Physics-Informed Neural Networks (PINNs) often suffer from slow convergence, training instability, and reduced accuracy on challenging partial differential equations due to the anisotropic and rapidly varying geometry of their loss landscapes. We propose a lightweight curvature-aware optimization framework that augments existing first-order optimizers with an adaptive predictive correction based on secant information. Consecutive gradient differences are used as a cheap proxy for local geometric change, together with a step-normalized secant curvature indicator to control the correction strength. The framework is plug-and-play, computationally efficient, and broadly compatible with existing optimizers, without explicitly forming second-order matrices. Experiments on diverse PDE benchmarks show consistent improvements in convergence speed, training stability, and solution accuracy over standard optimizers and strong baselines, including on the high-dimensional heat equation, Gray--Scott system, Belousov--Zhabotinsky system, and 2D Kuramoto--Sivashinsky system.

In finite-width deep neural networks, the empirical kernel $G$ evolves stochastically across layers. We develop a collective kernel effective field theory (EFT) for pre-activation ResNets based on a $G$-only closure hierarchy and diagnose its finite validity window. Exploiting the exact conditional Gaussianity of residual increments, we derive an exact stochastic recursion for $G$. Applying Gaussian approximations systematically yields a continuous-depth ODE system for the mean kernel $K_0$, the kernel covariance $V_4$, and the $1/n$ mean correction $K_{1,\mathrm{EFT}}$, which emerges diagrammatically as a one-loop tadpole correction. Numerically, $K_0$ remains accurate at all depths. However, the $V_4$ equation residual accumulates to an $O(1)$ error at finite time, primarily driven by approximation errors in the $G$-only transport term. Furthermore, $K_{1,\mathrm{EFT}}$ fails due to the breakdown of the source closure, which exhibits a systematic mismatch even at initialization. These findings highlight the limitations of $G$-only state-space reduction and suggest extending the state space to incorporate the sigma-kernel.

Decision-makers rely on weather forecasts to plant crops, manage wildfires, allocate water and energy, and prepare for weather extremes. Today, such forecasts enjoy unprecedented accuracy out to two weeks thanks to steady advances in physics-based dynamical models and data-driven artificial intelligence (AI) models. However, model skill drops precipitously at subseasonal timescales (2 - 6 weeks ahead), due to compounding errors and persistent biases. To counter this degradation, we introduce probabilistic bias correction (PBC), a machine learning framework that substantially reduces systematic error by learning to correct historical probabilistic forecasts. When applied to the leading dynamical and AI models from the European Centre for Medium-Range Weather Forecasts (ECMWF), PBC doubles the subseasonal skill of the AI Forecasting System and improves the skill of the operationally-debiased dynamical model for 91% of pressure, 92% of temperature, and 98% of precipitation targets. We designed PBC for operational deployment, and, in ECMWF's 2025 real-time forecasting competition, its global forecasts placed first for all weather variables and lead times, outperforming the dynamical models from six operational forecasting centers, an international dynamical multi-model ensemble, ECMWF's AI Forecasting System, and the forecasting systems of 34 teams worldwide. These probabilistic skill gains translate into more accurate prediction of extreme events and have the potential to improve agricultural planning, energy management, and disaster preparedness in vulnerable communities.

Sam Altman speaks during the BlackRock infrastructure summit on 11 March in Washington DC. Sam Altman speaks during the BlackRock infrastructure summit on 11 March in Washington DC. How a fiery attack on Sam Altman's home unfolded Molotov cocktail attack on OpenAI CEO's home comes amid growing discontent against artificial intelligence I n the early hours of 10 April, a man approached the gate of OpenAI CEO Sam Altman's house in San Francisco and hurled a molotov cocktail at the building before fleeing. Federal and California state authorities have charged Moreno-Gama with a range of crimes including attempted arson and attempted murder. His parents issued a statement this week saying that their son had recently suffered a mental health crisis.

Panic says the Playdate Catalog won't accept games made with generative AI Using it for coding assistance is still OK, but you can't generate art, music, text or story elements. Panic's Playdate console displaying Season Two games. Panic, the company behind the tiny and excellent Playdate console, is taking a stand on generative AI. The company has published an AI disclosure that says as of this month, the Playdate Catalog "will no longer accept titles that use'Generative AI' for art, audio, music, text, or dialog." Panic does allow for developers to use AI assistance for coding, but also says that "we will flag any title as such and specify the extent that it was used (for example, "Lua debugging") so the customer can decide whether to support it or not."

