Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel--Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.

Constraint-based causal discovery is widely used for learning causal structures, but heavy reliance on conditional independence (CI) testing makes it computationally expensive in high-dimensional settings. To mitigate this limitation, many divide-and-conquer frameworks have been proposed, but most assume causal sufficiency, i.e., no latent variables. In this paper, we show that divide-and-conquer strategies can be theoretically generalized beyond causal sufficiency to settings with latent variables. Specifically, we propose a recursive decomposition framework, termed DiCoLa, that enables divide-and-conquer causal discovery in the presence of latent variables. It recursively decomposes the global learning task into smaller subproblems and integrates their solutions through a principled reconstruction step to recover the global structure. We theoretically establish the soundness and completeness of the proposed framework. Extensive experiments on synthetic data demonstrate that our approach significantly improves computational efficiency across a range of causal discovery algorithms, while experiments on a real-world dataset further illustrate its practical effectiveness.

Large language models are increasingly deployed with test-time strategies: sample $N$ responses, score them with a reward model or verifier, and return the best. This deployment rule exposes a mismatch in post-training: standard objectives optimize the mean reward of a single response, whereas best-of-$N$ performance is governed by the upper tail of the reward distribution. Recent test-time-aware objectives partly address this mismatch, but typically assume that training can use the same per-prompt rollout budget as deployment, which is impractical when post-training must cover many prompts while deployment can allocate much larger per-prompt test-time compute. We study this budget-mismatch regime, where only $m\ll N$ per-prompt rollouts are available during training but the target objective is best-of-$N$ deployment. Under structural assumptions on the reward tails, we show that the policy gradient of the best-of-$N$ objective can be approximated from a much smaller rollout group by extrapolating upper-tail statistics. This yields a family of Tail-Extrapolated estimators for best-of-$N$-oriented post-training: a simple direct estimator, Tail-Extrapolated Advantage (TEA), and a fixed-order debiased Prefix-TEA estimator based on moment cancellation. Experiments on instruction-following tasks show that TEA and Prefix-TEA improve best-of-$N$ performance across different language models, reward models and datasets under various training and test-time budget settings.

Security around Beijing's historic Tiananmen Square has been heightened for days, with rumours on social media swirling of a special parade or some big, choreographed event. Preparations for this major event have started with a whisper, but China appears ready to put on a show for US President Donald Trump. The visit will include talks, a banquet, and a visit to the Temple of Heaven, a complex of imperial temples where emperors would pray for a good harvest. And both Trump and Chinese President Xi Jinping will be hoping the visit will bear fruit. This summit between the world's two most powerful leaders is set to be one of the most consequential encounters for years.

We can now add cybercrimes to the list of growing concerns associated with artificial intelligence. Google's Threat Intelligence Group (GTIG) said it discovered, for the first time ever, a threat actor using a zero-day exploit that it believes was developed by AI. Zero-day vulnerabilities are often the most dangerous since they're unknown to the targets, leaving them with zero days to prepare for the attack. Google said in the report the threat actor was planning to use it in a mass exploitation event, but its proactive discovery may have prevented its use. Google added that it doesn't believe its own Gemini models were used, but still has high confidence an AI model was part of discovering the vulnerability and weaponizing an exploit.

The game, developed by the group of anonymous artists known as Secret Handshake, is available online and in person in Washington, DC. A new video game about President Donald Trump's war in Iran features fights with the pope and New York City mayor Zohran Mamdani . It's impossible to win, and that's the point. The game,, was developed by Secret Handshake, an anonymous group of artists behind a handful of satirical works mocking the Trump administration. The group previously installed a gold statue of Trump and Jeffrey Epstein on the National Mall; it portrayed Trump holding onto Epstein in a pose reminiscent of Jack and Rose from the movie .

Since 2023, Moderna and Korea University have been developing a new mRNA vaccine for hantavirus. The work has been promising so far, but a finished product isn't likely coming any time soon. US-based pharmaceutical company Moderna confirmed that it has been working on the development of hantavirus vaccines in collaboration with the Vaccine Innovation Center of Korea University College of Medicine (VIC-K). This comes after an outbreak of hantavirus occurred on a Dutch cruise ship that sailed from Argentina and disembarked its passengers and crew in the Canary Islands on May 10. At least three people aboard the MV died, and several cases were reported as serious.

When you purchase through links in our articles, we may earn a small commission. Who actually manufactures AmazonBasics batteries? AmazonBasics batteries are so cheap, you might be skeptical of their quality and performance. When AmazonBasics launched back in 2009, batteries were among the initial line-up of products--and they're still one of the best, most classic impulse buys of this white-label brand. Mainly sold in packs ranging from 8 to 300 batteries at extremely affordable prices, they've become the go-to value battery brand for day-to-day needs.

PCWorld warns that high DDR5 RAM demand is driving counterfeiters to sell fake modules with plastic chips glued onto circuit boards. These fraudulent listings often appear as'junk' or'untested' items to prevent returns, with buyers discovering the scam only after installation causes system failures. This mirrors previous scams involving fake GPUs and re-lidded CPUs, highlighting the need for extreme caution when purchasing PC components from secondary markets.

What are Russia's gains from the Iran war? 'We are not losers; we are winners' Who is Gerhard Schroeder, Putin's pick for Ukraine peace talks mediation? Russian President Vladimir Putin has suggested that former German Chancellor Gerhard Schroeder could coordinate talks with the European Union to secure a peace deal in Ukraine - a proposal met with scepticism by EU officials. European Council President Antonio Costa said recently he believed there was "potential" for the EU to negotiate with Russia and to discuss the future of Europe's security architecture. A day later, the Russian leader said the four-year-old war may be "coming to an end", adding that he was ready to hold direct talks with his Ukrainian counterpart, Volodymyr Zelenskyy, in Moscow or a neutral country. Speaking after Saturday's celebrations for Victory Day, which marks Russia's victory over Nazi Germany in 1945 at the end of World War II, Putin added he would be willing to meet Zelenskyy only once the terms of a peace agreement had already been settled.