Mechanism-Aware Ensemble Conditioning for Data-Limited Emulation of Extreme Events

Isabella S. Thiel; Juan Bello-Rivas; Yannis G. Kevrekidis; Themistoklis P. Sapsis

cs.LGSep 25, 2026

Mechanism-Aware Ensemble Conditioning for Data-Limited Emulation of Extreme Events

Authors: Isabella S. Thiel, Juan Bello-Rivas, Yannis G. Kevrekidis, Themistoklis P. Sapsis

Organizations: Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA 02139 · Department of Applied Mathematics and Statistics, Johns Hopkins University Baltimore, MD 21218 · Department of Chemical and Biomolecular Engineering, Johns Hopkins University Baltimore, MD 21218

Abstract

Extreme events in chaotic systems are difficult to learn from short trajectories because they are controlled by transient finite-time instability rather than by frequently observed bulk dynamics. We propose a mechanism-aware conditioning plug-in framework that turns a nudged coarse ensemble into a non-intrusive sensor of local instability geometry. In the small-noise regime, the ensemble covariance aggregates the same finite-time deformation kernels that govern local instability, providing a Jacobian-free proxy for the local amplification structure around a synchronized coarse trajectory. A small FiLM module injects statistics of this ensemble geometry into an otherwise unchanged backbone while leaving the coarse simulator unchanged. We demonstrate this interface in two distinct pipelines: a Transformer-style residual-attention corrector for a controlled low-dimensional chaotic system and a probabilistic recurrent STORN corrector for topographic two-layer quasi-geostrophic (QG) flow. In the low-dimensional benchmark, ensemble covariance directions co-activate with OTD modes and FiLM conditioning improves 99th-percentile exceedance-frequency errors over an identical no-context Transformer baseline. In QG, a fixed ensemble-conditioned FiLM-STORN model trained on only $50$ time units substantially improves long-horizon rare-event statistics in the data-limited regime, including density-tail errors, exceedance frequencies, and spatial exceedance-area distributions relative to an unconditioned STORN trained on the same data; on averaged high-threshold exceedance diagnostics, it also outperforms the baseline STORN trained with $20$ times more high-resolution data. These results show that local instability geometry is not merely interpretable post hoc, but an actionable conditioning signal for data-efficient rare-event emulation.

Figures & tables

Figure 1: Schematic of the ensemble-based correction framework. At each time instant, a nudged-coarse trajectory is constrained toward the reference trajectory. An ensemble of stochastically perturbed trajectories (dashed) is evolved around this nudged solution, whose spread at each instant approximates the leading-order unstable subspace (pink ellipsoids).

Figure 2: Plug-in backbone instantiation. The nudged ensemble summary entering a FiLM module attached to baseline network.

Figure 3: Low-dimensional validation of the ensemble-derived instability proxy comparing the component-wise magnitudes of the leading OTD mode $q_{\mathrm{OTD}}(t)$ and the leading ensemble-covariance mode $q_{\mathrm{ens}}(t)$ near a finite-time instability burst ( $\tau=8.0$ , $\sigma=5\cdot 10^{-6}$ ). The objective is not pointwise amplitude matching, but temporal localization of the unstable episode. The ensemble-derived mode activates over the same finite-time window as the OTD mode, indicating that the ensemble detects the relevant instability.

Diagnostic	No-FiLM attention	Best FiLM setting	Best FiLM value	Reduction
$L_{1_{\log}}(z_{3})$	$6.903\pm 0.100$	eigvec, $\sigma=10^{-6}$ , $M=2$	$6.727\pm 0.067$	$2.5\%$
$E_{99}(z_{3})$	$0.0056\pm 0.0017$	mean/std/eigs, $\sigma=10^{-5}$ , $M=2$	$0.0041\pm 0.0016$	$26.3\%$

Table 1: Sweep-selected low-dimensional

z_{3}

tail diagnostics. Entries are mean

\pm

standard error over five independent runs. Training was conducted on a trajectory of length

T_{\rm train}=150

while being tested on an independent trajectory of length

T_{\rm test}=500

. This table is a sensitivity summary, not the primary model-selection protocol. See Tables 6 and 7 for more detailed results.

