This work composition the observables'likelihoods in per-task free-routed last-layer beliefs on a shared backbone absorbs unit-dependent loss scaling into likelihood parameters learned in the same gradient pass, and results land where theory puts them.
Abstract
Forecasting a stochastic dynamical system rarely means a single number: one wants several observables---future state, threshold event, regime label---each with its own likelihood. Standard multi-task recipes balance per-task losses, tuned or learned. We instead compose the observables'likelihoods in per-task free-routed last-layer beliefs on a shared backbone; this absorbs unit-dependent loss scaling into likelihood parameters learned in the same gradient pass. Stochastic dynamics supply what static benchmarks cannot: computable ground truth for the predictive variance. Results land where theory puts them: on the well-specified, homoscedastic Ornstein--Uhlenbeck process the learned predictive law recovers the analytic kernel and correctly specified baselines tie. On heteroscedastic systems (stochastic Lorenz-63, real air-quality data) the belief's input-dependent variance separates: best single-run NLL on the state and regime tasks, calibration matched only by arms whose NLL it beats, at a fraction of the tuned grids'cost. On the real series the state margin holds across five rolling origins.
A calibrated stochastic world model can reveal how uncertain a future is without revealing why it branches. The same conditional future law can arise because an observation aliases physical states or because dynamics remain random after the declared full state is fixed. We prove that ordinary transitions cannot identify these two sources, even for a perfect probabilistic predictor. ClosurePairs makes them identifiable by crossing compatible microstates with repeated exogenous disturbances and estimating state, noise, and state-noise interaction variance. The central consequence is operational: under finite hierarchical sampling, forecast difficulty governs the useful compute scale, while the alias/process composition provides complementary information about its direction-resolving the current state or sampling future randomness. ClosurePairs recovers source attribution at unchanged likelihood, reduces equal-budget decomposition error in a nonlinear interaction benchmark, and supports observation-only routing. On exact-marginal MetaWorld twins, an output-only allocator is at chance while a Closure-supervised probe on frozen JEPA-WM features routes 89.8-100%. In an independent ManiSkill PushCube confirmation, a stochastic RSSM's outputs and latents remain at chance, whereas an RGB-only Closure probe routes 100% under both ID and geometry/camera OOD over five seeds, matching direct allocation rather than exceeding it. Across five unseen allocation menus, the same Closure probe routes 92.5%/90.4% ID/OOD with no new oracle labels, versus 37.9%/32.9% for a frozen direct allocator. ClosurePairs is therefore an identifiable, reusable mechanism target that cannot be recovered from forecast quality alone.
While stochastic diffusion samplers such as DDPM better preserve the enstrophy spectrum during rollouts in the stochastic setting, deterministic samplers such as DDIM and DPM-2 show better spectral preservation in the deterministic setting.
S. Pfister, Benjamin J. Holzschuh, Nils Thürey· 0 citations
Machine-learning weather models (MLWMs) now match or outperform operational numerical weather prediction (NWP) at global medium-range forecasting, at far lower inference cost. Many deployed MLWMs are deterministic, producing a single forecast with no estimate of its own uncertainty, whereas a growing family of trained-probabilistic models generate calibrated ensembles directly, at the price of a dedicated training run. We ask instead how much uncertainty can be extracted from a deterministic checkpoint that already exists, without retraining it. Where physical ensembles represent model uncertainty by stochastically perturbing parametrisation tendencies, we perturb the network's raw weight tensors at inference time, a scheme we call stochastically perturbed weights (SPW). We also ask whether it works, where and on which scales to inject the noise, and where it fails. A three-phase ablation across four deterministic backbones, Aurora, GraphCast, SFNO, and AIFS, selects one production baseline per model, benchmarked against the trained-probabilistic AIFS-ENS, FourCastNet 3 and Atlas as well as the operational ECMWF ensemble (IFS-ENS) over 112 initialisation times. At a 240 h (10-day) lead time the SPW ensembles reach continuous ranked probability skill scores (CRPSS) between 0.04 and 0.13 below the best trained-probabilistic baseline, at zero marginal training cost. No injection site works across models: the productive tensor group is architecture-specific, so SPW is at present a tuning procedure rather than a plug-and-play recipe. Its main failure mode is a coherent whole-field offset that overdisperses the domain mean, and restricting the noise to coarse scales or perturbing the initial conditions each repair part of it.
Simon Adamov, O. Fuhrer, R. Knutti et al.· 0 citations
The Observable Matrix Dynamics (OMD) approach monitors the time development of complex non-linear systems through the trajectory of a fixed-size distance matrix and its spectrum. We apply it to the S\&P 500 cross section over three crisis decades, the 2001 dot-com bust, the 2007--2008 financial crisis, and the 2020 Covid crash, with three fixed-size observables on a fixed universe. The arccos distance matrix of the rolling return correlations reads the correlation geometry: its effective dimension collapses at the 2008 and 2020 crises, while the 2001 bust is a dispersed unwind. Read against machine-learning distance matrices, its spectrum stays in the un-relaxed, pre-learning regime with no low-dimensional manifold, so the market never learns its correlation structure or relaxes to a stationary geometry. Subtracting the market factor exposes a coherent sector rotation, whose name-level attribution identifies which stocks drive each crisis and in what order. At a short lookback these signals resolve precursors and forecast the endogenous 2008 crisis, though not the exogenous 2020 shock. The other two observables model the daily return and volatility rankings as Markov chains on their ranking spaces. The return chain has persistent, defensive-led bellwethers and near-reversible dynamics. The volatility chain is far more persistent, led by the financial sector, and is the only one to carry a weak, episodic arrow of time, flaring at market stress and matching volatility clustering and the Zumbach effect. All three matrices show coherent changes during market crashes.
Stochastic volatility models provide a useful benchmark for mechanistic interpretability under noisy latent dynamics and partial observability, and output-head replacement shows that part of the degradation under noisy MSE training arises from readout misalignment rather than representation failure.
Susceptible Architectures (SUSA), a reservoir-design principle for volatility forecasting, and its two concrete implementations, based on complex-valued open-chain and periodic reservoirs and regime-conditioned experts to interpret reservoir features across calm, onset, recovery, and persistent-stress states are introduced.
Aliaksei Kaliutau· arXiv.org· 0 citations
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