Physics-informed neural networks (PINNs) provide a mesh-free framework for solving PDE-constrained inverse problems, but their extension to Bayesian inversion still faces a fundamental difficulty: prior distributions are typically defined in the weight space of neural networks, whereas physically meaningful prior assumptions are more naturally expressed in function space. In this study, we introduce a unified framework, termed functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks (fpBPINN), to incorporate functional priors into Bayesian PINN-based inversion. We consider two complementary approaches. The first is a functional-prior-informed Bayesian PINN (FPI-BPINN), in which a neural network weight prior is learned to be consistent with a prescribed functional prior, and Bayesian inference is subsequently performed in weight space. The second is function-space particle-based variational inference for PINNs (fParVI-PINN), which performs Bayesian estimation using ParVI directly in function space. We also show that random Fourier features (RFF) play an important role in representing Gaussian functional priors with neural networks and in improving posterior approximation. We applied the proposed approaches to one-dimensional seismic traveltime tomography and two-dimensional Darcy-flow permeability inversion. These numerical experiments showed that both approaches accurately estimated posterior distributions, highlighting the significance of introducing physically interpretable functional priors into Bayesian PINN-based inverse problems. We also identified the contrasting advantages of FPI-BPINN and fParVI-PINN, namely flexibility and accuracy, respectively.

Quantum machine learning (QML) aims to accelerate machine learning tasks by exploiting quantum computation. Previous work studied a QML algorithm for selecting sparse subnetworks from large shallow neural networks. Instead of directly solving an optimization problem over a large-scale network, this algorithm constructs a sparse subnetwork by sampling hidden nodes from an optimized probability distribution defined using the ridgelet transform. The quantum algorithm performs this sampling in time $O(D)$ in the data dimension $D$, whereas a naive classical implementation relies on handling exponentially many candidate nodes and hence takes $\exp[O(D)]$ time. In this work, we construct and analyze a quantum-inspired fully classical algorithm for the same sampling task. We show that our algorithm runs in time $O(\operatorname{poly}(D))$, thereby removing the exponential dependence on $D$ from the previous classical approach. Numerical simulations show that the proposed sampler achieves empirical risk comparable to exact sampling from the optimized distribution and substantially lower than sampling from the non-optimized uniform distribution, while also exhibiting exponentially improved runtime scaling compared with the conventional classical implementation. These successful dequantization results show that sparse subnetwork selection via optimized sampling can be achieved classically with polynomial data-dimension scaling on conventional computers without quantum hardware, providing an alternative to the existing quantum algorithm.

Tabular data underpins most high-value prediction problems in science and industry, and TabPFN has driven the foundation model revolution for this modality. Designed with feedback from our users, TabPFN-3 builds on this foundation to scale state-of-the-art performance to datasets with 1M training rows and substantially reduce training and inference time. Pretrained exclusively on synthetic data from our prior, TabPFN-3 dramatically pushes the frontier of tabular prediction and brings substantial gains on time series, relational, and tabular-text data. On the standard tabular benchmark TabArena, a forward pass of TabPFN-3 outperforms all other models, including tuned and ensembled baselines, by a significant margin, and pareto-dominates the speed/performance frontier. On more diverse datasets, TabPFN-3 ranks first on datasets with many classes, and beats 8-hour-tuned gradient-boosted-tree baselines on datasets up to 1M training rows and 200 features. TabPFN-3 introduces test-time compute scaling to tabular foundation models. Our API offering TabPFN-3-Plus (Thinking) exploits this to beat all non-TabPFN models by over 200 Elo on TabArena, rising to 420 Elo on the largest data subset, and outperforms AutoGluon 1.5 extreme while being 10x faster, without using LLMs, real data, internet search or any other model besides TabPFN. TabPFN-3 extends the capabilities of our models, enabling SOTA prediction on relational data (new SOTA foundation model on RelBenchV1) and tabular-text data (SOTA on TabSTAR via TabPFN-3-Plus); and improves existing integrations: a specialized checkpoint, TabPFN-TS-3, ranks 2nd on the time-series benchmark fev-bench, and SHAP-value computation is up to 120x faster. TabPFN-3 achieves this performance while being up to 20x faster than TabPFN-2.5. In addition, a reduced KV cache and row-chunking scale to 1M rows on one H100 with fast inference speed.

