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A Lens That Takes Derivatives US Patent Basis: US8610839B2 - Optical Processing System for Computing Derivatives. In a 4f Optical Processor, the first lens takes the incoming field and forms its Fourier spectrum. At that middle plane, a tiny optical mask multiplies the spectrum by iξ. This is the...

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Mathelirium

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A Lens Is a Physical Fourier Computer A lens does not only bend light. It computes with the wave. Send several Gaussian beams into the lens, and the field after the glass is no longer just focused. The lens adds a quadratic phase, lets the wave propagate, and converts angle into position at the focal plane. That detector plane is the Fourier plane.
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A Lens Is a Physical Fourier Computer A lens does not only bend light. It computes with the wave. Send several Gaussian beams into the lens, and the field after the glass is no longer just focused. The lens adds a quadratic phase, lets the wave propagate, and converts angle into position at the focal plane. That detector plane is the Fourier plane.

Mathelirium

34,713 Aufrufe • vor 3 Monaten

A lens can perform the Fourier transform! 🔍 This is the 4F system: the first lens converts an image into its frequency components, and the second lens reconstructs it. By adding an aperture at the Fourier plane, we can filter out high frequencies. #Optics #Physics #b3d
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A lens can perform the Fourier transform! 🔍 This is the 4F system: the first lens converts an image into its frequency components, and the second lens reconstructs it. By adding an aperture at the Fourier plane, we can filter out high frequencies. #Optics #Physics #b3d

Hans Chiu

220,371 Aufrufe • vor 1 Jahr

U.S. Air Force Under Secretary Matthew Lohmeier discusses accounts of UAP penetrating solid walls to “observe”, scientific investigation into whether ‘beings exist in the invisible realm’, and a possible connection to The Bible. “If you take a look at the Earth, or the Sun, or any number of things that exist in space using a different lens – not the optical lens, but ultraviolet or infrared – What you’ll see is something that looks like a sea creature flapping its wings swimming through The Great Deep.” “Even though in this, what we used to call the ‘vacuum of space’, it seems to our optical lens as if it’s empty – in fact, it’s filled with plasma. Plasma is highly conductive. So there’s this energy that’s instant and direct and immediate between all of the stars that seems to connect all of us.”
6:26

U.S. Air Force Under Secretary Matthew Lohmeier discusses accounts of UAP penetrating solid walls to “observe”, scientific investigation into whether ‘beings exist in the invisible realm’, and a possible connection to The Bible. “If you take a look at the Earth, or the Sun, or any number of things that exist in space using a different lens – not the optical lens, but ultraviolet or infrared – What you’ll see is something that looks like a sea creature flapping its wings swimming through The Great Deep.” “Even though in this, what we used to call the ‘vacuum of space’, it seems to our optical lens as if it’s empty – in fact, it’s filled with plasma. Plasma is highly conductive. So there’s this energy that’s instant and direct and immediate between all of the stars that seems to connect all of us.”

UAP James

112,736 Aufrufe • vor 1 Jahr

OPTICAL AIDS CAUSE 28 DAY MONTHS? NCS does not recommend that we use optical aids for a moonsighting system in Britain for many reasons. The primary reason is that we are seeking a unified solution, and unity is in the sunnah - local, naked eye sightings. Not everybody believes that optical aids are permissible, but naked eye sightings are wholly uncontroversial. Another issue is that optical aids would lead to more and more examples of the moon being “seen” a day early, and out of phase, on the 28th of a lunar month. This has ALREADY HAPPENED to those who use optical aids in the UK - yet has never happened to naked eye moonsighters for over 100 months. Whilst there are procedures from the sunnah of what to do if you were to see a moon on the 28th, we cannot really justify asking people to end their fast a day early on the 28th because of a moon which is invisible to the eye and only visible through optical aids. Hence our stance. Remember - we ourselves use optical aids - and had them on our livestream. We are experienced in them, and it is from our experience we share with you that we have found that it would leave to problems. In any case why are you in a rush to end Ramadan, using technology, and away from the sunnah?
1:02

