Data assimilation (DA) addresses the problem of sequentially estimating the state of a dynamical system from noisy and incomplete observations. In this work, we employ a diffusion model as a world model to simulate and predict the system's dynamics. Recently, score-based diffusion models have learned global diffusion priors that effectively model (stochastic) dynamics, revealing strong potential for data assimilation. In this paper, we investigate how information from noisy observations can be incorporated to enable continuous correction and refinement of the predicted system state when using a diffusion prior. Motivated by particle filtering methods, we represent the posterior distribution using a set of particles. After receiving noisy observations, the diffusion model is guided using the observation likelihood to steer the generation process toward observation-consistent states. Nevertheless, such guidance does not guarantee sampling from the true posterior. We therefore employ a Sequential Monte Carlo approach over the diffusion trajectory, viewed as a path measure, to reweight and resample particles, thereby correcting the generation process and ensuring convergence toward the desired posterior distribution. This leads to an unbiased particle filtering method that rigorously fuses observational data with diffusion model simulations.

Time series forecasting drives operational decisions in areas like finance, transportation, and energy. While supervised learning approaches achieve strong performance, they require domain-specific training, feature engineering, and ongoing maintenance. Large-scale foundation models have recently emerged as a zero-shot alternative, avoiding task-specific training much like LLMs. In this work, we evaluate foundation models against standard supervised approaches. Rather than focusing solely on aggregate accuracy, we analyze performance across four operational regimes: periodic human-centric systems, physically constrained processes, stochastic financial markets, and heterogeneous demand forecasting. Our results characterize optimal deployment areas. Foundation models perform well in domains with transferable periodic structures and are efficient for cold-start or long-tail scenarios. Conversely, supervised specialists maintain higher precision in systems governed by strict physical constraints. In financial domains, newer foundation models are rapidly closing the performance gap with supervised specialists. We further quantify trade-offs in inference latency, data drift adaptability, and deployment constraints. Finally, we propose a Complexity Router that assigns each series to the optimal model class using empirical features. We demonstrate that this selective routing achieves higher accuracy and significantly lower inference costs compared to deploying a universal foundation model, providing a practical framework for balancing generalization and efficiency.

Microsimulation models used by ministries of finance and central banks rely on parametric processes for lifetime earnings that capture only first and second moments of the conditional distribution and miss long-range nonlinear structure. We propose SAGA, a decoder-only transformer for irregular tabular panel sequences, paired with a split conformal calibration wrapper that delivers individual-level prediction intervals with finite-sample marginal coverage guarantees. Trained on the longitudinal Swedish LISA register over 1990 to 2022, comprising 2,143,817 individuals and 61,284,903 person-years, the model forecasts annual labor earnings at horizons of one to thirty years and aggregates them by Monte Carlo into present-discounted lifetime earnings distributions. Against the canonical Guvenen, Karahan, Ozkan, and Song parametric process and tabular and recurrent baselines, SAGA reduces continuous ranked probability score by 31.9 percent at the ten-year horizon and mean absolute error by 37.7 percent at the twenty-year horizon. Conformal intervals achieve nominal coverage to within 0.4 percentage points marginally and within 2.4 percentage points on the worst-case demographic subgroup. The reconstructed lifetime earnings Gini coefficient is 0.327 against the partially observed truth of 0.341 and the GKOS estimate of 0.378. Model weights, calibration tables, and a synthetic equivalent dataset are released for replication outside the protected SCB MONA environment.

We introduce DeRegiME -- Deep Regime Mixture of Experts -- a direct multi-horizon probabilistic forecaster that separates latent uncertainty regimes from the underlying signal and softly assigns each forecast location to learned recurring regimes using a sparse variational Gaussian process (GP) whose nonstationary regime-mixing kernel and Student-t likelihood combine per-regime sub-kernels and noise processes via a shared gate. This yields a single sparse-GP posterior, not a mixture of GP experts. DeRegiME addresses a key limitation of neural forecasters: point forecasts discard residual uncertainty, and probabilistic heads -- whether single marginals, uninterpreted mixtures, quantile sets, or diffusion samples -- rarely expose the regime structure of the residual. Yet distribution shift in noisy heteroskedastic time series may be abrupt, gradual, or horizon-dependent and often appears in residual uncertainty rather than the conditional mean. DeRegiME yields an interpretable mean-residual-noise decomposition with a direct-sum feature-space representation that anchors regimes as clusters of residual similarity whose transitions surface as implicit changepoints. The effective number of regimes is pruned by the stick-breaking gate. We prove kernel validity and predictive-density propriety, and across ten benchmarks and three encoder grids DeRegiME improves negative log predictive density (NLPD) by 20.3% over the strongest encoder-matched baseline, a DeepAR/GluonTS-style dynamic Student-t head, with parallel gains on CRPS (3.0%) and MSE (4.7%). Improvements are consistent across all datasets, which span abrupt, gradual, and seasonal shifts.

Irregular multivariate time series impose a trade-off for long-horizon forecasting: discrete methods can distort temporal structure via re-gridding, while continuous-time models often require sequential solvers prone to drift. To bridge this gap, we present Latent Laplace Diffusion (LLapDiff), a generative framework that models the target as a low-dimensional latent trajectory, enabling horizon-wide generation without step-by-step integration over physical time. We guide the reverse process utilizing a stable modal parameterization motivated by stochastic port-Hamiltonian dynamics, and parameterize its mean evolution in the Laplace domain via learnable complex-conjugate poles, enabling direct evaluation over irregular timestamps. We also link continuous dynamics to irregular observations through renewal-averaging analysis, which maps sampling gaps to effective event-domain poles and motivates a gap-aware history summarizer. Extensive experiments show that LLapDiff improves over baselines in long-horizon forecasting, and its continuous-time generative nature supports missing-value imputation by querying the same model at historical timestamps. Code is available at https://github.com/pixelhero98/LLapDiffusion.

