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If you work in Physics, Machine Learning, Engineering, or any field with a serious mathematical component, Space Mapping is one of those ideas worth adding to your toolkit.

Mathelirium

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27,265 views • 3 months ago •via X (Twitter)

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Physics-Informed Neural Operators: Learning The Solver, Not Just One Solution Our PINN scene learned one solution field for one PDE setup. A Physics-Informed Neural Operator learns the map from input fields, like material coefficients or source terms, to the full solution across a whole family of PDE problems. So, the goal is no longer just one approximate answer, but a reusable solver-shaped object guided by the physics itself.

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The Chinese Physicist Who Turned Force Into Geometry In 1954, Chinese physicist Chen-Ning Yang, together with Robert Mills, introduced a new idea that changed modern physics: Maybe a force is not just something that pushes or pulls. Maybe a force can come from geometry. In Yang-Mills Theory, every point in space carries a hidden internal direction. As you move through space, that direction can twist. If you go around a closed loop and the direction does not return exactly the same, the field has curvature. That curvature is what we see as force. In this animation, the glowing surface shows where the gauge field is strongest. The changing colours show how the hidden internal direction varies across space. The moving rectangular loops are Wilson loops. They travel through the field and measure how much the internal direction changes after completing a closed path. The key equation behind the scene is Fₓᵧ = ∂ₓAᵧ - ∂ᵧAₓ + Aₓ × Aᵧ. The final term, Aₓ × Aᵧ, is the special part. It means the field can interact with itself. That self-interaction is one reason Yang-Mills theory became one of the foundations of particle physics. #China #ChinesePhysics #YangMills #GaugeTheory #QuantumFieldTheory #NonAbelianGeometry #WilsonLoops #ModernPhysics #STEM

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Why Are Some People Still Taking String Theory Seriously? String theory is controversial, but it is still one of the few ideas that tries to describe Quantum Physics and Gravity in one picture. This animation points to one of its best-known ideas: Holography. What happens on the boundary helps determine what happens in the interior, so the edge seems to shape the bulk.

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Ever feel like you've worked hard, learned a lot, yet the leap to innovative originality still feels out of reach? Pure Mathematics is the solution! Pure Mathematics feels optional until you reach areas where the central objects are mathematical structures and not just equations. In Physics, gravity does this because General Relativity is a theory of Spacetime itself...a Topological Manifold. In Engineering it shows up in Optimization when you need to know what assumptions make a method stable, fast or even correct. In Computer Science, it shows up when you stop treating Algorithms as black boxes and start asking WHY they behave the way they do, especially when they surprise you.

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Mathelirium

19,011 views • 4 months ago

String Theory is one of the few frameworks that tries to describe Quantum Physics and Gravity in the same mathematical language. In this animation, the glowing loop is the string, the shifting color along it represent its quantum vibration modes, and the surrounding fabric is a proxy for Spacetime responding. When you quantize a string, one of its vibration modes behaves like a graviton. So, geometry shows up as part of the same quantum system. #StringTheory #QuantumGravity #GeneralRelativity #QuantumPhysics #TheoreticalPhysics #PhysicsVisualization

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The Machine That Learns The Law Behind The Data A very very interesting US Patent US10963540B2 - Physics Informed Learning Machine describes a learning system that does not begin with data alone. It begins with a physical model, usually written as a differential equation (or PDE) dx/dt = f(x,t) A normal Machine Learning model sees scattered data and tries to fit it. A physics-informed learning machine starts with a law. Then it treats the data as evidence that updates what the model believes about the physical system. For this application, I use the patent idea on NASA C-MAPSS Turbofan engine data. The machine watches multivariate telemetry from a degrading engine and infers a hidden health state that is not measured directly. From that posterior belief, it estimates the engine’s remaining useful life. In the main 3D scene, the engine lifetime is turned into a tunnel. The spiral ribbons are real sensor channels evolving over cycle-time. The glowing core is the inferred health state. The surrounding cloud is uncertainty. The orange wall ahead is the predicted failure horizon. So the big picture is: sensor evidence comes in, posterior belief tightens, and the machine moves from uncertainty toward a concrete failure prediction. The inset posteriors make that explicit. The health posterior shows where the model believes the hidden engine condition sits at the current moment, and how sharply it believes it. The RUL posterior shows the same idea for remaining life... early on it is broad, later it shifts left and narrows as the machine becomes more certain about how close failure is. This idea is not limited to engines. The same idea can apply to data centers, CPUs, GPUs, cooling systems, power grids, robotics, batteries, and any machine that produces telemetry while obeying physical constraints. In an age where machine learning runs on massive hardware infrastructure, this kind of model matters: it can turn noisy sensor streams into early warnings before expensive systems fail.

