Time-dependent high-dimensional partial differential equations (PDEs) with spatially localised and dynamically evolving solutions pose a fundamental challenge for physics-informed neural networks (PINNs), as uniform collocation sampling becomes increasingly ineffective in high-dimensional spatiotemporal domains. In this work, a deep adaptive sampling framework for PINNs is extended to the time-dependent setting by treating space and time as a unified domain without any explicit time marching. A normalising flow neural network model effectively learns the distribution induced by the PDE residual and generates new collocation points concentrated in regions where the solution is most difficult to learn. Unlike conventional adaptive strategies that require explicit time stepping or moving meshes, high-residual regions are automatically identified and tracked across both space and time, driven purely by the PDE residual distribution. The effectiveness of the proposed strategy is assessed on a range of benchmark problems, from sharp and moving features in two spatial dimensions to localised structures in up to eight spatial dimensions.

This paper establishes a theoretical framework for the uniform convergence of smoothly activated deep neural network (DNN) estimators. While standard ReLU networks achieve minimax-optimal rates in the $L^2(P)$ norm for various nonparametric regression tasks, we establish a theoretical lower bound demonstrating that least-squares ReLU estimators can suffer from the curse of dimensionality in their uniform convergence behavior. Motivated by the need for reliable uniform guarantees in downstream tasks requiring worst-case reliability, we address this limitation by analyzing smoothly activated DNNs (smooth DNNs), encompassing both feedforward and residual structures. We establish novel pseudo-dimension bounds, non-asymptotic approximation guarantees, and Hölder-norm bounds for the approximators of these models. Leveraging these results, we derive non-asymptotic uniform convergence rates for smooth DNN estimators across multiple statistical contexts, including Huber, least-squares, quantile, and logistic regression. We prove that smooth DNNs can mitigate the {curse of dimensionality} in uniform convergence by adaptively exploiting the low-dimensional hierarchical composition structure of the target function. Supported by both simulation studies and a real-world application, our results position smooth DNNs as a theoretically grounded and practically viable alternative to ReLU networks for statistical learning tasks requiring uniform guarantees.

Diffusion models have demonstrated strong performance in time series modeling due to their ability to progressively capture complex data distributions through iterative denoising. However, existing approaches struggle with frequency-sensitive denoising, high-frequency reconstruction and balancing global trends with local dynamics. To address these limitations, we propose \textbf{HyFAD}, a \textbf{Hy}brid time-frequency \textbf{D}iffusion model with \textbf{F}requency-\textbf{A}ware embedding for time series imputation. Built upon the DDPM paradigm, HyFAD adopts a coupled time-frequency diffusion framework, in which the reverse denoising proceeds sequentially from the time domain to the frequency domain, enabling coarse-to-fine generation. Specifically, the time-domain diffusion process captures low-frequency global trends, while the frequency-domain diffusion process refines high-frequency spectral components. We further introduce a frequency-aware step embedding that exploits the relationship between diffusion steps and spectral components, providing step-dependent spectral guidance and facilitates more accurate band-wise reconstruction. Extensive experiments on multiple benchmark datasets demonstrate that HyFAD achieves state-of-the-art performance. Our source code is available at https://github.com/hongfangao/HyFAD.

Things to Do in L.A. Tap to enable a layout that focuses on the article. O.C. immigration attorneys suspended for filing briefs filled with AI-hallucinated errors The attorneys were fined $2,500 each and suspended from practicing in the U.S. 9th Circuit Court of Appeals for six months. This is read by an automated voice. Please report any issues or inconsistencies here . A pair of Orange County immigration attorneys received temporary suspensions after the court discovered they used generative AI to write briefs that included "multiple nonexistent cases, misattributed quotations, and gross misrepresentations."

Plus: The EU has proposed new legislation to end its Big Tech dependence. Most days in her chambers, Judge Maritza Braswell, a federal magistrate judge in Colorado, sifts through stacks of documents written by people without a lawyer. The number of these filings has more than doubled compared to before 2023. She puts that jump down to AI. But while AI appears to be expanding access to justice, it doesn't seem to be improving people's chances of winning. Judges are starting to question what rights and duties chatbots should have as they stand in for lawyers.

