Forecasting the life-cycle trajectory of a newly launched product is important for launch planning, resource allocation, and early risk assessment. This task is especially difficult in the pre-launch and early post-launch phases, when product-specific outcome history is limited or unavailable, creating a cold-start problem. In these phases, firms must make decisions before demand patterns become reliably observable, while early signals are often sparse, noisy, and unstable We propose the Conditional Diffusion Life-cycle Forecaster (CDLF), a conditional generative framework for forecasting new-product life-cycle trajectories under cold start. CDLF combines three sources of information: static descriptors, reference trajectories from similar products, and newly arriving observations when available. Here, static descriptors refer to structured pre-launch characteristics of the product, such as category, price tier, brand or organization identity, scale, and access conditions. This structure allows the model to condition forecasts on relevant product context and to update them adaptively over time without retraining, yielding flexible multi-modal predictive distributions under extreme data scarcity. The method satisfies consistency with a horizon-uniform distributional error bound for recursive generation. Across studies on Intel microprocessor stock keeping unit (SKU) life cycles and the platform-mediated adoption of open large language model repositories, CDLF delivers more accurate point forecasts and higher-quality probabilistic forecasts than classical diffusion models, Bayesian updating approaches, and other state-of-the-art machine-learning baselines.

We introduce Prior-Fitted Functional Flows, a generative foundation model for pharmacokinetics that enables zero-shot population synthesis and individual forecasting without manual parameter tuning. We learn functional vector fields, explicitly conditioned on the sparse, irregular data of an entire study population. This enables the generation of coherent virtual cohorts as well as forecasting of partially observed patient trajectories with calibrated uncertainty. We construct a new open-access literature corpus to inform our priors, and demonstrate state-of-the-art predictive accuracy on extensive real-world datasets.

Time series foundation models (TSFMs) are widely used as generic feature extractors, yet the notion of non-stationarity in their embedding spaces remains poorly understood. Recent work often conflates non-stationarity with distribution shift, blurring distinctions fundamental to classical time-series analysis and long-standing methodologies such as statistical process control (SPC). In SPC, non-stationarity signals a process leaving a stable regime - via shifts in mean, variance, or emerging trends - and detecting such departures is central to quality monitoring and change-point analysis. Motivated by this diagnostic tradition, we study how different forms of distributional non-stationarity - mean shifts, variance changes, and linear trends - become linearly accessible in TSFM embedding spaces under controlled conditions. We further examine temporal non-stationarity arising from persistence, which reflects violations of weak stationarity due to long-memory or near-unit-root behavior rather than explicit distributional shifts. By sweeping shift strength and probing multiple TSFMs, we find that embedding-space detectability of non-stationarity degrades smoothly and that different models exhibit distinct, model-specific failure modes.

Synthetic augmentation is increasingly used to mitigate data scarcity in financial machine learning, yet its statistical role remains poorly understood. We formalize synthetic augmentation as a modification of the effective training distribution and show that it induces a structural bias--variance trade-off: while additional samples may reduce estimation error, they may also shift the population objective whenever the synthetic distribution deviates from regions relevant under evaluation. To isolate informational gains from mechanical sample-size effects, we introduce a size-matched null augmentation and a finite-sample, non-parametric block permutation test that remains valid under weak temporal dependence. We evaluate this framework in both controlled Markov-switching environments and real financial datasets, including high-frequency option trade data and a daily equity panel. Across generators spanning bootstrap, copula-based models, variational autoencoders, diffusion models, and TimeGAN, we vary augmentation ratio, model capacity, task type, regime rarity, and signal-to-noise. We show that synthetic augmentation is beneficial only in variance-dominant regimes, such as persistent volatility forecasting-while it deteriorates performance in bias-dominant settings, including near-efficient directional prediction. Rare-regime targeting can improve domain-specific metrics but may conflict with unconditional permutation inference. Our results provide a structural perspective on when synthetic data improves financial learning performance and when it induces persistent distributional distortion.

We study adaptive pooling under predictive heterogeneity in high-dimensional multivariate time series forecasting, where global models improve statistical efficiency but may fail to capture heterogeneous predictive structure, while naive specialization can induce negative transfer. We formulate adaptive pooling as a statistical decision problem and propose a validation-driven framework that determines when and how specialization should be applied. Rather than grouping series based on representation similarity, we define partitions through out-of-sample predictive performance, thereby aligning data organization with predictive risk, defined as expected out-of-sample loss and approximated via validation error. Cluster assignments are iteratively updated using validation losses for both point (Huber) and probabilistic (pinball) forecasting, improving robustness to heavy-tailed errors and local anomalies. To ensure reliability, we introduce a leakage-free fallback mechanism that reverts to a global model whenever specialization fails to improve validation performance, providing a safeguard against performance degradation under a strict training-validation-test protocol. Experiments on large-scale traffic datasets demonstrate consistent improvements over strong baselines while avoiding degradation when heterogeneity is weak. Overall, the proposed framework provides a principled and practically reliable approach to adaptive pooling in high-dimensional forecasting problems.

Machine learning has opened new frontiers in purely data-driven algorithms for data assimilation in, and for forecasting of, dynamical systems; the resulting methods are showing some promise. However, in contrast to model-driven algorithms, analysis of these data-driven methods is poorly developed. In this paper we address this issue, developing a theory to underpin data-driven methods to solve smoothing problems arising in data assimilation and forecasting problems. The theoretical framework relies on two key components: (i) establishing the existence of the mapping to be learned; (ii) the properties of the operator learning architecture used to approximate this mapping. By studying these two components in conjunction, we establish novel universal approximation theorems for purely data driven algorithms for both smoothing and forecasting of dynamical systems. We work in the continuous time setting, hence deploying neural operator architectures. The theoretical results are illustrated with experiments studying the Lorenz `63, Lorenz `96 and Kuramoto-Sivashinsky dynamical systems.

