This chapter introduces the Bayesian reflex -- an analogy with the autonomic nervous system -- as a unifying framework for online learning in AI. Bayesian online algorithms automatically maintain equilibrium in dynamic environments via three mechanisms: belief maintenance through probabilistic representations, sequential updating via Bayes' theorem, and uncertainty-driven action balancing exploration and exploitation. We survey online Bayesian methods, highlighting two computational principles: the look-up table principle for sequential inference in function space, and the ellipsoidal decomposition framework for nearly exact i.i.d. sampling from arbitrary posteriors. These principles are generalized across dynamic emulation, nonparametric state-space models, circular time series, inverse regression for climate model evaluation, and deep architectures via Recursive Gaussian Processes. Decision-making is explored via Thompson sampling and restless bandits. We extend the framework to assess infinite series convergence (applied to climate dynamics and the Riemann Hypothesis), model prime number distributions leading to the discovery of 184 strong Mersenne prime candidates, detect stationarity, and characterize point processes. The Bayesian reflex provides a foundational infrastructure for adaptive AI that continuously learns in a complex world.

OpenAI's cofounder and president revealed in federal court on Monday that he's one of the largest individual stakeholders in the AI lab. Two days before the Musk v. Altman trial began, Elon Musk asked OpenAI cofounder and president Greg Brockman about reaching a settlement. When Brockman suggested both sides drop their claims, Musk responded, "By the end of this week, you and Sam [Altman] will be the most hated men in America. If you insist, so be it." The message --which OpenAI's lawyers made public on Sunday, and which Judge Yvonne Gonzalez Rogers subsequently refused to let the jury hear about--underscores what may be Musk's larger goal in this trial.

PCWorld explores OpenAI's new Codex AI pets, which provide visual status indicators for desktop AI agents through customizable on-screen companions. These pets address a key user experience issue by displaying red clocks when agent approval is needed and green checks upon task completion. The feature enhances multitasking efficiency by keeping users informed of AI agent activity without constant monitoring of the main interface. Whether I'm using Claude's desktop Cowork application or OpenAI's Codex coding app, I prefer that my AI agents check back with me before making high-stakes decisions. But while that makes for a safer setup, it also means my agents are often waiting around, twiddling their thumbs as they wait for me to approve their next steps. Now, if I'm sitting and watching the Cowork or Codex apps in action, I'll see right away when an agent is awaiting my approval. But if I'm working in another window or multitasking, I could easily miss the fact that an idled Cowork or Codex agent is sitting around, staring vacantly into space.

Independent sample generation is the prevailing paradigm in modern diffusion-based generative models of AI. We ask a different question: can samples coordinate through shared population statistics to transport probability mass more efficiently? We introduce Mean-Field Path-Integral Diffusion (MF-PID), a framework in which samples are promoted to interacting agents whose drift depends self-consistently on the evolving population density. We identify two analytically tractable regimes: a Linear-Quadratic-Gaussian (LQG) benchmark in which the infinite-dimensional mean-field system reduces to a finite set of Riccati and linear ODEs, and a Gaussian-mixture regime governed by a piecewise-constant protocol that preserves closed-form solvability. For a quadratic interaction potential with schedule βt and zero base drift we prove that the self-consistent MF guidance is the exact linear interpolant between initial and target global means -- a result that holds for arbitrary initial and target densities and any βt. Applied to demand-response control of energy systems, where agents aggregated into an ensemble are energy consumers (e.g. The energy saving is independent of the number of zones per building (d = 1-32 tested), confirming that the linear guidance formula broadcasts a single d-vector with O(d) communication and grows mildly in compute (sub-cubically for d 32, asymptotically O(d3) for d 1). Introduction Generative AI has been transformed by diffusion models, which frame sample generation as a stochastic process steered from noise to data [1-3]. A key structural feature of these models -- shared with other generative models, e.g. Similarly, stochastic optimal transport (SOT) and Schrödinger bridge formulations [6-8] cast distribution matching as an independent-particle path optimization, yielding tractable convolutions of Green functions but discarding inter-particle information; stochastic interpolants [9] construct flexible transport bridges between arbitrary densities via tunable continuous-time stochastic processes, recovering the Schrödinger bridge as a special limit -- again in an independent-particle framework.