OPTICAL AIDS CAUSE 28 DAY MONTHS? NCS does not recommend that we use optical aids for a moonsighting system in Britain for many reasons. The primary reason is that we are seeking a unified solution, and unity is in the sunnah - local, naked eye sightings. Not everybody believes that optical aids are permissible, but naked eye sightings are wholly uncontroversial. Another issue is that optical aids would lead to more and more examples of the moon being “seen” a day early, and out of phase, on the 28th of a lunar month. This has ALREADY HAPPENED to those who use optical aids in the UK - yet has never happened to naked eye moonsighters for over 100 months. Whilst there are procedures from the sunnah of what to do if you were to see a moon on the 28th, we cannot really justify asking people to end their fast a day early on the 28th because of a moon which is invisible to the eye and only visible through optical aids. Hence our stance. Remember - we ourselves use optical aids - and had them on our livestream. We are experienced in them, and it is from our experience we share with you that we have found that it would leave to problems. In any case why are you in a rush to end Ramadan, using technology, and away from the sunnah?

New Crescent Society

18,343 Aufrufe • vor 4 Monaten

You’ve Seen First and Second Derivatives, But Have You Ever Seen a 0.5 Derivative? You know the first derivative measures slope, the second measures curvature, the third tracks how curvature changes, and so on. What exactly is a fractional derivative?
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You’ve Seen First and Second Derivatives, But Have You Ever Seen a 0.5 Derivative? You know the first derivative measures slope, the second measures curvature, the third tracks how curvature changes, and so on. What exactly is a fractional derivative?

Mathelirium

66,439 Aufrufe • vor 4 Monaten

Evaluating Neural Networks at the Speed of Light (with Light!). See live optical inference in the video below. Excited to share recent academic work on optical neural networks as a collection of computing elements embedded in the camera lens! These elements perform computation optically even before an image is captured, using the photons in the scene instead of GPU computation after the capture. We were able to achieve ImageNet classification more than two orders of magnitude faster than conventional neural networks on today's GPUs at almost no power consumption! To do this, we developed an array of metalenses that perform this computation on light from the scene. Project: Paper: Amazing collaboration with Kaixuan Wei, Xiao Li, Johannes Froech, Praneeth Chakravarthula, James Whitehead, Ethan Tseng , Arka Majumdar .
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Evaluating Neural Networks at the Speed of Light (with Light!). See live optical inference in the video below. Excited to share recent academic work on optical neural networks as a collection of computing elements embedded in the camera lens! These elements perform computation optically even before an image is captured, using the photons in the scene instead of GPU computation after the capture. We were able to achieve ImageNet classification more than two orders of magnitude faster than conventional neural networks on today's GPUs at almost no power consumption! To do this, we developed an array of metalenses that perform this computation on light from the scene. Project: Paper: Amazing collaboration with Kaixuan Wei, Xiao Li, Johannes Froech, Praneeth Chakravarthula, James Whitehead, Ethan Tseng , Arka Majumdar .

Felix Heide

29,904 Aufrufe • vor 1 Jahr

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics
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The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

Mathelirium

66,892 Aufrufe • vor 7 Monaten

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics
0:46

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

Mathelirium

133,454 Aufrufe • vor 5 Monaten

This is huge! A UCLA team managed to build an optical generative model that runs on light instead of GPUs. In their demo, a shallow encoder maps noise into phase patterns, which a free-space optical decoder then transforms into images—digits, fashion, butterflies, faces, even Van Gogh–style art—without any computation during synthesis. ⚡ The results rival digital diffusion models, pointing to ultra-fast, energy-efficient AI powered by photonics. Optical generative models | Nature Paper:
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This is huge! A UCLA team managed to build an optical generative model that runs on light instead of GPUs. In their demo, a shallow encoder maps noise into phase patterns, which a free-space optical decoder then transforms into images—digits, fashion, butterflies, faces, even Van Gogh–style art—without any computation during synthesis. ⚡ The results rival digital diffusion models, pointing to ultra-fast, energy-efficient AI powered by photonics. Optical generative models | Nature Paper:

机器之心 JIQIZHIXIN

173,639 Aufrufe • vor 10 Monaten

Spherical Harmonics 🤝 Fourier Transform I’m modeling the cell surface as a time-varying field u(θ,φ). Budding is energy shifting into higher angular frequencies. Decompose u into spherical harmonics (the Fourier basis on a sphere), evolve the coefficients aₗₘ(t) where growth/smoothing is clean, then inverse-transform back into a breathing, budding 3D surface. #SphericalHarmonics #FourierTransform #HarmonicAnalysis #SpectralMethods #SignalProcessing #AppliedMath #MathAnimation #ScientificVisualization #Biophysics
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Spherical Harmonics 🤝 Fourier Transform I’m modeling the cell surface as a time-varying field u(θ,φ). Budding is energy shifting into higher angular frequencies. Decompose u into spherical harmonics (the Fourier basis on a sphere), evolve the coefficients aₗₘ(t) where growth/smoothing is clean, then inverse-transform back into a breathing, budding 3D surface. #SphericalHarmonics #FourierTransform #HarmonicAnalysis #SpectralMethods #SignalProcessing #AppliedMath #MathAnimation #ScientificVisualization #Biophysics

