We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives. A FNN maps the input from a possibly infinite-dimensional weighted manifold to the real-valued hidden layer, on which a non-linear scalar activation function is applied, and then returns the output into a Banach space via some linear readouts. By proving a weighted Nachbin theorem, we establish a universal approximation theorem for differentiable maps, which goes beyond the usual formulation on compact sets and also includes the approximation of the derivatives. This leads us to approximation results for non-anticipative functionals including the horizontal and vertical derivatives. As a further application, we show that linear functions of the signature are able to approximate path space functionals including their directional derivatives.

Recent progress in flow-based generative modeling has led to models that output high-quality samples while using only a small number of function evaluations. However, at present, there is a lack of similar advances in estimating the model likelihood. In particular, most existing methods either rely on restrictive architectures that enable exact calculations, or use stochastic approximations such as Hutchinson's trace estimator that introduce substantial variance. In this work, we introduce SCAlable LikeLihood distillation of flOw maPs ( SCALLOP). SCALLOP builds on the recently proposed F2D2, a likelihood flow map model that can generate samples and their densities in a small number of function evaluations. While F2D2 uses Hutchinson's estimator during training, we introduce an alternative and more scalable likelihood distillation objective that is Hutchinson-free and admits a vectorized formulation. Empirically, we demonstrate the effectiveness of SCALLOP as a Boltzmann generator in molecular science, and further validate its benefit on image datasets. SCALLOP significantly reduces both training variance and training time while consistently improving performance compared to F2D2, and is competitive with the state-of-the-art while achieving up to 10 inference speedup over the fastest baseline.

The Ring star Daveigh Chase's autopsy reveals actress died from AIDS after painful health battle Clint Eastwood's son reveals shocking on-set spat with entitled Hollywood star: 'They think the world owes them' I thought my drinking was harmless until I realized I couldn't go a day without it. Then I discovered a $3 miracle pill that killed all my alcohol cravings... I'm completely cured I thought I knew the secret to great sex... then one man proved me so wrong: JANA HOCKING is mind-blown by trick that women over 40 are loving Hollywood nepo baby, 17, shows she has her father's unique style with edgy turn on red carpet... who is she? How well do you REALLY know America? Take our ultimate history quiz to find out... Stay-alert warnings issued as sharks return to one of America's busiest beaches Harry DOES want to bring Archie and Lili to the UK - but not without'proportionate protective security', team Sussex say: Duke and Duchess lay out demands after'state-funded guards turned down at 11th hour' Boy, 12, reveals how brother's quick thinking saved him from shark bite while on gorgeous Bahamas vacation Former FBI agent believes there's sinister motive behind new Nancy Guthrie ransom note... as desperation seeps in At 45 I was plagued by muscle pain, brain fog and memory loss... but it wasn't the menopause. I caught a disease while sitting on my sofa.

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With democracy on the decline in both the UK and US, Mehdi Hasan makes the case for independent journalism. Mehdi Hasan has had a front seat to US and UK politics for decades. With Britain facing yet another change in prime minister a decade after Brexit and the US looking ahead to its next vote with the midterm elections in November, we get his take on this moment and why independent journalism is needed more than ever. How did Colombia's election split a nation in two? Who's being left out of the World Cup? How is China using AI in the classroom?

In Russia, the impact of Ukraine's missile and drone strikes on energy infrastructure from Moscow to the Black Sea and beyond has long been evident. Drivers in the Russia-annexed Ukrainian peninsula of Crimea banned from filling their tanks so priority can be given to military vehicles. But such is the gravity of the situation it has now been explicitly acknowledged by President Vladimir Putin for the first time. Over the weekend, Russia's president discussed the crisis with senior officials and oil executives. And in public remarks, he was unusually frank. You're well aware that problems persist for both motorists and businesses, he told the meeting.

NFL great Chris Johnson reveals common first symptom of paralyzing ALS that's left him unable to speak or move: 'He thought it was a pinched nerve' Why Thylane Blondeau renounced her'most beautiful girl in the world' title after struggling with childhood honour - as she prepares to take on new name after marrying French DJ Ben Attal Stay-alert warnings issued as sharks return to one of America's busiest beaches I thought my drinking was harmless until I realized I couldn't go a day without it. Then I discovered a $3 miracle pill that killed all my alcohol cravings... I'm completely cured I thought I knew the secret to great sex... then one man proved me so wrong: JANA HOCKING is mind-blown by trick that women over 40 are loving How well do you REALLY know America? Take our ultimate history quiz to find out... Prince Harry vows to'explore every option to bring Archie and Lilibet to the UK' after hitting out at'bizarre decision' to deny armed security At 45 I was plagued by muscle pain, brain fog and memory loss... but it wasn't the menopause. I caught a disease while sitting on my sofa. I escaped Europe's dying democracies and discovered the real secret to America's enduring success: AYAAN HIRSI ALI Physical therapist who killed newborn and threw her in dumpster was finally snared 17 years later thanks to COSTCO receipt... as she's given incredibly light sentence for manslaughter I feel terrible saying my husband's penis isn't big enough for me.

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Usernames Are Coming to WhatsApp Soon. Even if you only use WhatsApp sometimes, you might want to snag your username now to stop giving out your phone number. One of WhatsApp's most in-demand features is finally coming out of beta. Later this year, the messaging app used by over 3 billion people plans to add usernames. It's an additional (and more privacy-friendly) way for WhatsApp users to connect without sharing phone numbers.