Multiplicative gating is widely used in neural architectures and has recently been applied to attention layers to improve performance and training stability in large language models. Despite the success of gated attention, the mathematical implications of gated attention mechanisms remain poorly understood. We study attention through the geometry of its representations by modeling outputs as mean parameters of Gaussian distributions and analyzing the induced Fisher--Rao geometry. We show that ungated attention operator is restricted to intrinsically flat statistical manifolds due to its affine structure, while multiplicative gating enables non-flat geometries, including positively curved manifolds that are unattainable in the ungated setting. These results establish a geometric expressivity gap between ungated and gated attention. Empirically, we show that gated models exhibit higher representation curvature and improved performance on tasks requiring nonlinear decision boundaries whereas they provide no consistent advantage on tasks with linear decision boundaries. Furthermore, we identify a structured regime in which curvature accumulates under composition, yielding a systematic depth amplification effect.

Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square stability of ZO methods depends on the entire Hessian spectrum. Since computing the full Hessian spectrum is infeasible in practical neural network training, we further derive tractable stability bounds that depend only on the largest eigenvalue and the Hessian trace. Empirically, we find that full-batch ZO methods operate at the edge of stability: ZO-GD, ZO-GDM, and ZO-Adam consistently stabilize near the predicted stability boundary across a range of deep learning training problems. Our results highlight an implicit regularization effect specific to ZO methods, where large step sizes primarily regularize the Hessian trace, whereas in FO methods they regularize the top eigenvalue.

Data-driven operations management often relies on parameters estimated from costly human-generated labels. Recent advances in large language models (LLMs) and other AI systems offer inexpensive auxiliary data, but introduce a new challenge: AI outputs are not direct observations of the target outcomes, but could involve high-dimensional representations with complex and unknown relationships to human labels. Conventional methods leverage AI predictions as direct proxies for true labels, which can be inefficient or unreliable when this relationship is weak or misspecified. We propose Generative Augmented Inference (GAI), a general framework that incorporates AI-generated outputs as informative features for estimating models of human-labeled outcomes. GAI uses an orthogonal moment construction that enables consistent estimation and valid inference with flexible, nonparametric relationship between LLM-generated outputs and human labels. We establish asymptotic normality and show a "safe default" property: relative to human-data-only estimators, GAI weakly improves estimation efficiency under arbitrary auxiliary signals and yields strict gains whenever the auxiliary information is predictive. Empirically, GAI outperforms benchmarks across diverse settings. In conjoint analysis with weak auxiliary signals, GAI reduces estimation error by about 50% and lowers human labeling requirements by over 75%. In retail pricing, where all methods access the same auxiliary inputs, GAI consistently outperforms alternative estimators, highlighting the value of its construction rather than differences in information. In health insurance choice, it cuts labeling requirements by over 90% while maintaining decision accuracy. Across applications, GAI improves confidence interval coverage without inflating width. Overall, GAI provides a principled and scalable approach to integrating AI-generated information.

Lion optimizer is a popular learning-based optimization algorithm in machine learning, which shows impressive performance in training many deep learning models. Although convergence property of the Lion optimizer has been studied, its generalization analysis is still missing. To fill this gap, we study generalization property of the Lion via algorithmic stability based on the mathematical induction. Specifically, we prove that the Lion has a generalization error of $O(\frac{1}{Nτ^T})$, where $N$ is training sample size, and $τ>0$ denotes the smallest absolute value of non-zero element in gradient estimator, and $T$ is the total iteration number. In addition, we obtain an interesting byproduct that the SignSGD algorithm has the same generalization error as the Lion. To enhance generalization of the Lion, we design a novel efficient Cautious Lion (i.e., CLion) optimizer by cautiously using sign function. Moreover, we prove that our CLion has a lower generalization error of $O(\frac{1}{N})$ than $O(\frac{1}{Nτ^T})$ of the Lion, since the parameter $τ$ generally is very small. Meanwhile, we study convergence property of our CLion optimizer, and prove that our CLion has a fast convergence rate of $O(\frac{\sqrt{d}}{T^{1/4}})$ under $\ell_1$-norm of gradient for nonconvex stochastic optimization, where $d$ denotes the model dimension. Extensive numerical experiments demonstrate effectiveness of our CLion optimizer.

Imagine a chatbot that actually knows how to talk to you. Earlier this year, when I walked into a renovated loft in downtown San Francisco, the couches and tables were littered with flyers advertising an "emotionally intelligent real-time AI coach." They were for Amotions AI--one of several start-ups that had gathered that day to pitch investors, entrepreneurs, and tech workers. Pianpian Xu Guthrie, Amotions AI's founder, was eager to tell me more. The AI model observes video calls on your computer, she said, and gives you real-time tips based on the other person's tone and facial expression.