Method	$T_{\mathrm{train}}$	$(\sigma,M)$	$D_{\mathrm{KL}}\downarrow$	$L_{\log}\downarrow$	$\psi_{2}\,\bar{E}_{\mathrm{AOT}}$ $\downarrow$	$\psi_{2}\,\bar{E}_{\mathrm{freq}}$ $\downarrow$	Avg. $\bar{E}_{\mathrm{AOT}}$ $\downarrow$	Avg. $\bar{E}_{\mathrm{freq}}$ $\downarrow$
STORN ∗	1000	—	0.00393	5.3454	0.00501	0.10564	0.01195	0.11726
STORN	50	—	0.07079	11.970	0.00589	0.14660	0.01574	0.12118
FiLM-STORN (ours)	50	$(0.01,2)$	0.03646	8.2959	0.00391	0.06874	0.01047	0.07199

Table 2: Held-out QG comparison emphasizing high-threshold extreme-event diagnostics. Lower is better for all metrics. The STORN and FiLM-STORN models use the same previously validated STORN backbone Barthel Sorensen et al. (2024) , loss, optimizer and all training hyperparameters were taken from the baseline configuration optimized for the long-data STORN model with

T_{\mathrm{train}}=1000

. The FiLM variant does not retune the backbone or training procedure, and differs only by the addition of the ensemble-conditioned FiLM interface and the ensemble parameters

(\sigma,M)=(0.01,2)

. With only

T_{\mathrm{train}}=50

, FiLM-STORN improves the threshold-exceedance area and event-frequency errors relative to STORN trained on 20 times longer trajectory data. The full sweep over

\sigma

and

M

is reported in Appendix E.2 .

Panel of Figure 4 — Figure 4: Comparison of corrected trajectories across ensemble sizes and noise intensities for KL divergence 4(a) and log-L1 errors 4(b) , both normalized relative to the $T=1000$ baseline.

Diagnostic	Metric	CR	STORN $T=50$	STORN $T=1000$	FiLM-STORN $T=50$
$\psi_{2}$	$\overline{\mathcal{E}}_{\mathrm{AOT}}$	0.00970	0.00589	0.00501	0.00391
$\psi_{2}$	$\overline{\mathcal{E}}_{\mathrm{freq}}$	0.25215	0.14660	0.10564	0.06874
Avg. $\psi_{1},\psi_{2}$	$\overline{\mathcal{E}}_{\mathrm{AOT}}$	0.03115	0.01574	0.01195	0.01047
Avg. $\psi_{1},\psi_{2}$	$\overline{\mathcal{E}}_{\mathrm{freq}}$	0.40190	0.12118	0.11726	0.07199

Table 3: Extreme-event geometry errors over high thresholds

c\in\{1.0,1.5,1.75\}

\overline{\mathcal{E}}_{\mathrm{AOT}}

is the mean empirical Wasserstein-1 distance between the distributions of exceedance area

A_{c}(t)/A

, and

\overline{\mathcal{E}}_{\mathrm{freq}}

is the mean absolute error in event frequency

\mathbb{P}(A_{c}(t)>0)

. Lower is better. The lower layer

\psi_{2}

is reported separately because the topographic forcing acts directly in that layer; the two-layer average checks that the improvement is not isolated to one field. FiLM-STORN uses

T_{\mathrm{train}}=50

\sigma=0.01

, and

N_{\mathrm{add}}=2

Panel of Figure 5 — Figure 5: Threshold-exceedance area distributions across increasing amplitude thresholds. Each curve shows the empirical distribution of $\mathrm{AOT}_{c}(t)=A_{c}(t)/A$ , the fraction of the domain exceeding threshold $c$ . Black is the projected high-resolution reference, dashed black is the uncorrected coarse trajectory, blue is STORN trained with $T_{\mathrm{train}}=1000$ , orange is STORN trained with validation-selected configuration $T_{\mathrm{train}}=50$ , and green is FiLM-STORN trained with $T_{\mathrm{train}}=50$ , $\sigma=0.01$ , and $N_{\mathrm{add}}=2$ . The separation at the highest thresholds shows the rare-event regime: the short-data STORN baseline loses large exceedance regions, whereas ensemble conditioning recovers the reference tail more accurately.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Visualization of the 3D dynamical system with slow–fast dynamics. Left: reference system; right: coarse, damped system. Color corresponds to the $z_{3}$ axis magnitude.

c_{t}=\begin{cases}(\mu_{t},s_{t},\lambda_{1}(t),\lambda_{2}(t)),&\mathrm{ctx}=\mathrm{mean/std/eigs},\\ (v_{1}(t),\ldots,v_{r}(t)),&\mathrm{ctx}=\mathrm{eigvec}.\end{cases}

