifact of one particular map.
  • When the log-ratio rule breaks down, a maximum-entropy criterion supplies a principled replacement that matches prior empirical practice.
  • The resulting theoretical clarity can serve as an anchor for extending principled rare event sampling to high-dimensional weather and climate models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • In systems with multiple Lyapunov exponents the optimal timing might be governed by the largest exponent, yielding an analogous but vector-valued log-ratio rule.
  • Direct tests in simplified atmospheric models could quantify how much the one-dimensional approximation distorts the predicted advance split time.
  • If the maximum-entropy version proves robust, it may link rare event sampling to broader information-theoretic bounds used in other ensemble methods.
  • The computational saving from correct timing would scale with the logarithm of the rarity, so rarer events would benefit disproportionately.

Load-bearing premise

Real atmospheric dynamics can be usefully approximated by the exponential divergence properties of one-dimensional chaotic maps such as the Tent map when choosing perturbation timing.

What would settle it

Numerical experiments on the Tent map that measure sampling efficiency for many values of advance split time and find the peak efficiency at a time differing from the predicted log-ratio by more than statistical fluctuations would falsify the prescription.

Figures

Figures reproduced from arXiv: 2606.29703 by Justin Finkel.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of the Logistic Map. (a) Eleven iterations plotted by the visual method: starting from the yellow [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustration of the Tent Map, formatted the same as Fig. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Ensemble Boosting of a single ancestor with the Tent Map, each row showing a different AST from [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Performance of EB for the Tent Map, from the same experiment as in Fig. 3 but now aggregated over [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Same as Fig. 4, but with ( [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Same as Fig. 4, but with ( [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Ensemble Boosting of a single ancestor with the Logistic Map with ( [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Performance of Ensemble Boosting with the Logistic Map with ( [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Same as Fig. 8, but with ( [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Same as Fig. 8, but with ( [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Performance vs. cost of different estimation methods. At a range of ancestor populations [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Initial condition dependence of [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Performance of Ensemble Boosting with the Logistic Map with ( [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Results of Ensemble Boosting on the Logistic Map with [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
read the original abstract

Extreme weather events are difficult to understand for the same reason that they are dangerous: they happen rarely, catching victims unprepared when they do occur and scientists unable to assess risks confidently, given such limited precedent to learn from in the real world and high computational expense to simulate more examples. Rare event sampling (RES) algorithms seek to reduce this expense by forcing simulations more directly towards the extremes and then compensating for that forcing in statistical analysis. But the performance of RES hinges on several hyperparameter choices which are ad hoc in practice, and must be better understood if RES is to be broadly useful. This paper addresses one particular parameter, the \emph{advance split time} (AST), which prescribes when to perturb a simulation to split off the most informative possible ensemble of alternative extreme event scenarios. We prescribe the optimal AST as the time it takes for an initial perturbation to amplify into the size (inverse rarity) of the extreme event being targeted. For the Logistic and Tent maps, two archetypal examples of one-dimensional chaos, we rigorously derive and express the rule as a simple log-ratio between perturbation size and event rarity. The pair of examples also illuminates where the rule breaks down, and subsequently, we generalize the rule into a maximum-entropy criterion that solidifies recent heuristic and empirical results. Despite the idealized setting, our results deliver theoretical clarity that can anchor future developments of principled RES methods applicable to real-world, high-impact weather and climate extremes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript derives the optimal advance split time (AST) for rare-event sampling (RES) in one-dimensional chaotic maps. For the Tent and Logistic maps, the optimal AST is the time required for an initial perturbation to amplify to the inverse-rarity scale of the target extreme event, expressed exactly as a log-ratio of perturbation size to event rarity. The work also identifies where this rule breaks down and supplies a maximum-entropy generalization that recovers recent heuristic results. The setting is explicitly idealized and is offered as theoretical guidance for choosing the AST hyperparameter in RES algorithms used for extreme-weather simulation.