Mathelirium

25,914 Aufrufe • vor 7 Monaten

The Black Sun and Lunar Phases: Conical Refraction in the Geocentric Earth Model Conical Refraction: An optical phenomenon where light bends into a cone-like shape when passing through certain materials. The theory is that Earth's atmosphere acts like a lens, bending light in a way that creates the appearance of Moon phases. Geocentrism: an ancient view of our universe where Earth is the stationary center, and everything else (the Sun, Moon, and stars) revolve around it. This contrasts with the modern heliocentric model, where planets orbit the Sun. Lunar phases are the result of conical refraction caused by the enigmatic Black Sun. In this geocentric study, we examine how distortions of light through Earth's atmospheric "lens" can create the lunar cycle. 🌑🌒🌓🌔🌝🌖🌗🌘🌑
11:22

The Black Sun and Lunar Phases: Conical Refraction in the Geocentric Earth Model Conical Refraction: An optical phenomenon where light bends into a cone-like shape when passing through certain materials. The theory is that Earth's atmosphere acts like a lens, bending light in a way that creates the appearance of Moon phases. Geocentrism: an ancient view of our universe where Earth is the stationary center, and everything else (the Sun, Moon, and stars) revolve around it. This contrasts with the modern heliocentric model, where planets orbit the Sun. Lunar phases are the result of conical refraction caused by the enigmatic Black Sun. In this geocentric study, we examine how distortions of light through Earth's atmospheric "lens" can create the lunar cycle. 🌑🌒🌓🌔🌝🌖🌗🌘🌑

Ben Wehrman

43,552 Aufrufe • vor 1 Monat

Footage of Polish soldiers suppression fire with a K6 heavy machine gun. 🇰🇷 : No burst, just shoot full-auto. 🇵🇱 : Ok 😉 The DCL-120 scope from DI Optical mounted on the K6 is impressive. DI Optical is famous for various high-end scopes. It is said that personnel from Romania, Poland, Australia, Vietnam, Egypt, and Turkey are being dispatched to the Korean Army to receive in-depth training in the operation of the K2 tank and K9 self-propelled howitzer.
0:29

Footage of Polish soldiers suppression fire with a K6 heavy machine gun. 🇰🇷 : No burst, just shoot full-auto. 🇵🇱 : Ok 😉 The DCL-120 scope from DI Optical mounted on the K6 is impressive. DI Optical is famous for various high-end scopes. It is said that personnel from Romania, Poland, Australia, Vietnam, Egypt, and Turkey are being dispatched to the Korean Army to receive in-depth training in the operation of the K2 tank and K9 self-propelled howitzer.

Mason ヨンハク

173,909 Aufrufe • vor 1 Jahr

Optical illusion or a portal to another dimension? 😲🤩⁣ This rare and majestic phenomenon is called a Subsun, also known as a "sun candle." It occurs when sunlight interacts with small, flat ice crystals found in the atmosphere, typically within cirrus clouds or ice fog. ⁣ Each ice crystal acts like a mirror, creating an optical illusion that reflects sunlight and produces a vertical beam of light resembling a candle flame. ⁣
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Optical illusion or a portal to another dimension? 😲🤩⁣ This rare and majestic phenomenon is called a Subsun, also known as a "sun candle." It occurs when sunlight interacts with small, flat ice crystals found in the atmosphere, typically within cirrus clouds or ice fog. ⁣ Each ice crystal acts like a mirror, creating an optical illusion that reflects sunlight and produces a vertical beam of light resembling a candle flame. ⁣