Appendix

Algorithm 2 FiLM residual-attention training with ensemble-derived context

Model setup
Parameter	Value / Description
State variable	$z=(z_{1},z_{2},z_{3})^{\top}$
Coarse model	reference vector field with additional damping $-\gamma z_{3}$ in the third equation
Physical parameters
$a_{1},a_{2}$	$2,2$
$\epsilon$	$0.05$

Appendix

Table 4: Configuration of the low-dimensional controlled system and ensemble probe. The coarse model is deliberately biased through additional damping in the intermittently growing

z_{3}

direction, creating a non-intrusive correction task while preserving the finite-time instability structure of the reference dynamics.

Backbone
Parameter	Value / Description
Architecture	residual temporal attention corrector
Input state	$z_{t}^{\rm c}\in\mathbb{R}^{3}$
Hidden width $d_{\rm hid}$	$96$
Attention heads	$4$
Attention layers	$3$

Appendix

Table 5: Architecture parameters for the low-dimensional residual-attention corrector. The no-FiLM and FiLM models share the same temporal backbone and output parameterization; the FiLM variants add only the ensemble-derived context interface.

$T_{\rm train}$	Diagnostic	No-FiLM attention	Best FiLM setting	Best FiLM value	Reduction
5	$L_{\log}(z_{3})$	$6.985\pm 0.135$	eigvec, $\sigma=5{\times}10^{-6}$ , $M=16$	$6.952\pm 0.195$	$0.5\%$
5	$E_{99}(z_{3})$	$0.0042\pm 0.0005$	eigvec, $\sigma=10^{-5}$ , $M=4$	$0.0039\pm 0.0011$	$6.8\%$
150	$L_{\log}(z_{3})$	$6.903\pm 0.100$	eigvec, $\sigma=10^{-6}$ , $M=2$	$6.727\pm 0.067$	$2.5\%$
150	$E_{99}(z_{3})$	$0.0056\pm 0.0017$	mean/std/eigs, $\sigma=10^{-5}$ , $M=2$	$0.0041\pm 0.0016$	$26.3\%$

Appendix

Table 6: Sweep-selected low-dimensional

z_{3}

tail diagnostics. Entries are mean

\pm

standard error over the number of repeats available in the aggregate file. This table is a sensitivity summary, not the primary model-selection protocol. The fixed-configuration result in Table 7 should be used as the main repeated-run ablation.

Coordinate	Model	$D_{\rm KL}$ $\downarrow$	$L_{\log}$ $\downarrow$	$E_{99}$ $\downarrow$
$z_{1}$	Residual attention	$2.336\pm 0.243$	$7.103\pm 0.444$	$0.0084\pm 0.0023$
$z_{1}$	FiLM attention, mean/std/eigs	$\mathbf{1.884\pm 0.025}$	$\mathbf{6.237\pm 0.110}$	$0.0056\pm 0.0015$
$z_{1}$	FiLM attention, eigvec	$2.048\pm 0.341$	$6.745\pm 0.592$	$\mathbf{0.0048\pm 0.0007}$
$z_{2}$	Residual attention	$2.728\pm 0.064$	$5.755\pm 0.251$	$0.0156\pm 0.0017$
$z_{2}$	FiLM attention, mean/std/eigs	$2.430\pm 0.238$	$6.015\pm 0.455$	$0.0137\pm 0.0017$
$z_{2}$	FiLM attention, eigvec	$\mathbf{2.112\pm 0.107}$	$\mathbf{4.993\pm 0.205}$	$\mathbf{0.0112\pm 0.0018}$

Appendix

Table 7: Low-dimensional residual-attention ablation at

T_{\rm train}=50

\sigma=10^{-5}

, and

M=4

. All entries are mean

\pm

standard error over

n=5

independent training runs.

E_{99}

is the absolute 99th-percentile exceedance-frequency error. Lower is better. Bold denotes the best value among the no-FiLM baseline and the two FiLM-conditioned variants. The point of this table is not sweep selection: it fixes one ensemble configuration and shows that FiLM conditioning improves the residual-attention backbone across tail-density and exceedance diagnostics.

Figure 7: Bottom topography used in the quasi-geostrophic benchmark. The lower-layer forcing is constructed from seven Gaussian bumps, producing spatially inhomogeneous and anisotropic dynamics that are substantially more challenging than standard two-layer QG test cases without topography. This configuration is designed to probe whether the proposed method can recover extreme-event statistics in the presence of localized, topography-induced variability.

