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.

However, many real-world problems involve intrinsic uncertainties arising from incomplete physical knowledge, imperfect observations, environmental variability, and unresolved multiscale processes. These uncertainties may appear, for example, in initial or boundary conditions, unresolved physical processes, or heterogeneous material properties, and can significantly impact predictive accuracy. To obtain reliable and uncertainty-aware predictions, such effects are often incorporated directly into the mathematical formulation through random inputs, stochastic coefficients, or stochastic perturbations, leading to stochastic partial differential equations (SPDEs). Deriving numerical solutions to SPDEs has thus become a central focus of the uncertainty quantification (UQ) community, where extensive efforts have been devoted to developing efficient solvers that can accurately characterize and propagate uncertainty in high-dimensional, nonlinear dynamical systems (see, e.g., [1, 2, 12, 9, 13, 14, 18, 30, 40, 50, 56] and the references therein). Although traditional methods are effective for solving SPDEs and propagating uncertainty from stochastic input data to model predictions, they often require substantial computational cost, especially for time-dependent and multiscale problems [49, 51]. 1

Discrete diffusion has become a leading framework for generative modeling in various applications including language, vision, and biology. Existing convergence theory, however, exhibits fundamental limitations. KL-based analyses diverge under singular priors such as the masked distribution, while bounds in total variation (TV) depend on the state space size $S$ and become vacuous for modern language tasks, where vocabularies contain hundreds of thousands of tokens. We develop a unified adjoint-equation-based framework that establishes dimension-free convergence guarantees in any integral probability metric (IPM). To the best of our knowledge, our bounds are the first to be entirely free of $S$ and applicable to both masked and uniform priors. Importantly, our theory relies only on a single standard rate-matrix regularity assumption and is compatible with time-inhomogeneous schedules. Four novel techniques drive our improvements: working in the space of observables via adjoint equations rather than directly with probability measures, a regularity analysis that yields bounds on any IPM, a coupling argument that removes $S$-dependence under uniform transitions, and a score-marginal cancellation technique that removes $S$-dependence under masked transitions. Our framework thus sharply departs from prior analyses and avoids the shortcomings of pathspace-KL and existing TV-based approaches. Beyond convergence bounds, our framework provides a versatile toolkit for further theoretical study of discrete diffusion models.

While diffusion has drawn considerable recent attention from the language modeling community, continuous diffusion has appeared less scalable than discrete approaches. To challenge this belief we revisit Plaid, a likelihood-based continuous diffusion language model (DLM), and construct RePlaid by aligning the architecture of Plaid with modern discrete DLMs. In this unified setting, we establish the first scaling law for continuous DLMs that rivals discrete DLMs: RePlaid exhibits a compute gap of only $20\times$ compared to autoregressive models, outperforms Duo while using fewer parameters, and outperforms MDLM in the over-trained regime. We benchmark RePlaid against recent continuous DLMs: on OpenWebText, RePlaid achieves a new state-of-the-art PPL bound of $22.1$ among continuous DLMs and superior generation quality. These results suggest that continuous diffusion, when trained via likelihood, is a highly competitive and scalable alternative to discrete DLMs. Moreover, we offer theoretical insights to understand the advantage of likelihood-based training. We show that optimizing the noise schedule to minimize the ELBO's variance naturally yields linear cross-entropy (information loss) over time. This evenly distributes denoising difficulty without any case-specific time reparameterization. In addition, we find that optimizing embeddings via likelihood creates structured geometries and drives the most significant likelihood gain.

Elon Musk, left, and Sam Altman. Elon Musk, left, and Sam Altman. How Sam Altman's victory over Elon Musk clears way for OpenAI's trillion-dollar ambitions OpenAI's plans now seem all but guaranteed, given that the world's richest man couldn't put a stop to them On Monday morning, a jury in Oakland, California, handed a resounding victory to Sam Altman and OpenAI in their long, bitter courtroom battle with Elon Musk. The federal jury found Altman, OpenAI and its president, Greg Brockman, not liable for Elon Musk's claims that they unjustly enriched themselves and broke a founding contract made with Musk when founding the startup. The unanimous verdict, delivered after less than two hours of deliberation, is a stark rebuke of Musk and his lawyer's claims that Altman "stole a charity" through his leadership of OpenAI.

The federal jury in Oakland, California, found Altman, OpenAI and its president, Greg Brockman, not liable for Elon Musk's claims that they unjustly enriched themselves and broke a founding contract made with Musk when founding the startup. The verdict, delivered after less than two hours of deliberation, is a stark rebuke of Musk and his lawyer's claims that Altman "stole a charity" through his leadership of OpenAI . It also provides the AI firm with a clear path ahead to pursue going public later this year at about a $1tn valuation . The jury's finding is a non-binding, advisory verdict that left Judge Yvonne Gonzalez Rogers with ultimate power to issue her own ruling in the case. Gonzalez Rogers immediately said that she would agree with the jury's decision and dismissed Musk's claims.

Elon Musk just lost another lawsuit. Elon Musk, the world's richest man, has not been winning in court lately. His loss on Monday in his lawsuit against OpenAI and its co-founder Sam Altman is the latest in a string of legal defeats or settlements. Late last year he agreed to settle with former Twitter executives and thousands of former employees of the social platform, which he has renamed X, after fighting for years to pay them nothing. Then in March, he lost a case brought against him by investors of Twitter, who claimed they were misled by public statements he made during the takeover.

A new report alleges foreign influence from China, including state media and funded nonprofits like CodePink, is driving campaigns to block U.S. AI data center projects.

The pope's encyclical will address'the protection of the human person in the age of AI', the Vatican says In the first major text of his papacy, Pope Leo will address the rapid rise of artificial intelligence . The Chicago-born pontiff will present the document, known as an encyclical, at the Vatican next week during an event attended by Christopher Olah, the co-founder of Anthropic - a US-based AI firm that has clashed with Donald Trump's administration. The encyclical will address "the protection of the human person in the age of artificial intelligence", the Vatican said on Monday. In a break from tradition, Leo, who was elected pontiff in May last year, will launch the document during a public presentation on 25 May. He will be joined by lay speaker Olah of Anthropic, which is in the middle of a high-profile lawsuit with the Trump administration over the ethics of AI, as well as theologians Anna Rowlands and Léocadie Lushombo.

Campaigners say women and girls often struggle to get intimate images removed once they are shared online. Campaigners say women and girls often struggle to get intimate images removed once they are shared online. Ofcom to update codes of practice amid rise in'revenge porn' and AI-generated deepfakes targeting women and girls Social media, messaging platforms and online forums that publish intimate image abuse - often intended to humiliate women and girls - are being instructed to follow new guidelines to stop it spreading. Ofcom said it would change its codes of practice to force service providers to detect and quash intimate image abuse - sometimes called "revenge porn" - and crack down on AI-generated deepfakes. A wave of deepfakes emerged in January when Elon Musk's Grok AI was widely used to create sexualised videos of women in bikinis.