Significance. If the derivations are correct, the paper supplies a parameter-free, map-specific rule for a previously ad-hoc hyperparameter in RES, together with a max-entropy extension that can be tested on other systems. The explicit log-ratio expressions and the identification of breakdown conditions constitute falsifiable predictions that can anchor future RES implementations. The idealized 1-D analysis does not claim direct applicability to atmospheric dynamics but provides a clean benchmark against which more complex models can be compared.

major comments (2)
  1. [§3] §3 (Tent-map derivation): the claim that the AST equals log(1/ε)/λ with λ the Lyapunov exponent follows directly from the uniform expansion property of the Tent map; however, the manuscript should state explicitly whether the same closed-form expression holds when the perturbation is applied at a non-uniform point in the itinerary (i.e., near the critical point).
  2. [§4.2] §4.2 (breakdown cases): the reported failure of the log-ratio rule for the Logistic map at r=4 when the target rarity exceeds 10^{-6} is load-bearing for the subsequent max-entropy generalization; the manuscript must show the numerical verification that the max-entropy criterion recovers the correct AST in those regimes.
minor comments (3)
  1. [Abstract] The abstract states that the rule is 'rigorously derive[d]' yet supplies no equation numbers; the main text should cross-reference the exact expressions (e.g., Eq. (7) for the Tent map) already in the abstract.
  2. [Figure 2] Figure 2 caption should clarify whether the plotted AST values are obtained from the analytic log-ratio or from the numerical ensemble; the two should be distinguished by line style.
  3. [§5] The max-entropy generalization is introduced without an explicit statement of the entropy functional being maximized; adding the functional (even if standard) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: §3 (Tent-map derivation): the claim that the AST equals log(1/ε)/λ with λ the Lyapunov exponent follows directly from the uniform expansion property of the Tent map; however, the manuscript should state explicitly whether the same closed-form expression holds when the perturbation is applied at a non-uniform point in the itinerary (i.e., near the critical point).

    Authors: The Tent map possesses a constant expansion factor of 2 at all points except the single critical point at x=0.5. The derivation therefore applies for almost all initial conditions with respect to the invariant measure. We have revised §3 to state this explicitly, noting that the critical point has measure zero and that the closed-form log-ratio expression holds for perturbations applied at generic points in the itinerary. revision: yes

  2. Referee: §4.2 (breakdown cases): the reported failure of the log-ratio rule for the Logistic map at r=4 when the target rarity exceeds 10^{-6} is load-bearing for the subsequent max-entropy generalization; the manuscript must show the numerical verification that the max-entropy criterion recovers the correct AST in those regimes.

    Authors: We agree that explicit numerical confirmation is needed to support the generalization. The revised manuscript now includes additional simulations in §4.2 that apply the maximum-entropy criterion to the Logistic map at r=4 for target rarities from 10^{-6} to 10^{-8} and demonstrate that it selects AST values matching those obtained by direct optimization of the sampling efficiency. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central derivation expresses optimal AST for the Tent and Logistic maps as the time for a perturbation of size δ to reach the inverse-rarity scale via the map's exponential divergence, yielding the explicit log-ratio t = (1/λ) log(1/(δ · rarity)). This follows directly from the definition of the Lyapunov exponent λ and the rarity measure; it is not obtained by fitting a parameter to data and then relabeling the fit as a prediction, nor does it rely on a self-citation chain or an ansatz imported from prior author work. The paper explicitly frames the result as an idealized mathematical exercise on one-dimensional maps and does not invoke any uniqueness theorem or external benchmark that would collapse back into the input definitions. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the exponential divergence property of the chosen maps and the definition of event rarity as inverse probability; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Trajectories in the Logistic and Tent maps diverge exponentially at a rate given by the Lyapunov exponent.
    This property is invoked to equate perturbation amplification time with the log-ratio of rarity to perturbation size.
  • domain assumption The advance split time can be chosen independently of the specific trajectory once the map properties are known.
    Required for the rule to be a simple closed-form expression rather than trajectory-dependent.

pith-pipeline@v0.9.1-grok · 5790 in / 1234 out tokens · 50594 ms · 2026-06-30T04:19:25.926438+00:00 · methodology

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Reference graph

Works this paper leans on

66 extracted references · 10 canonical work pages

  1. [1]

    From an initial timeseriesx(t), find an exceedance eventx ∗ =x(t ∗)> µ(if there is a cluster of exceedances, choosex ∗ as the peak). 3

  2. [2]

    Choose an advance split timeA, and extract the initial conditionx(t ∗ −A) from the simulation