What are Russia's gains from the Iran war? 'We are not losers; we are winners' UN aid convoy hit by drone strikes in Ukraine's Kherson NewsFeed UN aid convoy hit by drone strikes in Ukraine's Kherson A UN humanitarian convoy delivering aid to the city of Kherson was hit twice by drones, despite prior coordination with Ukrainian and Russian forces. No injuries were reported, and the UN has not attributed the attack to either side. 'China is gaining from what the US is doing in Iran' Iran's FM urges BRICS states to condemn US-Israeli aggression

The President swept off to Beijing to court Xi Jinping. Back Stateside, it was non-Presidential motorcades, video games, and a languid vibe at the White House. The endless motorcades and wail of sirens in Washington this week made it seem as if the President were travelling non-stop around the city, or receiving a bevy of foreign dignitaries. As it happened, it was National Police Week, and the ceremonial convoys were carrying the families of police officers who had been killed in the line of duty. Donald Trump was leaving for China.

Closing arguments begin in Elon Musk's landmark lawsuit against OpenAI Lawyers for OpenAI and Elon Musk began closing arguments in a landmark trial that could impact the future of the ChatGPT maker. On Thursday, each side presented a concluding statement to jurors, who will decide whether OpenAI and its leaders profited from a venture that was meant to be a "charity". Musk sued OpenAI, its CEO Sam Altman and its president Greg Brockman, alleging that the company strayed from its founding mission to build AI that was safe and beneficial to humanity. Musk was not present for the closing statements on Thursday, as he is currently in China on a diplomatic visit with United States President Donald Trump. His lawyer, Steven Molo, used his final remarks to accuse OpenAI of breaching its charitable trust by enriching investors and insiders at the nonprofit's expense.

Jennifer Lopez's dress holds on for dear life at her Netflix event, Trump powers through some wine & Kay Adams Eric Church's uses his guitar to deliver one of the most powerful addresses ever at UNC commencement Ella Langley crushes online troll with just four words, reminds the internet she doesn't miss Victoria's Secret should sign Rachel Pizzolato to face Sydney Sweeney in lingerie war, Reds fan is dumb & MEAT Morgan Wallen fan goes viral kicking a phone out of someone's hand as she's being escorted out in handcuffs Are teams that insist on singing'Sweet Caroline' during games the worst thing in sports? If this hasn't been said before, it should've been -- you can't hide in the bushes at a bachelorette pool party Shakira cranks up the heat with a World Cup song that has people dancing, buy Elvis' rhinestone jock & BBQ Greg Gutfeld: Maybe Katie Couric is coming around, who knows! Gutfeld: We don't have this kind of person on the right because we have humility Trump pledges to raise Jimmy Lai's case with Xi amid Hong Kong publisher's 20-year prison sentence Jesse Watters: Once the US gets fast food into a country, that's it Brit Hume: China's attitude on Taiwan has'never changed' History'judges' Obama for Iran nuclear deal, Iranian-American activist says Lt Gen Keith Kellogg says Iran is led by a'bunch of thugs' Larry Kudlow: 'The chickens are coming home to roost' OutKick Japanese bear-fightin' robo-wolves are pure unleaded nightmare fuel but they're working A humanoid robot chasing a pack of wild boars yeah, this actually happened. There are so many things happening around the world, it's really tough to stay up to speed on everything. However, one thing that I can't believe isn't talked about more is the problem Japan is having with bear attacks . In fact, it has gotten so bad that they've turned to robotic wolves to deal with it.