We study trajectory forecasting under squared loss for time series with weak conditional structure, using highly expressive prediction models. Building on the classical characterization of squared-loss risk minimization, we emphasize regimes in which the conditional expectation of future trajectories is effectively degenerate, leading to trivial Bayes-optimal predictors (flat for prices and zero for returns in standard financial settings). In this regime, increased model expressivity does not improve predictive accuracy but instead introduces spurious trajectory fluctuations around the optimal predictor. These fluctuations arise from the reuse of noise and result in increased prediction variance without any reduction in bias. This provides a process-level explanation for the degradation of Transformerbased forecasts on financial time series. We complement these theoretical results with numerical experiments on high-frequency EUR/USD exchange rate data, analyzing the distribution of trajectory-level forecasting errors. The results show that Transformer-based models yield larger errors than a simple linear benchmark on a large majority of forecasting windows, consistent with the variance-driven mechanism identified by the theory.

Text-to-time-series generation is particularly important in meteorology, where natural language offers intuitive control over complex, multi-scale atmospheric dynamics. Existing approaches are constrained by the lack of large-scale, physically grounded multimodal datasets and by architectures that overlook the spectral-temporal structure of weather signals. We address these challenges with a unified framework for text-guided meteorological time-series generation. First, we introduce MeteoCap-3B, a billion-scale weather dataset paired with expert-level captions constructed via a Multi-agent Collaborative Captioning (MACC) pipeline, yielding information-dense and physically consistent annotations. Building on this dataset, we propose MTransformer, a diffusion-based model that enables precise semantic control by mapping textual descriptions into multi-band spectral priors through a Spectral Prompt Generator, which guides generation via frequency-aware attention. Extensive experiments on real-world benchmarks demonstrate state-of-the-art generation quality, accurate cross-modal alignment, strong semantic controllability, and substantial gains in downstream forecasting under data-sparse and zero-shot settings. Additional results on general time-series benchmarks indicate that the proposed framework generalizes beyond meteorology.

Test-time adaptation (TTA) enables neural forecasters to adapt to distribution shifts in streaming time series, but existing methods apply the same adaptation intensity regardless of the nature of the shift. We propose Regime-Guided Test-Time Adaptation (RG-TTA), a meta-controller that continuously modulates adaptation intensity based on distributional similarity to previously-seen regimes. Using an ensemble of Kolmogorov-Smirnov, Wasserstein-1, feature-distance, and variance-ratio metrics, RG-TTA computes a similarity score for each incoming batch and uses it to (i) smoothly scale the learning rate -- more aggressive for novel distributions, conservative for familiar ones -- and (ii) control gradient effort via loss-driven early stopping rather than fixed budgets, allowing the system to allocate exactly the effort each batch requires. As a supplementary mechanism, RG-TTA gates checkpoint reuse from a regime memory, loading stored specialist models only when they demonstrably outperform the current model (loss improvement >= 30%). RG-TTA is model-agnostic and strategy-composable: it wraps any forecaster exposing train/predict/save/load interfaces and enhances any gradient-based TTA method. We demonstrate three compositions -- RG-TTA, RG-EWC, and RG-DynaTTA -- and evaluate 6 update policies (3 baselines + 3 regime-guided variants) across 4 compact architectures (GRU, iTransformer, PatchTST, DLinear), 14 datasets (6 real-world multivariate benchmarks + 8 synthetic regime scenarios), and 4 forecast horizons (96, 192, 336, 720) under a streaming evaluation protocol with 3 random seeds (672 experiments total). Regime-guided policies achieve the lowest MSE in 156 of 224 seed-averaged experiments (69.6%), with RG-EWC winning 30.4% and RG-TTA winning 29.0%. Overall, RG-TTA reduces MSE by 5.7% vs TTA while running 5.5% faster; RG-EWC reduces MSE by 14.1% vs standalone EWC.

Probabilistic forecasting provides a principled framework for uncertainty quantification in dynamical systems by representing predictions as probability distributions rather than deterministic trajectories. However, existing forecasting approaches, whether physics-based or neural-network-based, remain fundamentally trajectory-oriented: predictive distributions are usually accessed through ensembles or sampling, rather than evolved directly as dynamical objects. A distribution-to-distribution (D2D) neural probabilistic forecasting framework is developed to operate directly on predictive distributions. The framework introduces a distributional encoding and decoding structure around a replaceable neural forecasting module, using kernel mean embeddings to represent input distributions and mixture density networks to parameterise output predictive distributions. This design enables recursive propagation of predictive uncertainty within a unified end-to-end neural architecture, with model training and evaluation carried out directly in terms of probabilistic forecast skill. The framework is demonstrated on the Lorenz63 chaotic dynamical system. Results show that the D2D model captures nontrivial distributional evolution under nonlinear dynamics, produces skillful probabilistic forecasts without explicit ensemble simulation, and remains competitive with, and in some cases outperforms, a simplified perfect model benchmark. These findings point to a new paradigm for probabilistic forecasting, in which predictive distributions are learned and evolved directly rather than reconstructed indirectly through ensemble-based uncertainty propagation.