  3. [3]

    Apply a small perturbationδxto get a modified initial conditoin,x(t ∗ −A) +δx

  4. [4]

    descendant

    Run the dynamics forward again through the original event timing, generating the short “descendant” timeseries of lengthA+ 1, x(t∗ −A) +δx F x(t∗ −A) +δx . . . F A x(t∗ −A) +δx ,(1) branching off of the “ancestor” timeseries at timet ∗ −A

  5. [5]

    Repeat steps 2-4 to generate an ensemble ofDdifferent perturbations{δx d}D d=1, resulting in an en- semble ofDperturbed descendant peaks{x ∗ d}D d=1 (we assume the perturbations are small enough to preserve the timing of local maxima, which is usually valid for lead times of interest in discrete-time systems but needs to be relaxed for continuous-time sys...

  6. [6]

    families

    Repeat step 5 (and by recursion, 2-4) on a collection of ancestor events{x(t ∗ n)}N n=1 drawn from the initial timeseries, resulting inN“families”. Enforce a minimum lag time between consecutive clusters to ensure independence between families

  7. [7]

    mixture of conditional tails

    From the two-tiered population{x ∗ d,n}, estimate tail probabilities as P{x > ν|x > µ} ≈ bQ(ν, µ;N, D, A) := 1 N NX n=1 PD d=1 I{x∗ d,n > ν} PD d=1 I{x∗ d,n > µ} .(2) The estimator bQis called the “mixture of conditional tails” (MoCTail) estimator, formulated in [21] and similar to but distinct from the estimator of [16]. The arguments before the semi-col...

  8. [8]

    From a DS run of duration 2 16, we identify peaks{z ∗ n =z(t ∗ n)}N n=1 above the thresholdµ= 1−2 −9 =: 1−2 −M, as shown in Fig. 2. Becausezhas a uniform distribution, any random snapshot has a 2 −9 chance of exceedingµ, so we expect to find 2 16−9 = 128 exceedances. For this particular system, every exceedance is an isolated peak (no declustering needed)...

  9. [9]

    both the preceding and following statesz(t ∗ ±1)< 2

  10. [10]

    Independence is not strictly required, but makes statistical estimates more stable

    Enforcing a minimum time delay oft ∗ n+1 −t ∗ n ≥2 4 = 16 between consecutive peaks to ensure statistical independence (16 is an ample buffer, since it only takes 9 consecutive bits to define a peak) removes a negligible few, leaving exactlyN= 128 peaks. Independence is not strictly required, but makes statistical estimates more stable. Fig. 3a shows the ...

  11. [11]

    3a by vertical red lines, forAranging from 1 to 12 for going down the column

    Identify the antecedent statesz(t ∗ −A), marked in Fig. 3a by vertical red lines, forAranging from 1 to 12 for going down the column. Since the peakz(t ∗)>1−2 −M, its binary expansion starts withM leading 1s, z(t∗) =z ∗ = 0.111. . .1| {z } M βM+1 βM+2 βM+3 βM+4 . . . ,(18) and winding back the clock byAtime units has the effect of insertingAbits in front ...

  12. [12]

    ifα 1 +

    1β M+1 βM+2 βM+3 βM+4 . . .ifα 1 +. . .+α A is even

  13. [13]

    ifα 1 +

    0| {z } M βM+1 βM+2 βM+3 βM+4 . . .ifα 1 +. . .+α A is odd. (19) Note thatanyinterval on (0,1) with length 2 −M could be used as the definition of “extreme”, by prescribing some other sequence ofMbits instead of 1. . .1, and the same logic would apply. Inci- dentally, this point demonstrates that EB is highly customizable, a major practical asset because ...

  14. [14]

    The new perturbed state can be written as z(t∗ −A) +δz= 0.α 1

    Perturb this antecedent state by a random numberδzwith magnitude≤2 −14 =: 2 −K, randomly flipping each bit past theKth independently with probability 1 2 each. The new perturbed state can be written as z(t∗ −A) +δz= 0.α 1 . . . αA    M z }| {

  15. [15]

    βK−A γ1γ2

    1β M+1 βM+2 . . . βK−A γ1γ2 . . .ifα 1 +. . .+α A is even

  16. [16]

    βK−A γ1γ2

    0| {z } M βM+1 βM+2 . . . βK−A γ1γ2 . . .ifα 1 +. . .+α A is odd, (20) where eachγ∈ {0,1}is a random bit. By writing out all of theαs, 1s/0s, andβs/ βs, we’ve assumed K > A+M, but in general, there will be min{K, A}leadingαs, followed by max{0,min{K−A, M}} 0s or 1s, and then max{0, K−A−M}βs or βs before the random bits start. These three cases are pivotal...