Model setup
Parameter	Value / Description
Spatial domain	$[0,2\pi]^{2}$ (doubly periodic)
Layers	2
State variables	$\psi_{1},\psi_{2},q_{1},q_{2}$
Physical parameters
Bottom drag $r$	$0.1$

Appendix

Table 8: Configuration of the two-layer quasi-geostrophic (QG) model with Gaussian bump topography. The system is evolved on a doubly periodic domain using a Fourier spectral discretization and fourth-order Runge–Kutta time stepping. Bottom topography consists of a superposition of seven localized Gaussian features, introducing spatial heterogeneity and dynamical complexity relative to usual setting without topology. Parameters are chosen to approximate mid- and high-latitude atmospheric circulation subject to an imposed shear current.

Figure 8: Runtime and peak memory versus number of additional ensemble members $N_{\mathrm{add}}$ . Vertical bars indicate variability across noise levels $\sigma$ .

Data and sequence setup
Component	Value / Description
Grid resolution	$N_{x}=N_{y}=24$ , two layers, $d=2N_{x}N_{y}=1152$
Window length $H$	100, (adaptively reduced if too few windows are available, i.e. $T_{\mathrm{train}}=50$
ML time step $\Delta t_{\mathrm{ML}}$	0.1, corresponding to stride $s=\Delta t_{\mathrm{ML}}/\Delta t_{\mathrm{DNS}}=10$
Training horizons	$T_{\mathrm{train}}\in\{50,200,1000\}$
Test horizon	$T_{\mathrm{test}}=50\,000$

Appendix

Table 9: Machine learning hyperparameters and architecture settings for the QG experiments.

$T$	baseline	$N_{\mathrm{add}}$	$\sigma=0.001$	$\sigma=0.01$	$\sigma=0.05$	$\sigma=0.1$	$\sigma=0.2$
$T=50$	0.07079	1	0.036477	0.036453	0.057637	0.071921	0.78613
		2	0.036464	0.036460	0.074342	0.063390	0.74884
		4	0.036473	0.051943	0.087202	0.051904	0.65616
		8	0.036442	0.053113	0.061234	0.071854	0.72615
$T=200$	0.10011	1	0.032722	0.066066	0.087975	0.094977	0.083387
		2	0.063504	0.045316	0.091538	0.15473	0.13318

Appendix

Table 10: Global KL divergence,

D_{\mathrm{KL}}

by training time trajectory length

T

, noise-level

\sigma

, and number of additional ensemble members

N_{\mathrm{add}}

. Baseline is the deterministic STORN model; the remaining columns are the FiLM modulated ensemble strategy.The reported values are averaged over five independent runs with negligible variation across runs.

$T$	baseline	$N_{\mathrm{add}}$	$\sigma=0.001$	$\sigma=0.01$	$\sigma=0.05$	$\sigma=0.1$	$\sigma=0.2$
$T=50$	11.970	1	8.4110	8.4510	7.9457	9.5331	36.176
		2	8.3122	8.2959	7.7896	8.8488	35.336
		4	8.1277	12.310	6.6458	11.571	32.948
		8	8.2253	15.228	8.4209	11.450	29.796
$T=200$	13.828	1	7.5888	9.2690	11.450	13.215	12.623
		2	9.4531	5.7103	14.582	18.820	12.733

Appendix

Table 11: Global

\log(L_{1})

by training time trajectory length

T

, noise-level

\sigma

, and number of additional ensemble members

N_{\mathrm{add}}

. Baseline is the deterministic STORN model; the remaining columns are the FiLM modulated ensemble strategy. Lower is better. The reported values are averaged over five independent runs with negligible variation across runs.

Diagnostic	Model	$\mathcal{E}_{\mathrm{AOT}}(1.0)$	$\mathcal{E}_{\mathrm{AOT}}(1.5)$	$\mathcal{E}_{\mathrm{AOT}}(1.75)$	$\mathcal{E}_{\mathrm{freq}}(1.0)$	$\mathcal{E}_{\mathrm{freq}}(1.5)$	$\mathcal{E}_{\mathrm{freq}}(1.75)$
$\psi_{1}$	CR	0.13839	0.01553	0.00389	0.70602	0.60802	0.34090
	STORN $T=50$	0.06560	0.00861	0.00259	0.02953	0.15682	0.10090
	STORN $T=1000$	0.04683	0.00766	0.00220	0.04715	0.20020	0.13932
	FiLM-STORN $T=50$	0.04564	0.00399	0.00144	0.00202	0.09209	0.13165
$\psi_{2}$	CR	0.02876	0.00032	$2.67{\times}10^{-5}$	0.72329	0.02798	0.00517
	STORN $T=50$	0.01732	0.00031	$2.66{\times}10^{-5}$	0.40766	0.02704	0.00511