AccuWeather

39,077 Aufrufe • vor 1 Jahr

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation
0:33

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation

Mathelirium

22,197 Aufrufe • vor 2 Monaten

Lecture 2 of our Physics-Informed Neural Networks mini-series. In Lecture 1 we made the idea visible...a neural network isn’t predicting a PDE solution, it is the candidate function uᵩ(x,t), and the PDE residual rᵩ(x,t) is the leash that keeps it honest. Now the natural question follows: How can a neural network be punished for breaking a PDE when nobody ever handed it the true solution, and the equation itself contains derivatives like uᵩₜₜ and uᵩₓₓ? Here’s the satisfying answer: A PINN doesn’t need the true answer to be corrected. It only needs a way to measure how wrong it is according to the PDE! The network outputs uᵩ(x,t). A software called "autodiff" is used to compute the derivatives (uᵩₓ, uᵩₜ, uᵩₓₓ, …) exactly by applying the chain rule through the network. Those derivatives get dropped into the PDE to produce rᵩ(x,t). If rᵩ is big at some point, the loss spikes there, and gradient descent pushes the parameters so that rᵩ shrinks. The math breakdown We want a function u(x,t) that satisfies a PDE on a domain Ω. In this lecture we keep a concrete nonlinear example in mind, the damped sine-Gordon equation uₜₜ(x,t) + γ uₜ(x,t) − c² uₓₓ(x,t) + sin(u(x,t)) = 0. A PINN replaces the unknown function u with a neural network uᵩ(x,t), where ᵩ means all the network parameters (weights and biases). Now we build the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = uᵩₜₜ(x,t) + γ uᵩₜ(x,t) − c² uᵩₓₓ(x,t) + sin(uᵩ(x,t)). If uᵩ were a true solution, rᵩ would be 0 everywhere. So we sample points (xⱼ,tⱼ) inside the domain. These are collocation points. At each one we evaluate rᵩ, and we define a physics loss L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|². This is the punishment mechanism. (Punish just means: if |rᵩ| is big, L_phys is big; training updates ᵩ to make L_phys smaller. Reward means the loss drops, so those parameter changes are kept.) The key question was where the derivatives come from. Since uᵩ is built out of differentiable operations, we can compute uᵩₜ(x,t), uᵩₜₜ(x,t), uᵩₓ(x,t), uᵩₓₓ(x,t), at any input (x,t) we choose. Imagine a simple differentiable model written as a sum of nonlinear features uᵩ(x,t) = Σₖ vₖ σ( wₖx x + wₖt t + bₖ ) + b₀. Then the derivatives are just chain rule uᵩₓ(x,t) = Σₖ vₖ σ′(·) wₖx uᵩₓₓ(x,t) = Σₖ vₖ σ″(·) (wₖx)² uᵩₜ(x,t) = Σₖ vₖ σ′(·) wₖt uᵩₜₜ(x,t) = Σₖ vₖ σ″(·) (wₖt)². So rᵩ(x,t) is an explicit computable number at every (x,t). For the damped sine-Gordon example, it’s the same story, just with one extra nonlinear term: rᵩ(x,t) = [uᵩₜₜ(x,t) + γ uᵩₜ(x,t) − c² uᵩₓₓ(x,t)] + sin(uᵩ(x,t)). A real PINN is a deeper composition of these same building blocks, but it’s still just a chain rule, and autodiff is the machinery that does that bookkeeping reliably for big graphs. Then we train by gradient descent on the total loss. Even if we use only physics for the moment, the update is conceptually just ᵩ ← ᵩ − η ∇ᵩ L_phys(ᵩ), with learning rate η. In practice we also include initial/boundary conditions or data, because PDEs aren’t uniquely determined without them L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ), where L_bc/ic(ᵩ) enforces things like uᵩ(x,0) ≈ u₀(x) and uᵩₜ(x,0) ≈ v₀(x), or boundary conditions at x = ±L. So Lecture 2’s punchline is simple: the PDE becomes a training signal. We keep differentiating uᵩ, measuring rᵩ, and updating ᵩ until the residual goes quiet across Ω. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #AutoDiff #Backpropagation #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics
1:00