ABSTRACT We study the problem of training diffusion and flow generative models to sample from target distributions defined by an exponential tilting of a base density; a formulation that subsumes both sampling from unnormalized densities and reward fine-tuning of pre-trained models. This problem can be approached from a stochastic optimal control (SOC) perspective, using adjoint-based or score matching methods, or from a non-equilibrium thermodynamics perspective. We provide a unified framework encompassing these approaches and make three main contributions: (i) bias-variance decompositions revealing that Adjoint Matching/Sampling and Novel Score Matching have finite gradient variance, while Target and Conditional Score Matching do not; (ii) norm bounds on the lean adjoint ODE that theoretically support the effectiveness of adjoint-based methods; and (iii) adaptations of the CMCD and NETS loss functions, along with novel Crooks and Jarzynski identities, to the exponential tilting setting. We validate our analysis with reward fine-tuning experiments on Stable Diffusion 1.5 and 3. 1 INTRODUCTION Recent advances in generative modeling have demonstrated the effectiveness of diffusion and flow matching models for learning complex data distributions (Song et al., 2021; Ho et al., 2020; Lipman et al., 2022; Albergo et al., 2023; Liu et al., 2023). In many applications, however, it is desirable to tailor the generative process to favor certain qualities, either by sampling from an unnormalized target distribution or by fine-tuning a pre-trained model with a reward function (Uehara et al., 2024; Domingo-Enrich et al., 2025; Zhang & Chen, 2022; Holdijk et al., 2023).

Dirac-Frenkel instantaneous residual minimization evolves nonlinear parametrizations of PDE solutions in time, but ill-conditioning can render the parameter dynamics non-unique. We interpret this non-uniqueness as a gauge freedom: nullspace directions that leave the time derivative unchanged can be used to select better-conditioned parameter velocities. Building on Onsager's minimum-dissipation principle, we introduce a history variable -- interpretable as momentum -- and inject it only along the nullspace directions. The resulting Dirac-Frenkel-Onsager dynamics preserve instantaneous residual minimization, in contrast to standard regularization that can introduce bias, while promoting temporally smooth parameter evolutions. Examples demonstrate that the approach leads to increased robustness in singular and near-singular regimes.

Riemannian neural networks have proven effective in solving a variety of machine learning tasks. The key to their success lies in the development of principled Riemannian analogs of fundamental building blocks in deep neural networks (DNNs). Among those, Riemannian batch normalization (BN) layers have shown to enhance training stability and improve accuracy. In this paper, we propose BN layers for neural networks on complex domains. The proposed layers have close connections with existing Riemannian BN layers. We derive essential components for practical implementations of BN layers on some complex domains which are less studied in previous works, e.g., the Siegel disk domain. We conduct experiments on radar clutter classification, node classification, and action recognition demonstrating the efficacy of our method.

OpenAI's Sam Altman, left, is a backer of Rain Neuromophics, one of the companies that received funds from the UK's Aria, the brainchild of Dominic Cummings, right OpenAI's Sam Altman, left, is a backer of Rain Neuromophics, one of the companies that received funds from the UK's Aria, the brainchild of Dominic Cummings, right Exclusive: Brainchild of Dominic Cummings, Aria is aimed at funding'crazy' scientific projects to benefit the UK Britain's "invention agency" has pledged £50m of UK taxpayer money to US tech companies and venture capital projects. Dreamed up by Dominic Cummings to fund "crazy" ideas, the Advanced Research and Invention Agency (Aria) is meant to " restore Britain's place as a scientific superpower ". But a joint investigation by the Guardian and Democracy for Sale, an investigative website, has established that more than an eighth of the agency's £400m in research and development funding over the past two years has gone to 14 US tech companies and venture capital groups, in some cases, with no clear return for the UK or Aria. One of these companies, Rain Neuromorphics, is also backed by the OpenAI chief executive, Sam Altman, and was reported to be near collapse last year, shortly after winning Aria money. It did not respond to a request for comment; two of its founders appear to have left the company.

Vibe coding just got a whole lot more adorable. OpenAI introduced AI-generated pets to the Codex app, its agentic tool that helps with coding. These optional animated companions don't do any coding themselves, but serve as a floating overlay that can tell you what Codex is working on, notify you when Codex completes a task or whether it needs your input on something. The new feature lets developers see Codex's active thread, without having to switch away from your current open app. Users can type /pet in to the Codex app to summon or dismiss the companion.

OpenAI made the perfect tool for scammers. Donald Trump is on TikTok doing his morning routine. "Get ready with me for a big day," reads the caption, as the president holds a makeup brush to his cheek. The scene is a still, ostensibly a screenshot of a TikTok clip. Like so much other AI-generated slop coursing through the internet, the image is fake and ridiculous.