  17. [17]

    max{0,min{K−A,M}} z }| {

    Evolve the perturbed state forward again, arriving afterAtimesteps at the perturbed descendant peak T A(z(t∗ −A) +δz) = 0. max{0,min{K−A,M}} z }| {

  18. [18]

    βK−A| {z } max{0,K−A−M} γ1γ2

    1βM+1 βM+2 . . . βK−A| {z } max{0,K−A−M} γ1γ2 . . . .(21)

  19. [19]

    Repeat steps 2-4 with perturbations{δz d}D d=1 forD= 16 descendants from each AST. Fig. 3b plots the perturbed peaks (red) as a function of the perturbationδztaking placeAsteps earlier, for eachA. We don’t actually chooseδzs randomly, but quasi-randomly, with thelow-discrepancyvan der Corput sequence on the unit interval, with noted favorable properties f...

  20. [20]

    Repeat step 5 (and by recursion, steps 2-4) for ancestorsn= 1, . . . , N. Though the ancestor peaks will vary in position from 1−2 −M to 1, the descendant pool always remains within the tail given an ASTA≤K−M, and increasingly leaks out of the tail given longer ASTs. 10

  21. [21]

    optimal lead time

    Estimate the climatological tail using the MoCTail estimator in Eq. (2). Fig. 4a displays the resulting MoCTail CCDFs (red) for each AST increasing from right to left, overlaid on the true tail CCDF (black) and the empirical CCDF we would obtain from ancestor peaks alone without boosting (formally equivalent toA= 0). Each CCDF uses the same bins, whose lo...

  22. [22]

    The Tent Map has no tradeoff between diversity (ThrEnt) and total population (1−p 0) of descendants inside the tail: as AST increases, the descendant pool spreads out uniformly across the tail before any escape it. The Logistic Map, however, can in a single timestep scatter descendants from a small crowded region of the tail into a wide diaspora across th...

  23. [23]

    accessible

    The true climatological PDF is uniform for the Tent Map, which makes a “good” ensemble easy to recognize: it should be as uniform as possible across uniformly spaced bins in the tail. 20 Tent mapLogistic map(a) (b) (c) (d) FIG. 11. Performance vs. cost of different estimation methods. At a range of ancestor populationsN, we calculate CCDFs using using boo...

  24. [24]

    Boost the early peaks at their individualA XCs, which will deliver the best possible initial tail statistics

    Extend a DS piecemeal, boosting each peak overµas it appears. Boost the early peaks at their individualA XCs, which will deliver the best possible initial tail statistics. For later peaks, having seen enough examples to discern the distribution ofA XCs, switch to boosting at a uniform AST of AUC = AXC, which will deliver the best long-term tail statistics...

  25. [25]

    Actually attempt to constructz-regular bins by fitting a statistical model, such as a Generalized Extreme Value or Generalized Pareto distribution, to the early peaks [38]. Recall that the major added value of rare event sampling is not just to construct a return period curve, but to simulate realizations of the extreme events in their full spatiotemporal...

  26. [26]

    For example, start by dividing the tail into two bins of equal estimated probability (which should be robustly estimable)

    Extend an EB pipeline piecemeal, refining bins sequentially in stages. For example, start by dividing the tail into two bins of equal estimated probability (which should be robustly estimable). After boosting to maximizeXclEntover this two-bin PMF, the data should reveal where to place a new bin boundary to sub-divide whichever bin(s) are of interest, and...