Appendix

Table 12: Threshold-resolved exceedance-area diagnostics corresponding to Table 3 . For threshold

c

\mathcal{E}_{\mathrm{AOT}}(c)

is the empirical Wasserstein-1 distance between the reference and model distributions of

A_{c}(t)/A

, while

\mathcal{E}_{\mathrm{freq}}(c)

is the absolute error in event frequency

\mathbb{P}(A_{c}(t)>0)

. Lower is better. FiLM-STORN denotes the ensemble-conditioned model trained with

T_{\mathrm{train}}=50

\sigma=0.01

, and

N_{\mathrm{add}}=2

. Bold entries mark the best value among the four reported models for each diagnostic and threshold.

Diagnostic	$T_{\mathrm{train}}$	Metric	Best ∗ $(\sigma,N_{\mathrm{add}})$	Best ∗ value
$\psi_{2}$	50	$\overline{\mathcal{E}}_{\mathrm{AOT}}$	$(0.1,8)$	0.00202
$\psi_{2}$	50	$\overline{\mathcal{E}}_{\mathrm{freq}}$	$(0.01,2)$	0.06874
Avg. $\psi_{1},\psi_{2}$	50	$\overline{\mathcal{E}}_{\mathrm{AOT}}$	$(0.05,1)$	0.00862
Avg. $\psi_{1},\psi_{2}$	50	$\overline{\mathcal{E}}_{\mathrm{freq}}$	$(0.1,4)$	0.06456
$\psi_{2}$	200	$\overline{\mathcal{E}}_{\mathrm{AOT}}$	$(0.001,1)$	0.00344
$\psi_{2}$	200	$\overline{\mathcal{E}}_{\mathrm{freq}}$	$(0.001,1)$	0.07810

Appendix

Table 13: Best FiLM-STORN configurations over the reported

(\sigma,N_{\mathrm{add}})

sweep, evaluated using high-threshold averages over

c\in\{1.0,1.5,1.75\}

. This table summarizes achievable performance under the sweep and is not used as the primary comparison, where

\sigma=0.01

and

N_{\mathrm{add}}=2

are fixed for the

T_{\mathrm{train}}=50

FiLM-STORN model. ∗ Best denotes the best configuration over the reported sweep, used here to summarize the achievable gain

Figure 9: Global log-PDF comparison for the fixed short-data configuration $T_{\mathrm{train}}=50$ , $\sigma=0.01$ , and $N_{\mathrm{add}}=2$ . The logarithmic scale highlights low-probability tails, where ordinary linear-scale PDFs can make substantial multiplicative errors appear visually small. Agreement in these plots supports tail-distribution fidelity, while the AOT diagnostics in Tables 12 – 13 test the corresponding spatial extent of thresholded events.

Figure 10: Regional log-PDF comparison for the upper stream-function $\psi_{1}$ for the fixed short-data configuration $T_{\mathrm{train}}=50$ , $\sigma=0.01$ , and $N_{\mathrm{add}}=2$ . The domain is partitioned into nine subregions, and each panel compares the projected reference, the uncorrected coarse trajectory, a STORN baseline trained on $T_{train}=1000$ , a STORN baseline trained on $T_{train}=50$ , and the FiLM-STORN correction trained on the same short-data configuration. These plots test whether the correction recovers spatially heterogeneous tail behavior rather than only matching a domain-averaged distribution.

Figure 11: Regional log-PDF comparison for the lower stream-function $\psi_{2}$ subject to topography for the fixed short-data configuration $T_{\mathrm{train}}=50$ , $\sigma=0.01$ , and $N_{\mathrm{add}}=2$ . The domain is partitioned into nine subregions, and each panel compares the projected reference, the uncorrected coarse trajectory, a STORN baseline trained on $T_{train}=1000$ , a STORN baseline trained on $T_{train}=50$ , and the FiLM-STORN correction trained on the same short-data configuration. These plots test whether the correction recovers spatially heterogeneous tail behavior rather than only matching a domain-averaged distribution.

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Abstract

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