Lecture 2 of our Physics-Informed Neural Networks mini-series. In Lecture 1 we made the idea visible...a neural network isn’t predicting a PDE solution, it is the candidate function uᵩ(x,t), and the PDE residual rᵩ(x,t) is the leash that keeps it honest. Now the natural question follows: How can a neural network be punished for breaking a PDE when nobody ever handed it the true solution, and the equation itself contains derivatives like uᵩₜₜ and uᵩₓₓ? Here’s the satisfying answer: A PINN doesn’t need the true answer to be corrected. It only needs a way to measure how wrong it is according to the PDE! The network outputs uᵩ(x,t). A software called "autodiff" is used to compute the derivatives (uᵩₓ, uᵩₜ, uᵩₓₓ, …) exactly by applying the chain rule through the network. Those derivatives get dropped into the PDE to produce rᵩ(x,t). If rᵩ is big at some point, the loss spikes there, and gradient descent pushes the parameters so that rᵩ shrinks. The math breakdown We want a function u(x,t) that satisfies a PDE on a domain Ω. In this lecture we keep a concrete nonlinear example in mind, the damped sine-Gordon equation uₜₜ(x,t) + γ uₜ(x,t) − c² uₓₓ(x,t) + sin(u(x,t)) = 0. A PINN replaces the unknown function u with a neural network uᵩ(x,t), where ᵩ means all the network parameters (weights and biases). Now we build the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = uᵩₜₜ(x,t) + γ uᵩₜ(x,t) − c² uᵩₓₓ(x,t) + sin(uᵩ(x,t)). If uᵩ were a true solution, rᵩ would be 0 everywhere. So we sample points (xⱼ,tⱼ) inside the domain. These are collocation points. At each one we evaluate rᵩ, and we define a physics loss L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|². This is the punishment mechanism. (Punish just means: if |rᵩ| is big, L_phys is big; training updates ᵩ to make L_phys smaller. Reward means the loss drops, so those parameter changes are kept.) The key question was where the derivatives come from. Since uᵩ is built out of differentiable operations, we can compute uᵩₜ(x,t), uᵩₜₜ(x,t), uᵩₓ(x,t), uᵩₓₓ(x,t), at any input (x,t) we choose. Imagine a simple differentiable model written as a sum of nonlinear features uᵩ(x,t) = Σₖ vₖ σ( wₖx x + wₖt t + bₖ ) + b₀. Then the derivatives are just chain rule uᵩₓ(x,t) = Σₖ vₖ σ′(·) wₖx uᵩₓₓ(x,t) = Σₖ vₖ σ″(·) (wₖx)² uᵩₜ(x,t) = Σₖ vₖ σ′(·) wₖt uᵩₜₜ(x,t) = Σₖ vₖ σ″(·) (wₖt)². So rᵩ(x,t) is an explicit computable number at every (x,t). For the damped sine-Gordon example, it’s the same story, just with one extra nonlinear term: rᵩ(x,t) = [uᵩₜₜ(x,t) + γ uᵩₜ(x,t) − c² uᵩₓₓ(x,t)] + sin(uᵩ(x,t)). A real PINN is a deeper composition of these same building blocks, but it’s still just a chain rule, and autodiff is the machinery that does that bookkeeping reliably for big graphs. Then we train by gradient descent on the total loss. Even if we use only physics for the moment, the update is conceptually just ᵩ ← ᵩ − η ∇ᵩ L_phys(ᵩ), with learning rate η. In practice we also include initial/boundary conditions or data, because PDEs aren’t uniquely determined without them L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ), where L_bc/ic(ᵩ) enforces things like uᵩ(x,0) ≈ u₀(x) and uᵩₜ(x,0) ≈ v₀(x), or boundary conditions at x = ±L. So Lecture 2’s punchline is simple: the PDE becomes a training signal. We keep differentiating uᵩ, measuring rᵩ, and updating ᵩ until the residual goes quiet across Ω. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #AutoDiff #Backpropagation #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

19,977 Aufrufe • vor 7 Monaten

Ask anyone who’s taken a course in Ordinary Differential Equations (ODEs) what a solution to an ODE represents geometrically, and most of them won’t have a clean answer. When I first took ordinary differential equations, the pattern was always the same. Early on it turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations. Then pretty quickly the course slides into hammer-picking. Spot the form, apply the recipe, move on. Too mechanical! And the real problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. That matters because in real modeling the equations you meet are rarely nice enough to reward memorised recipes. So you get trained to solve toy forms, while the actual subject stays blurry. The behavior. The flow. The shape of solutions. It wasn't until I watched the first lecture of Professor Arthur Mattuck that I realized I didn’t actually know what a solution to a differential equation represents geometrically. His point is almost embarrassingly simple. A first-order ODE is a slope field, and a solution is a curve that stays tangent to that field everywhere. The math breakdown: Write the ODE as dy/dx = f(x,y). At each point (x,y), attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the one line that ties both viewpoints together: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages. Once you see that, you stop obsessing over whether you can write y(x) in closed form. You start asking the questions that actually matter. Where do solutions flow. Where do they get trapped. Where do they blow up. Where does existence or uniqueness fail because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early. It’s also why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #AppliedMathematics #Mathematics #
20:13