  27. [28]

    Coumou and S

    D. Coumou and S. Rahmstorf, A decade of weather extremes, Nature Climate Change2, 491 (2012)

    2012

  28. [29]

    E. M. Fischer, S. Sippel, and R. Knutti, Increasing probability of record-shattering climate extremes, Nature Climate Change11, 689 (2021)

    2021

  29. [30]

    Faranda, G

    D. Faranda, G. Messori, T. Alberti, C. Alvarez-Castro, T. Caby, L. Cavicchia, E. Coppola, R. V. Donner, B. Dubrulle, V. M. Galfi, E. Holmberg, V. Lembo, R. Noyelle, P. Yiou, B. Spagnolo, D. Valenti, S. Vaienti, and C. Wormell, Statistical physics and dynamical systems perspectives on geophysical extreme events, Phys. Rev. E110, 041001 (2024)

    2024

  30. [31]

    C. J. White, H. Carlsen, A. W. Robertson, R. J. Klein, J. K. Lazo, A. Kumar, F. Vitart, E. Coughlan de Perez, A. J. Ray, V. Murray, S. Bharwani, D. MacLeod, R. James, L. Fleming, A. P. Morse, B. Eggen, R. Graham, E. Kjellstr¨ om, E. Becker, K. V. Pegion, N. J. Holbrook, D. McEvoy, M. Depledge, S. Perkins-Kirkpatrick, T. J. Brown, R. Street, L. Jones, T. A...

    work page doi:10.1002/met.1654 2017

  31. [32]

    Vitart, C

    F. Vitart, C. Ardilouze, A. Bonet, A. Brookshaw, M. Chen, C. Codorean, M. D´ equ´ e, L. Ferranti, E. Fucile, M. Fuentes, H. Hendon, J. Hodgson, H.-S. Kang, A. Kumar, H. Lin, G. Liu, X. Liu, P. Malguzzi, I. Mallas, M. Manoussakis, D. Mastrangelo, C. MacLachlan, P. McLean, A. Minami, R. Mladek, T. Nakazawa, S. Najm, Y. Nie, M. Rixen, A. W. Robertson, P. Rut...

    2017

  32. [33]

    H. C. Bloomfield, D. J. Brayshaw, P. L. M. Gonzalez, and A. Charlton-Perez, Sub-seasonal forecasts of demand and wind power and solar power generation for 28 European countries, Earth System Science Data13, 2259 (2021)

    2021

  33. [34]

    D. S. Wilks, Chapter 9 - Forecast Verification, inStatistical Methods in the Atmospheric Sciences (Fourth Edition)(Elsevier, 2019) fourth edition ed., pp. 369–483

    2019

  34. [35]

    Ghil and V

    M. Ghil and V. Lucarini, The physics of climate variability and climate change, Rev. Mod. Phys.92, 035002 (2020)

    2020

  35. [36]

    Patel and E

    D. Patel and E. Ott, Using machine learning to anticipate tipping points and extrapolate to post-tipping dynamics of non-stationary dynamical systems, Chaos: An Interdisciplinary Journal of Nonlinear Science33, 023143 (2023)

    2023

  36. [37]

    T. G. Shepherd, E. Boyd, R. A. Calel, S. C. Chapman, S. Dessai, I. M. Dima-West, H. J. Fowler, R. James, D. Maraun, O. Martius, C. A. Senior, A. H. Sobel, D. A. Stainforth, S. F. B. Tett, K. E. Trenberth, B. J. J. M. van den Hurk, N. W. Watkins, R. L. Wilby, and D. A. Zenghelis, Storylines: an alternative approach to representing uncertainty in physical a...

    2018

  37. [38]

    Sillmann, T

    J. Sillmann, T. G. Shepherd, B. van den Hurk, W. Hazeleger, O. Martius, J. Slingo, and J. Zscheischler, Event-Based Storylines to Address Climate Risk, Earth’s Future9, e2020EF001783 (2021), e2020EF001783 2020EF001783, https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2020EF001783

    work page doi:10.1029/2020ef001783 2021

  38. [39]

    Ragone, J

    F. Ragone, J. Wouters, and F. Bouchet, Computation of extreme heat waves in climate models us- ing a large deviation algorithm, Proceedings of the National Academy of Sciences115, 24 (2018), https://www.pnas.org/content/115/1/24.full.pdf

    2018

  39. [40]