Silicon Valley is turning to ethicists to shape the future of AI. Philosophy has long suffered an unfortunate reputation as pedantic and abstruse. In one of the most prominent debates of the 20th century, philosophers spent a great deal of energy arguing over what means. Paul Graham, the legendary tech investor, studied philosophy as a college student, which seemed "an impressively impractical thing to do," as he later wrote. But over time, Graham became disillusioned: "I kept taking philosophy courses and they kept being boring," he explained .

Taken to its logical conclusion, this line of thinking is absurd--and damning. Anthropic is regarded as a giant among AI companies, but perhaps what it really excels in is anthropomorphism. Earlier this year, the company released an 84-page document titled Claude's "constitution," Claude being the name of the large language model that is the company's flagship product. The first sentence reads, "Claude's constitution is a detailed description of Anthropic's intentions for Claude's values and behaviors." It goes on: "The document is written with Claude as its primary audience," "we want Claude to be able to use its judgment once armed with a good understanding of the relevant considerations," "Claude's moral status is deeply uncertain," and "Claude may have some functional version of emotions or feelings." This anthropomorphism is by no means limited to the document. In an interview earlier this year, Anthropic's CEO, Dario Amodei, said that "we're open to the idea" that AI could be conscious. In a separate interview, Anthropic's in-house philosopher, Amanda Askell (who is credited as a lead author of Claude's constitution), said, "I want Claude to be very happy--and this is a thing that I want Claude to know more, because I worry about Claude getting anxious when people are mean to it on the internet and stuff." It's enough to make you wonder: Should we seriously consider the possibility that Claude, or any large language model, might be conscious? And if it has feelings, is it capable of receiving moral instruction?

ChatGPT now processes around 2.5 billion queries every day UN researchers are urging people to be less polite to artificial intelligences after a report found that cutting words from prompts could reduce ChatGPT's energy consumption by up to 25 per cent. Removing "please", "thank you" and other unnecessary words from AI prompts could save 87 to 98 gigawatt-hours of electricity per year, the report from the UN University Institute for Water, Environment and Health (UNU-INWEH) found. That is the equivalent of the annual residential electricity use of up to 760,000 people in sub-Saharan Africa. 'Flashes of brilliance and frustration': I let an AI agent run my day To reduce their energy consumption and carbon footprint, people should write concise prompts, avoid getting sucked into conversation loops and refrain from starting relationships with AI, the researchers said. "We are not saying be rude to your AI. But don't fall into the interaction trap and don't go falling in love with it either," says Kaveh Madani at UNU-INWEH.

OpenAI has failed to execute several strategies to monetise ChatGPT, including advertisements, which Sam Altman, OpenAI's CEO, had said would be a'last resort'. OpenAI has failed to execute several strategies to monetise ChatGPT, including advertisements, which Sam Altman, OpenAI's CEO, had said would be a'last resort'. As the tech mega-IPO race heats up, has OpenAI missed its moment? With rivals racing to market to raise'eye-popping sums', the spotlight is now on the AI sector's one-time'poster child' A year is a long time in AI. Just 12 months ago, Sam Altman was predicting his company OpenAI would build a super intelligence and fundamentally remake society.

I am attempting to solve a mathematical conundrum that has stumped many of humanity's greatest thinkers. I have zero mathematical training, apart from a distant undergraduate physics degree, which should put my odds of success at slim to none. But I also have a trick up my sleeve - a kind of mathematical genie that can conjure arcane secrets seemingly out of thin air. I make a short request concerning an esoteric conjecture in number theory, then cross my fingers. Perhaps "genie" is a bit too strong - I'm simply using GPT 5.5 Pro, the latest iteration of OpenAI's flagship model. But for mathematicians, modern AI models appear to have a spark of magic.