Ask anyone who’s taken a course in Ordinary Differential Equations (ODEs) what a solution to an ODE represents geometrically, and most of them won’t have a clean answer. When I first took ordinary differential equations, the pattern was always the same. Early on it turns into a speedrun of methods: separation of variables, integrating factors, variation of parameters, Bernoulli, exact equations. Then pretty quickly the course slides into hammer-picking. Spot the form, apply the recipe, move on. Too mechanical! And the real problem is what you don’t walk away with. You leave with a toolkit, but without a feel for what a differential equation even is, especially geometrically. That matters because in real modeling the equations you meet are rarely nice enough to reward memorised recipes. So you get trained to solve toy forms, while the actual subject stays blurry. The behavior. The flow. The shape of solutions. It wasn't until I watched the first lecture of Professor Arthur Mattuck that I realized I didn’t actually know what a solution to a differential equation represents geometrically. His point is almost embarrassingly simple. A first-order ODE is a slope field, and a solution is a curve that stays tangent to that field everywhere. The math breakdown: Write the ODE as dy/dx = f(x,y). At each point (x,y), attach a tiny line segment with slope f(x,y). A function y = y₁(x) is a solution exactly when its graph follows those slopes. At every x, the slope of the curve equals the slope prescribed by the field at the point on the curve. That’s the one line that ties both viewpoints together: y₁′(x) = f(x, y₁(x)). So solving the ODE and drawing an integral curve are the same statement in two languages. Once you see that, you stop obsessing over whether you can write y(x) in closed form. You start asking the questions that actually matter. Where do solutions flow. Where do they get trapped. Where do they blow up. Where does existence or uniqueness fail because the field isn’t even defined? That’s the perspective shift I wish every ODE course forces early. It’s also why I keep pairing math with animation. #DifferentialEquations #ODEs #VectorFields #AppliedMathematics #Mathematics #

Mathelirium

40,739 Aufrufe • vor 6 Monaten

Jensen Huang is investing in every photonics company he can find and the reason why tells you everything about where AI is headed (Save this). Lip-Bu Tan, the CEO of Intel says, when he looks for investment opportunities, he looks for the bottleneck and right now, the bottleneck is the interconnect, the pipes that move data between chips inside an AI data center. That is why he backed Credo Semiconductor, Astera Labs and Celestial AI on the optical side. Here is the simple version of what the interconnect bottleneck actually means. Think of an AI data center like a city, the GPUs are the buildings where all the work happens but for those buildings to function, you need roads connecting them, fast roads that can carry enormous traffic without congestion. And those roads are now the single biggest constraint on AI performance. As clusters scale to hundreds of thousands of GPUs, traditional copper wiring is hitting its physical limits and that is where this entire sector comes in. Credo Semiconductor (CRDO) is the most direct pure play on this theme, Credo makes high speed cables and optical chips that connect GPUs inside data center racks. Their revenue tripled in fiscal 2026 to $1.3 billion, growing 272% year over year at its peak and four of the world's largest hyperscalers each individually account for more than 10% of Credo's revenue. Astera Labs (ALAB) solves the connection problem between different chip types. Astera makes the PCIe and connectivity chips that manage data flow between GPUs, CPUs, and memory without errors or slowdowns. Their revenue grew 93% year over year to $308 million in Q1 2026 alone. The optical companies are where the longer-term and potentially larger opportunity lives. Copper has physical limits, you can only push electrical signals so far before the signal degrades, the heat spikes and power consumption explodes. The solution is light, fiber optic connections that move data using photons instead of electrons which is faster, cooler and far more energy efficient. Jensen Huang made this clear at Computex 2026 because copper works as long as physically possible but at greater distances and larger scale, optics takes over. Coherent (COHR) is the most established optical company in this space. Coherent makes the lasers, transceivers, and optical components at the foundation of all fiber optic communications. Nvidia signed a multibillion-dollar purchase commitment and invested $2 billion directly into the company and their customer order books are already extending out to 2028. Marvell (MRVL) is the most comprehensive bet across the entire connectivity stack. Marvell makes chips for optical networking, PCIe switching and custom AI silicon. Jensen Huang called Marvell the next trillion dollar company at Computex 2026 and backed it with a $2 billion Nvidia investment. Marvell also acquired Celestial AI, the exact company Lip-Bu Tan backed for $3.25 billion, gaining photonic fabric technology delivering 16 terabits per second of bandwidth. Lumentum (LITE), Corning (GLW), and Ciena (CIEN) round out the major public names. Lumentum received a $2 billion Nvidia investment for laser and photonics components. Corning known mostly for phone glass received $500 million from Nvidia for optical connectivity work and is up over 100% year to date. Ciena runs the optical networking systems between data centers and is seeing analyst price targets raised on the back of the AI optics boom. Every time a hyperscaler spends a billion dollars on Nvidia GPUs, the surrounding infrastructure, cables, switches, transceivers, optical components has to be upgraded to match. The smarter the GPU gets, the more the interconnect matters. Nvidia has committed at least $6.5 billion to photonics companies in the past 4 months alone and the companies building the roads between the GPUs may end up being just as valuable as the companies building the GPUs themselves. Follow me Melvin for more AI, semis and the next big market themes.
0:29