Many decision-facing stochastic systems are observed through aggregate distributions rather than scalar trajectories: queue occupancies, mobility shares, publichealth mixtures, generation-source shares, ecological compositions, and air-quality severity profiles all live on the probability simplex and evolve over time. We study causal (time-respecting online) forecasting for these distribution-valued time series and argue that the transition operator itself should be structured around the simplex. We introduce CAST (Causal Anchored Simplex Transport), a successor-local operator that (i) retrieves empirical successors from causal context, (ii) stabilizes them with a persistence anchor, and (iii) applies a bounded local stochastic transport on ordered supports; every stage preserves the simplex by construction. We identify a structural failure mode, latent transition-kernel aliasing, where similar observed distributions evolve differently under different contextual regimes, and prove that any forecaster depending only on an aliased summary incurs an irreducible weighted Jensen-Shannon excess-risk lower bound, while the CAST hypothesis class contains the regime-aware Bayes successor; for ordered supports an additional Pinsker separation holds whenever the transported successor lies outside the no-transport anchor hull. On a suite of eleven public and simulated benchmarks spanning ecology, energy, diet, mortality, employment, air quality, severe weather, mobility, and G/G/1, Gt/G/1 queue occupancy, CAST achieves the best average rank on both one-step KL (1.27) and autoregressive rollout JSD (1.91), winning 8/11 sections on each metric against a broad statistical, compositional, recurrent, convolutional, Transformer, and modern time-series baseline set, and top-2 on all 11 sections for offline KL. Component ablations and a controlled synthetic aliasing experiment corroborate the theory. The code release is available at this link.

Use of AI is a valuable tool for weather prediction but only when it's trained with ample data, experts say Mon 18 May 2026 08.00 EDTLast modified on Mon 18 May 2026 08.01 EDT As the US prepares for hurricane season and a summer of record-breaking heat, experts fear the Trump administration's cuts to climate and weather data programming could make the federal government's weather forecasts less reliable when they are needed most. The National Oceanic and Atmospheric Administration (Noaa) late last year launched a suite of artificial intelligence-powered global weather forecast models which it said would improve "speed, efficiency, and accuracy". In March, an agency official said those models are being trained with centuries of weather data. Artificial intelligence is a valuable tool for weather prediction, but only when it is well-trained with ample data, said Monica Medina, who served as Noaa's principal deputy undersecretary of commerce for oceans and atmosphere from 2009 to 2012. Under Trump, climate and weather data collection has declined, said Medina.

Nearest-neighbor methods are fundamental to classical and modern machine learning, yet their geometric properties are typically analyzed under independent sampling. In this paper, we study the nearest-neighbor radii under dependent sampling. We consider strong mixing dependent observations and ask whether dependence changes the scale of nearest-neighbor neighborhoods. We establish distribution-free almost sure convergence under polynomial mixing and sharp non-asymptotic moment bounds under geometric mixing. The moment bounds depend on the local intrinsic dimension rather than the ambient dimension, making the results applicable to high-dimensional data concentrated near lower-dimensional manifolds. Synthetic experiments and real-world time-series benchmarks support the theory, showing that nearest-neighbor geometry remains informative under dependence sampling.

We propose a novel adaptive Mixture-of-Experts (MoE) framework for time series forecasting that enhances expert specialization by incorporating expert-specific loss information directly into the training process. Notably, the overall objective comprises the base forecasting loss and expert-specific losses, allowing expert-level prediction errors to jointly shape training alongside the global forecasting loss. This framework is further combined with a partial online learning strategy, enabling incremental updates of both the gating mechanism and expert parameters. This approach significantly reduces computational cost by eliminating the need for repeated full model retraining. By integrating expert-level loss awareness with efficient online optimization, the proposed method achieves improved learning efficiency while maintaining strong predictive performance. Empirical results across economic, tourism, and energy datasets with varying frequencies demonstrate that the proposed approach generally outperforms both statistical methods and state-of-the-art neural network models, such as Transformers and WaveNet, in forecasting accuracy and computational efficiency. Furthermore, ablation studies confirm the effectiveness of the expert-specific loss integration strategy, highlighting its contribution to enhancing predictive performance.

Accurate trend forecasting in healthcare time series is essential for planning and resource allocation. This paper proposes a Bayesian framework for predicting oncology demand trends, modeling weekly appointments as a Poisson process with a Gamma prior to the demand rate. To enhance adaptability and capture persistent directional patterns, we incorporate a residual-based boosting mechanism grounded in a Gamma-Log-Normal conjugate structure. This boosting approach allows the model to track both short- and long-term trend shifts while maintaining the analytical tractability of conjugate Bayesian updating. The methodology was evaluated on real oncology service data from Cariri, Ceara, Brazil, and compared against established baselines, including linear regression, ARIMA, naive forecasting, LSTM neural networks, and XGBoost. Results showed that the proposed model outperforms competing methods in trend detection accuracy, with gains in terms of percentage of correct direction of 38.25% in relation to the second best approach in some cases.