Mathelirium

17,758 views • 2 months ago

Elon Musk Elon Musk on the importance of physics: “I am a big believer in physics. Everything else is a recommendation, pretty much, but physics is the law. Like if you break physics... the rocket will explode and the car will not work. So you have to adhere to physics in order to make cars that work, to make rockets that work, to make space internet work. Because if you're wrong, then reality is a very harsh judge. Reality is merciless. Because if you make a device that doesn't work, violates physics, it simply doesn't work, and then you rocket blows up and the satellites don't work and the cars break. In order to make the technology work, you must have an extremely rigorous pursuit of the truth. Because, otherwise reality will bite you in the a**.”

Elon Musk Elon Musk on the importance of physics: “I am a big believer in physics. Everything else is a recommendation, pretty much, but physics is the law. Like if you break physics... the rocket will explode and the car will not work. So you have to adhere to physics in order to make cars that work, to make rockets that work, to make space internet work. Because if you're wrong, then reality is a very harsh judge. Reality is merciless. Because if you make a device that doesn't work, violates physics, it simply doesn't work, and then you rocket blows up and the satellites don't work and the cars break. In order to make the technology work, you must have an extremely rigorous pursuit of the truth. Because, otherwise reality will bite you in the a**.”

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Can Spacetime Move Faster Than Light? In 1994, Miguel Alcubierre proposed a strange solution to Einstein’s field equations: A Spacetime geometry where a compact region could be transported without the object inside locally breaking the speed of light. The idea is not that a ship accelerates through space faster than light. The local interior remains comparatively calm. The violent part is the surrounding geometry. Space contracts in front of the bubble, expands behind it, and forms a moving transport shell that carries the interior region forward. A simplified form of the Alcubierre metric is ds² = −dt² + [dx − vₛ f(rₛ)dt]² + dy² + dz² where f(rₛ) shapes the warp bubble and vₛ describes the motion of its centre. The physics is still highly speculative, especially because the original construction requires exotic stress-energy, but the mathematical idea is powerful. #GeneralRelativity #AlcubierreDrive #WarpDrive #Spacetime #EinsteinEquations #Relativity #Physics #Astrophysics #Cosmology

Can Spacetime Move Faster Than Light? In 1994, Miguel Alcubierre proposed a strange solution to Einstein’s field equations: A Spacetime geometry where a compact region could be transported without the object inside locally breaking the speed of light. The idea is not that a ship accelerates through space faster than light. The local interior remains comparatively calm. The violent part is the surrounding geometry. Space contracts in front of the bubble, expands behind it, and forms a moving transport shell that carries the interior region forward. A simplified form of the Alcubierre metric is ds² = −dt² + [dx − vₛ f(rₛ)dt]² + dy² + dz² where f(rₛ) shapes the warp bubble and vₛ describes the motion of its centre. The physics is still highly speculative, especially because the original construction requires exotic stress-energy, but the mathematical idea is powerful. #GeneralRelativity #AlcubierreDrive #WarpDrive #Spacetime #EinsteinEquations #Relativity #Physics #Astrophysics #Cosmology

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37,986 views • 9 days ago

We know gravity pulls things together in space… but have you ever watched it work in phase space? In this scene we’re looking at a thin probability sheet in (x, v) position-velocity space evolving under its own self-gravity. In the collisionless limit, Liouville’s theorem says the sheet can stretch, fold, and swirl, but it can’t tear and it can’t change area. Any collapse in x is paid for by stretching in v. Numerically this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line to get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are the places the sheet keeps revisiting dense, sharp caustics carved into phase space. Gravity here isn’t stuff falling down. It’s area-preserving origami in (x, v). #Physics #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow #Mathematics

We know gravity pulls things together in space… but have you ever watched it work in phase space? In this scene we’re looking at a thin probability sheet in (x, v) position-velocity space evolving under its own self-gravity. In the collisionless limit, Liouville’s theorem says the sheet can stretch, fold, and swirl, but it can’t tear and it can’t change area. Any collapse in x is paid for by stretching in v. Numerically this is a 1D cold-dark-matter toy universe: a sheet of particles evolved under the Vlasov-Poisson system. We solve Poisson on a periodic line to get the self-consistent gravitational field, then integrate Hamilton’s equations so the sheet folds into a multi-layer spiral. The glowing filaments are the places the sheet keeps revisiting dense, sharp caustics carved into phase space. Gravity here isn’t stuff falling down. It’s area-preserving origami in (x, v). #Physics #VlasovPoisson #PhaseSpace #Liouville #DarkMatter #HamiltonianFlow #ProbabilityFlow #Mathematics

Mathelirium

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Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

Phase Space Is the Real Stage of Dynamics. So, Stop Watching the Object and Watch the State. Ordinary Space tells you where something is. Phase Space tells you what state it is in. Position alone is not enough because you also need the variable that tells you where the system is trying to go. In the animation, the left panel shows the particle moving in Ordinary Space, while the right panel shows the same system as a moving point in Phase Space. For one degree of freedom, the state is (x, p) and as time evolves, that state traces a trajectory t ↦ (x(t), p(t)) Instead of thinking of a differential equation as just a formula, Phase Space lets you see it as a flow on the space of states. For this lecture we use ẋ = p ṗ = x − x³ with Hamiltonian H(x,p) = 0.5 p² + 0.25 x⁴ − 0.5 x² #PhaseSpace #DynamicalSystems #NonlinearDynamics #HamiltonianMechanics #Mathematics

Mathelirium

30,340 views • 3 months ago

Geometry of Machine Learning Models - Gaussian Process Kernel In 1948 Norbert Wiener framed prediction as a correlation problem, and in the 1970s George Wahba clarified that picking a smoothness preference is the same as picking a kernel. The motivation is that whenever data i s sparse and noisy, you don't just want a fit, you want a fit with calibrated uncertainty.