Jensen Huang is investing in every photonics company he can find and the reason why tells you everything about where AI is headed (Save this). Lip-Bu Tan, the CEO of Intel says, when he looks for investment opportunities, he looks for the bottleneck and right now, the bottleneck is the interconnect, the pipes that move data between chips inside an AI data center. That is why he backed Credo Semiconductor, Astera Labs and Celestial AI on the optical side. Here is the simple version of what the interconnect bottleneck actually means. Think of an AI data center like a city, the GPUs are the buildings where all the work happens but for those buildings to function, you need roads connecting them, fast roads that can carry enormous traffic without congestion. And those roads are now the single biggest constraint on AI performance. As clusters scale to hundreds of thousands of GPUs, traditional copper wiring is hitting its physical limits and that is where this entire sector comes in. Credo Semiconductor (CRDO) is the most direct pure play on this theme, Credo makes high speed cables and optical chips that connect GPUs inside data center racks. Their revenue tripled in fiscal 2026 to $1.3 billion, growing 272% year over year at its peak and four of the world's largest hyperscalers each individually account for more than 10% of Credo's revenue. Astera Labs (ALAB) solves the connection problem between different chip types. Astera makes the PCIe and connectivity chips that manage data flow between GPUs, CPUs, and memory without errors or slowdowns. Their revenue grew 93% year over year to $308 million in Q1 2026 alone. The optical companies are where the longer-term and potentially larger opportunity lives. Copper has physical limits, you can only push electrical signals so far before the signal degrades, the heat spikes and power consumption explodes. The solution is light, fiber optic connections that move data using photons instead of electrons which is faster, cooler and far more energy efficient. Jensen Huang made this clear at Computex 2026 because copper works as long as physically possible but at greater distances and larger scale, optics takes over. Coherent (COHR) is the most established optical company in this space. Coherent makes the lasers, transceivers, and optical components at the foundation of all fiber optic communications. Nvidia signed a multibillion-dollar purchase commitment and invested $2 billion directly into the company and their customer order books are already extending out to 2028. Marvell (MRVL) is the most comprehensive bet across the entire connectivity stack. Marvell makes chips for optical networking, PCIe switching and custom AI silicon. Jensen Huang called Marvell the next trillion dollar company at Computex 2026 and backed it with a $2 billion Nvidia investment. Marvell also acquired Celestial AI, the exact company Lip-Bu Tan backed for $3.25 billion, gaining photonic fabric technology delivering 16 terabits per second of bandwidth. Lumentum (LITE), Corning (GLW), and Ciena (CIEN) round out the major public names. Lumentum received a $2 billion Nvidia investment for laser and photonics components. Corning known mostly for phone glass received $500 million from Nvidia for optical connectivity work and is up over 100% year to date. Ciena runs the optical networking systems between data centers and is seeing analyst price targets raised on the back of the AI optics boom. Every time a hyperscaler spends a billion dollars on Nvidia GPUs, the surrounding infrastructure, cables, switches, transceivers, optical components has to be upgraded to match. The smarter the GPU gets, the more the interconnect matters. Nvidia has committed at least $6.5 billion to photonics companies in the past 4 months alone and the companies building the roads between the GPUs may end up being just as valuable as the companies building the GPUs themselves. Follow me Melvin for more AI, semis and the next big market themes.

