Stable probabilistic emulation of partially observed dynamical systems

Mario Massy; Sara Shamekh; Fabrizio Falasca; Matthieu Blanke

cs.LGNov 6, 2026

Stable probabilistic emulation of partially observed dynamical systems

Authors: Mario Massy, Sara Shamekh, Fabrizio Falasca, Matthieu Blanke

Organizations: École polytechnique, Institut polytechnique de Paris · Courant Institute, New York University · LEAP NSF STC · LMD, ISPL, ENS, Université PSL

Abstract

Predicting the long-term behavior of partially observed dynamical systems such as the coupled atmosphere-ocean system is central to societal decision-making. Motivated by the success of deep learning emulators in weather forecasting, data-driven climate models represent a promising approach. However, climate emulation poses distinct challenges: models must account for unresolved fast variables introduced by unavoidable coarse-graining, reproduce long-term statistics, and remain stable over arbitrarily long rollouts and under external perturbations. These requirements motivate stochastic data-driven emulators, but modern generative models lack stability guarantees. We introduce forced-dissipative stochastic interpolants, a generative framework for learning stochastic reduced dynamics. Its architecture uses structural constraints inspired by turbulent dynamics, which provide stability by design while retaining the flexibility of stochastic interpolants to represent complex probabilistic dynamics. Building on the theory of Markov-chain stability, we establish a Foster--Lyapunov drift condition and prove geometric ergodicity for our emulator, together with stability under steady external forcing. We evaluate the framework on the two-timescale Lorenz-96 system, a prototype of partially observed multiscale turbulence. The resulting emulator remains stable over long integrations and under finite-amplitude forcing, with accurate prediction of forced responses of both the mean and variance.

Figures & tables

Panel of Figure 1 — Figure 1: Stationary distributios and response to large perturbations for the single-layer system.

Panel of Figure 2 — Figure 2: Stationary distributions and response to large perturbations for the two-layer system.

Figure 3: Emulation error as a function of the generative model’s capacity. The

\infty

symbol denotes rollout instability.

Figure 4: Mean and variance at the forcing site under an oscillating forcing.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Experiment	Emulator	Size	Parameters	$N_{s}$
Single-layer, stationary	Unconstrained / projected	$L=4$ , $w=120$	$48{,}400$	$100$
	Convolutional	$c=128$	$1{,}665$	$100$
	Forced-dissipative	$L=3$ , $w=32$ , $r=2$ , $M=2$	$48{,}630$	$100$
Single-layer, response	Unconstrained / projected	$L=4$ , $w=1254$	$3{,}349{,}474$	$50$
	Convolutional	$c=128$	$1{,}665$	$50$
	Forced-dissipative	$L=4$ , $w=256$ , $r=16$ , $M=8$	$3{,}347{,}688$	$50$

Appendix

Table 1: Emulators of each experiment.

L

is the number of layers,

w

the width,

c

the number of channels,

r

the rank and

M

the number of nonlinear modes.

N_{s}

is the number of sampler steps per emulator step. All forced-dissipative models use

\gamma=1

Forced-dissipative			Diffusion
$w$	$(r,M)$	Parameters	$w$	Parameters
$4$	$(1,2)$	$5{,}064$	$32$	$5{,}092$
$8$	$(2,3)$	$12{,}409$	$64$	$12{,}196$
$16$	$(3,4)$	$32{,}146$	$128$	$32{,}548$
$32$	$(4,6)$	$94{,}216$	$256$	$97{,}828$
$88$	$(7,8)$	$487{,}860$	$640$	$490{,}276$

Appendix

Table 2: Sizes of the capacity study. Both architectures have

L=3

layers of width

w

Figure 5: Ensemble mean and variance of the slow variables over all sites and times, under the oscillating forcing.

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A-UTE: Advection Informed, Uncertainty Aware Temperature Emulator

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University of New South Wales, Sydney, NSW, Australia

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Open paper AskPDF arXiv

Physics-based Earth system models (ESMs) are essential for attributing climate change and generating scenario projections, yet their reliance on high-resolution numerical integration makes multi-decadal experiments expensive. In parallel, deep learning has delivered strong gains in short-range weather forecasting; however, auto-regressive roll-outs can accumulate error and become unstable when extended to decade-scale climate emulation. We introduce A-UTE: Advection Informed, Uncertainty Aware Temperature Emulator, aimed at stable multi-year emulation across heterogeneous climate models and grid resolutions. A-UTE is trained on various physics-based models at varying spatial resolutions to emulate temperature fields over a 10-year horizon. A-UTE formulates climate emulation as a forward-time stochastic dynamical system. We propose an auto-regressive ODE-SDE surrogate in which transport dynamics are constrained by an advection consistent ODE component, while a learned neural SDE term models coarse-grained variability and cross-model discrepancy at monthly resolution. We train A-UTE under negative log-likelihood objective for principled uncertainty estimates and probabilistic evaluation. Experiments across 20 climate models show that A-UTE improves long roll-out stability and accuracy relative to relevant baselines, advancing data-driven climate emulation with explicit physical structure and uncertainty-aware predictions.

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