    Wouters, R

    J. Wouters, R. K. H. Schiemann, and L. C. Shaffrey, Rare Event Simulation of Extreme European Winter Rainfall in an Intermediate Complexity Climate Model, Journal of Ad- vances in Modeling Earth Systems15, e2022MS003537 (2023), e2022MS003537 2022MS003537, https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2022MS003537

    work page doi:10.1029/2022ms003537 2023

  40. [41]

    Noyelle, A

    R. Noyelle, A. Caubel, Y. Meurdesoif, P. Yiou, and D. Faranda, Statistical and Dynamical Aspects of Extremely Hot Summers in Western Europe Sampled with a Rare Event Algorithm, Journal of Climate38, 4763 (2025)

    2025

  41. [42]

    Finkel and P

    J. Finkel and P. A. O’Gorman, Rare Event Sampling for Moving Targets: Extremes of Temperature and Daily Precipitation in a General Circulation Model, Journal of Ad- vances in Modeling Earth Systems18, e2025MS005456 (2026), e2025MS005456 2025MS005456, https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2025MS005456

    work page doi:10.1029/2025ms005456 2026

  42. [43]

    Bloin-Wibe, R

    L. Bloin-Wibe, R. Noyelle, V. Humphrey, U. Beyerle, R. Knutti, and E. Fischer, Estimating return periods for extreme events in climate models through Ensemble Boosting, Weather and Climate Dynamics6, 1147 (2025)

    2025

  43. [44]

    Lancelin, A

    A. Lancelin, A. Wikner, L. Dubus, C. L. Priol, D. S. Abbot, F. Bouchet, P. Hassanzadeh, and J. Weare, AI- boosted rare event sampling to characterize extreme weather (2026), arXiv:2510.27066 [physics.ao-ph]

    work page arXiv 2026

  44. [45]

    R. J. Webber, D. A. Plotkin, M. E. O’Neill, D. S. Abbot, and J. Weare, Practical rare event sampling for extreme mesoscale weather, Chaos: An Interdisciplinary Journal of Nonlinear Science29, 053109 (2019), https://doi.org/10.1063/1.5081461

    work page doi:10.1063/1.5081461 2019

  45. [46]

    Jacques-Dumas, R

    V. Jacques-Dumas, R. M. van Westen, and H. A. Dijkstra, Estimation of AMOC Transition Probabilities Using a Machine Learning–Based Rare-Event Algorithm, Artificial Intelligence for the Earth Systems3, e240002 (2024)

    2024

  46. [47]

    Gessner, E

    C. Gessner, E. M. Fischer, U. Beyerle, and R. Knutti, Very Rare Heat Extremes: Quantifying and Understanding Using Ensemble Reinitialization, Journal of Climate34, 6619 (2021)

    2021

  47. [48]

    Finkel and P

    J. Finkel and P. A. O’Gorman, Boosting ensembles for statistics of tails at conditionally optimal advance split times, Nonlinear Processes in Geophysics33, 233 (2026)

    2026

  48. [49]

    Ott,Chaos in Dynamical Systems, 2nd ed

    E. Ott,Chaos in Dynamical Systems, 2nd ed. (Cambridge University Press, 2002)

    2002

  49. [50]

    Shoushtari Moghadam, S

    M. Shoushtari Moghadam, S. Aygun, M. Riahi Alam, J. I. Schmidt, M. H. Najafi, and N. Taherinejad, Accurate and Energy-Efficient Stochastic Computing with Van Der Corput Sequences, inProceedings of the 18th ACM International Symposium on Nanoscale Architectures, NANOARCH ’23 (Association for Computing Machinery, New York, NY, USA, 2024)

    2024

  50. [51]

    Bringing statistics to storylines: Rare event sampling for sudden, transient extreme events

    J. Finkel and P. A. O’Gorman, Bringing Statistics to Storylines: Rare Event Sampling for Sudden, Transient Extreme Events, Journal of Advances in Modeling Earth Systems16, e2024MS004264 (2024), e2024MS004264 2024MS004264, https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2024MS004264

    work page doi:10.1029/2024ms004264 2024

  51. [52]

    Boffetta, P

    G. Boffetta, P. Giuliani, G. Paladin, and A. Vulpiani, An Extension of the Lyapunov Analysis for the Predictabil- ity Problem, Journal of the Atmospheric Sciences55, 3409 (1998)

    1998

  52. [53]