Marco Rubio warns China of'repercussions' as he reveals what really happened during closed-door Trump and Xi meeting Ex-Yankees star Carl Pavano'peed in shampoo bottles and soiled the bed,' ex-wife claims as bitter prenup feud takes disgusting twist Fury as Kash Patel SNORKELS at sacred war tomb where 900 sailors still lie... then jets off to Las Vegas Glamorous Texas Democrat's secret KINK exposed: Congressional candidate's past life returns to haunt her After theater groping shame, Lauren Boebert is being bankrolled by America's cringiest ex-congressman... and it exposes a MASSIVE hypocrisy Horrifying final days of killer dad Chris Watts' pregnant wife before she was slaughtered alongside their daughters. Read all the chilling texts and receipts in full for first time: 'My eyes burn from crying' RHOBH star Diana Jenkins denies claims she put Hayden Panettiere in bed with'undressed man' when she was 18 Trump reveals Xi's offer to break Iran's Hormuz chokehold... as China's price for the rescue looms Mystery blonde Trump aide with unfettered access to President's phone sparks White House friction: Real reason his posts contain random capital letters... and shadowy team behind them unmasked Despicable crimes paid for couple's lavish lifestyle that they flaunted online while gold chain-wearing husband fleeced $1BILLION from taxpayers New DNA analysis of Christopher Columbus reveals truth about explorer's origins that rewrites history Bitter cat fight erupts over DHS'sugar baby' scandal: Veteran female intelligence officer launches explosive new accusations that go right to top of counterterror HQ I lost 9lb in two weeks by making one simple tweak to my lifestyle. I didn't use Mounjaro, diet or change how I exercise and I couldn't believe the results... anyone can do it too I'm godfather to Candace Owens' daughter and Charlie Kirk was my friend... so I know the real reason she's attacking Erika - and I'll never publicly condemn her Britney Spears seen'barking and carrying knife' during chaotic restaurant visit I've had acid reflux all my life. Target customers threaten to boycott store after controversial'upgrade' to shopping cart New DNA analysis of Christopher Columbus reveals truth about explorer's origins that rewrites history A new DNA analysis of remains belonging to several direct descendants of Christopher Columbus may have uncovered a history-changing truth about the explorer's origins. For centuries, historians have believed the explorer was born in Genoa, Italy, rising from humble beginnings to persuade the Catholic Monarchs to finance what many considered an impossible voyage across the Atlantic.

US Air Force footage showed the rescue of 11 people after a private plane from the Bahamas crashed into the Atlantic Ocean off Florida. Survivors were found in a life raft as a storm approached and were airlifted to hospital. Iran's FM urges BRICS states to condemn US-Israeli aggression

Today on, we discuss how Donald Trump's visit to China could influence conversations between world leaders at a moment when the economic and foreign policy stakes couldn't be higher. This week on, the team dives into Trump's selected entourage for his high-stakes visit to China, ranging from Silicon Valley's tech billionaires to director Brett Ratner. We also break down the latest developments in Elon Musk's lawsuit against Sam Altman, alleging that OpenAI abandoned its original nonprofit mission for profit-driven goals, and whether either side is actually gaining an edge in the trial. Plus, Leah shares with us some of the most outlandish conspiracy theories that have been swirling around the hantavirus outbreak. Elon Musk Had'Hair-Raising' Idea of Passing OpenAI On to His Kids, Sam Altman Says Write to us at [email protected] . You can always listen to this week's podcast through the audio player on this page, but if you want to subscribe for free to get every episode, here's how: If you're on an iPhone or iPad, open the app called Podcasts, or just tap this link . The high profile testimonies we've heard this week, including from OpenAI CEO, Sam Altman himself, have resurfaced a lot of past events and a lot of drama, but we're asking will this actually be consequential to the trial's verdict? He's accompanied by a select number of Silicon Valley's top CEOs. We'll discuss how their presence could influence conversations between world leaders at a moment when the economic and foreign policy stakes could not be higher for the US. A lot of them have been recycling very similar conspiracy theories from the Covid-19 pandemic . We're going to tell you what they're sharing and also how to spot this kind of harmful misinformation.