Melvin

152,406 Aufrufe • vor 1 Monat

Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.
0:33

Erwin Schrödinger’s 1926 equation changed the game by turning Quantum Mechanics into wave dynamics. That move gave Physics a new way to think. Instead of forcing a particle onto one sharp path, it lets a complex wavefunction evolve in time, with its shape and phase holding the structure of the phenomenon. What makes this so striking is that Schrödinger’s equation does not start from vague mystery. It starts from a precise and daring idea The state of a particle is a complex field ψ(x,t), and the dynamics must push ψ forward in a way that preserves total probability.

Mathelirium

54,730 Aufrufe • vor 4 Monaten

NEW: Inside PsiQuantum's Silicon Photonic Chipset *Never-Before-Seen* With Er-Xuan Ping, SVP, Barium Titanate (BTO) Development Backdrop: the U.S. government just announced a $2B push into domestic quantum computing manufacturing - but what does that actually fund? The U.S. Department of Commerce Department recently awarded PsiQuantum $100M to accelerate development of BTO, giving a rare look into the underlying manufacturing stack required to scale fault-tolerant quantum computing. We profiled PsiQuantum last September following its $1B Series E, including how the company produces the world’s highest-performing optical switch - a core component of its silicon photonics platform. PsiQuantum is currently the only company or institution in the world manufacturing this BTO material for optical switches at 300mm scale. While the company built this silicon photonics platform for fault-tolerant quantum computing (FTQC)*, the implications may extend far beyond quantum itself. As AI data centers increasingly shift from copper to optical networking, PsiQuantum’s photonics stack could also become foundational infrastructure for next-generation AI systems. *Fault-tolerant quantum computing (FTQC) refers to quantum computers that can continue operating accurately even when individual quantum bits (“qubits”) are noisy or error-prone. Shoutout to Pete Shadbolt
13:45

NEW: Inside PsiQuantum's Silicon Photonic Chipset *Never-Before-Seen* With Er-Xuan Ping, SVP, Barium Titanate (BTO) Development Backdrop: the U.S. government just announced a $2B push into domestic quantum computing manufacturing - but what does that actually fund? The U.S. Department of Commerce Department recently awarded PsiQuantum $100M to accelerate development of BTO, giving a rare look into the underlying manufacturing stack required to scale fault-tolerant quantum computing. We profiled PsiQuantum last September following its $1B Series E, including how the company produces the world’s highest-performing optical switch - a core component of its silicon photonics platform. PsiQuantum is currently the only company or institution in the world manufacturing this BTO material for optical switches at 300mm scale. While the company built this silicon photonics platform for fault-tolerant quantum computing (FTQC)*, the implications may extend far beyond quantum itself. As AI data centers increasingly shift from copper to optical networking, PsiQuantum’s photonics stack could also become foundational infrastructure for next-generation AI systems. *Fault-tolerant quantum computing (FTQC) refers to quantum computers that can continue operating accurately even when individual quantum bits (“qubits”) are noisy or error-prone. Shoutout to Pete Shadbolt

Molly O’Shea

36,749 Aufrufe • vor 2 Monaten

Asmongold thinks the Supreme Court ruling on Birthright Citizenship should be ignored by the federal government "I don't think there's any reason that we should be listening to a Supreme Court that tells u a Chinese communist that has a baby here, takes the baby back to China for 35 years..baby comes back and then it can run for president of the US? Nah that's ridiculous" "We should totally ignore the court bc this is just simply not functional right? The court has no enforcement mechanism..the biggest threat to American democracy is the judicial system..has been hijacked by ideologues and other forms of political activists"
3:23

Asmongold thinks the Supreme Court ruling on Birthright Citizenship should be ignored by the federal government "I don't think there's any reason that we should be listening to a Supreme Court that tells u a Chinese communist that has a baby here, takes the baby back to China for 35 years..baby comes back and then it can run for president of the US? Nah that's ridiculous" "We should totally ignore the court bc this is just simply not functional right? The court has no enforcement mechanism..the biggest threat to American democracy is the judicial system..has been hijacked by ideologues and other forms of political activists"

yeet

386,229 Aufrufe • vor 1 Monat