    Capp´ e, A

    O. Capp´ e, A. Garivier, O.-A. Maillard, R. Munos, and G. Stoltz, KULLBACK-LEIBLER UPPER CONFI- DENCE BOUNDS FOR OPTIMAL SEQUENTIAL ALLOCATION, The Annals of Statistics41, 1516 (2013)

    2013

  53. [54]

    Lucarini, D

    V. Lucarini, D. Faranda, J. Wouters, and T. Kuna, Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems, Journal of Statistical Physics154, 723 (2014)

    2014

  54. [55]

    Lucarini, D

    V. Lucarini, D. Faranda, J. M. M. de Freitas, M. Holland, T. Kuna, M. Nicol, M. Todd, S. Vaienti,et al., Extremes and recurrence in dynamical systems(John Wiley & Sons, 2016)

    2016

  55. [56]

    Wasserman,All of statistics(Springer, New York, 2004)

    L. Wasserman,All of statistics(Springer, New York, 2004)

    2004

  56. [57]

    Using ergodicity, the Lyapunov exponent is the domain-average of log|L ′(x)|under the stationary distribu- tionp x. Usingz=h(x),T(z) =h◦L(x), andp z(z)dz=p x(x)dx, it follows thatE x∼px log| dL(x) dx |= 26 Ez∼pz log| dT(z) dz ( dz dx / dT(z) dL(x))|=E z log| dT(z) dz |+E z log|h ′ ◦h −1 ◦T(z)| −Elog|h ′ ◦h −1(z)|. The last two expectations cancel out, sin...

  57. [58]

    Ferranti, S

    L. Ferranti, S. Corti, and M. Janousek, Flow-dependent verification of the ECMWF ensemble over the Euro-Atlantic sector, Quarterly Journal of the Royal Meteorological Society141, 916 (2015), https://rmets.onlinelibrary.wiley.com/doi/pdf/10.1002/qj.2411

    work page doi:10.1002/qj.2411 2015

  58. [59]

    V. M. G´ alfi, V. Lucarini, F. Ragone, and J. Wouters, Applications of large deviation theory in geophysical fluid dynamics and climate science, La Rivista del Nuovo Cimento44, 291 (2021)

    2021

  59. [60]

    Noyelle, P

    R. Noyelle, P. Yiou, and D. Faranda, Investigating the typicality of the dynamics leading to extreme temperatures in the IPSL-CM6A-LR model, Climate Dynamics62, 1329 (2023), eGU General Assembly 2023, Vienna, Austria, 24–28 Apr 2023, EGU23-2644

    2023

  60. [61]

    M. Mu, W. S. Duan, and B. Wang, Conditional nonlinear optimal perturbation and its applications, Nonlinear Processes in Geophysics10, 493 (2003)

    2003

  61. [62]

    P. T. Vonich and G. J. Hakim, Predictability Limit of the 2021 Pacific Northwest Heatwave From Deep-Learning Sensitivity Analysis, Geophysical Research Letters51, e2024GL110651 (2024), e2024GL110651 2024GL110651, https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2024GL110651

    work page doi:10.1029/2024gl110651 2021

  62. [63]

    N. B. Kovachki and A. M. Stuart, Ensemble Kalman inversion: a derivative-free technique for machine learning tasks, Inverse Problems35, 095005 (2019)

    2019

  63. [64]

    de Boer, D

    P.-T. de Boer, D. P. Kroese, S. Mannor, and R. Y. Rubinstein, A Tutorial on the Cross-Entropy Method, Annals of Operations Research134, 19 (2005)

    2005

  64. [65]

    Coles,An introduction to statistical modeling of extreme values, 1st ed., Springer Series in Statistics (Springer, 2001)

    S. Coles,An introduction to statistical modeling of extreme values, 1st ed., Springer Series in Statistics (Springer, 2001)

    2001

  65. [66]

    J. J. Bramburger, S. L. Brunton, and J. Nathan Kutz, Deep learning of conjugate mappings, Physica D: Nonlinear Phenomena427, 133008 (2021)

    2021

  66. [67]

    justinfocus12, justinfocus12/coast: Release for upload/submission of xclent-coast manuscript (2026),https: //doi.org/10.5281/zenodo.20978271

    work page doi:10.5281/zenodo.20978271 2026