Geometry of Machine Learning Models - Gaussian Process Kernel In 1948 Norbert Wiener framed prediction as a correlation problem, and in the 1970s George Wahba clarified that picking a smoothness preference is the same as picking a kernel. The motivation is that whenever data i s sparse and noisy, you don't just want a fit, you want a fit with calibrated uncertainty.

Mathelirium

15,625 views • 4 months ago

In @Magicpathai every component is a conversation. Edit and refine, either manually or with AI. Design is the art of keeping good decisions. MagicPath lets you copy and paste interactive components across the canvas to explore and perfect your ideas.

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In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

Mathelirium

45,750 views • 3 months ago

This isn't VFX visual effects...it's actual freakin physics! 🤯🤩🤗🙌🏾 You're watching the physics of space and matter coevolve...a small window into the kind of world Wolfram Physics Wolfram predicts. Every ripple, collision, and shimmer you see is a causal event in a living hypergraph. Space itself is an active network of nodes and links...springs and diagonals...constantly stretching, relaxing, and rewriting as the system evolves. Each interaction you see...a nutrient diffusing through the medium, a burrower tunneling, a tentacle feeling drag...is a computation of spacetime itself. Every connection that forms or breaks is an update to the causal structure: a new link, a new moment in the unfolding hypergraph universe. Our environment behaves like an adaptive fabric...the lattice tightens where organisms stir it, loosens where they pass. Chemical fields spread through those same links, feeding back into motion and growth. When the cluster pulsates, it's literally reorganizing local causal geometry...matter and space changing together!👌🏾 We even test causal invariance frame by frame, swapping operation orders to ensure that spacetime remains consistent no matter the update path. That's not animation logic...that's a computational physics experiment running live!🥳 Tentacles feel friction, burrowers dig through gradients, and behind them the lattice stiffens, storing history as structure. #Mathematics #Physics #WolframPhysics #Mathelirium

This isn't VFX visual effects...it's actual freakin physics! 🤯🤩🤗🙌🏾 You're watching the physics of space and matter coevolve...a small window into the kind of world Wolfram Physics Wolfram predicts. Every ripple, collision, and shimmer you see is a causal event in a living hypergraph. Space itself is an active network of nodes and links...springs and diagonals...constantly stretching, relaxing, and rewriting as the system evolves. Each interaction you see...a nutrient diffusing through the medium, a burrower tunneling, a tentacle feeling drag...is a computation of spacetime itself. Every connection that forms or breaks is an update to the causal structure: a new link, a new moment in the unfolding hypergraph universe. Our environment behaves like an adaptive fabric...the lattice tightens where organisms stir it, loosens where they pass. Chemical fields spread through those same links, feeding back into motion and growth. When the cluster pulsates, it's literally reorganizing local causal geometry...matter and space changing together!👌🏾 We even test causal invariance frame by frame, swapping operation orders to ensure that spacetime remains consistent no matter the update path. That's not animation logic...that's a computational physics experiment running live!🥳 Tentacles feel friction, burrowers dig through gradients, and behind them the lattice stiffens, storing history as structure. #Mathematics #Physics #WolframPhysics #Mathelirium

Mathelirium

20,090 views • 8 months ago

The ultimate Engineering flex is making almost any shape fly.😁 AUREOLE - PVRA TriDisc is our experimental drone concept that replaces spinning propellers with Perimetric Vectoring Ring Arrays (PVRA). We place micro-nozzle belts around tri-lobed ring. These tiny jets push air to generate lift, lateral motion, and rotational control for the craft. This scene is a proof-of-concept rigid body dynamics demo rather than a full CFD. We juste wanted to see if the shape can fly. But everything is still grounded in the language of flight...control systems, aerodynamics, physics and mathematics.

The ultimate Engineering flex is making almost any shape fly.😁 AUREOLE - PVRA TriDisc is our experimental drone concept that replaces spinning propellers with Perimetric Vectoring Ring Arrays (PVRA). We place micro-nozzle belts around tri-lobed ring. These tiny jets push air to generate lift, lateral motion, and rotational control for the craft. This scene is a proof-of-concept rigid body dynamics demo rather than a full CFD. We juste wanted to see if the shape can fly. But everything is still grounded in the language of flight...control systems, aerodynamics, physics and mathematics.

Mathelirium

44,691 views • 4 months ago