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➡️ Machine Learning from Scratch, part 40: is linear regression machine learning? Linear regression is machine learning, because it learns a weight and a bias from examples with known answers, which makes it supervised learning in its simplest form. What makes it special is that the best weight and...

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MachineLearningToGo

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if you're struggling on where to start learning ML, here’s a playlist of 30 youtube videos to learn machine learning fundamentals from scratch "Machine Learning: Teach by Doing" is a solid choice to learn both theory and code. (1) Introduction to Machine Learning Teach by Doing: (2) What is Machine Learning? History of Machine Learning: (3) Types of ML Models: (4) 6 steps of any ML project: (5) Install Python and VSCode and run your first code: (6) Linear Classifiers Part 1: (7) Linear Classifiers Part 2: (8) Jupyter Notebook, Numpy and Scikit-Learn: (9) Running the Random Linear Classifier Algorithm in Python: (10) The oldest ML model - Perceptron: (11) Coding the Perceptron: (12) Perceptron Convergence Theorem: (13) Magic of features in Machine Learning: (14) One hot encoding: (15) Logistic Regression Part 1: (16) Cross Entropy Loss: (17) How gradient descent works: (18) Logistic Regression from scratch in Python: (19) Introduction to Regularization: (20) Implementing Regularization in Python: (21) Linear Regression Introduction: (22) Ordinary Least Squares step by step implementation: (23) Ridge regression fundamentals and intuition: (24) Regression recap for interviews: (25) Neural network architecture in 30 minutes: (26) Backpropagation intuition: (27) Neural network activation functions: (28) Momentum in gradient descent: (29) Hands on neural network training in Python: (30) Introduction to Convolutional Neural Networks (CNNs):
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if you're struggling on where to start learning ML, here’s a playlist of 30 youtube videos to learn machine learning fundamentals from scratch "Machine Learning: Teach by Doing" is a solid choice to learn both theory and code. (1) Introduction to Machine Learning Teach by Doing: (2) What is Machine Learning? History of Machine Learning: (3) Types of ML Models: (4) 6 steps of any ML project: (5) Install Python and VSCode and run your first code: (6) Linear Classifiers Part 1: (7) Linear Classifiers Part 2: (8) Jupyter Notebook, Numpy and Scikit-Learn: (9) Running the Random Linear Classifier Algorithm in Python: (10) The oldest ML model - Perceptron: (11) Coding the Perceptron: (12) Perceptron Convergence Theorem: (13) Magic of features in Machine Learning: (14) One hot encoding: (15) Logistic Regression Part 1: (16) Cross Entropy Loss: (17) How gradient descent works: (18) Logistic Regression from scratch in Python: (19) Introduction to Regularization: (20) Implementing Regularization in Python: (21) Linear Regression Introduction: (22) Ordinary Least Squares step by step implementation: (23) Ridge regression fundamentals and intuition: (24) Regression recap for interviews: (25) Neural network architecture in 30 minutes: (26) Backpropagation intuition: (27) Neural network activation functions: (28) Momentum in gradient descent: (29) Hands on neural network training in Python: (30) Introduction to Convolutional Neural Networks (CNNs):

ℏεsam

109,764 Aufrufe • vor 1 Jahr

a playlist of 30 youtube videos to learn machine learning fundamentals from scratch if you're struggling on where to start learning ML, this list goes this "Machine Learning: Teach by Doing" is a solid choice to learn both theory and code. (1) Introduction to Machine Learning Teach by Doing: (2) What is Machine Learning? History of Machine Learning: (3) Types of ML Models: (4) 6 steps of any ML project: (5) Install Python and VSCode and run your first code: (6) Linear Classifiers Part 1: (7) Linear Classifiers Part 2: (8) Jupyter Notebook, Numpy and Scikit-Learn: (9) Running the Random Linear Classifier Algorithm in Python: (10) The oldest ML model - Perceptron: (11) Coding the Perceptron: (12) Perceptron Convergence Theorem: (13) Magic of features in Machine Learning: (14) One hot encoding: (15) Logistic Regression Part 1: (16) Cross Entropy Loss: (17) How gradient descent works: (18) Logistic Regression from scratch in Python: (19) Introduction to Regularization: (20) Implementing Regularization in Python: (21) Linear Regression Introduction: (22) Ordinary Least Squares step by step implementation: (23) Ridge regression fundamentals and intuition: (24) Regression recap for interviews: (25) Neural network architecture in 30 minutes: (26) Backpropagation intuition: (27) Neural network activation functions: (28) Momentum in gradient descent: (29) Hands on neural network training in Python: (30) Introduction to Convolutional Neural Networks (CNNs):
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ℏεsam

117,748 Aufrufe • vor 1 Jahr

➡️Machine Learning from Scratch, part 51: dropout. At every training step each hidden neuron flips its own coin and about half of them are switched off, a new random half every step. No neuron can lean on its neighbours, so the knowledge spreads over many neurons, and only patterns that survive these small tests get stronger. After training, dropout is switched off and all neurons work together, like the average of many smaller networks that share the same weights. The code (dropout_reel.py, 16 lines of numpy) trains the same 1-64-1 tanh network twice on the same 12 noisy points of sin(3x), 60,000 steps of plain gradient descent, once the plain way and once with inverted dropout (chance one half, survivors multiplied by two). On 99 new points of the same curve the plain network ends at an error of 0.135 (its training error is almost zero, and its test error grows from 0.026 at step 100), the dropout network at 0.045: three times smaller, with a training error of 0.069. Sources: Hinton et al. 2012, arXiv:1207.0580; Srivastava et al. 2014, JMLR 15. #machinelearning #ai #coding #neuralnetworks #deeplearning
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➡️Machine Learning from Scratch, part 51: dropout. At every training step each hidden neuron flips its own coin and about half of them are switched off, a new random half every step. No neuron can lean on its neighbours, so the knowledge spreads over many neurons, and only patterns that survive these small tests get stronger. After training, dropout is switched off and all neurons work together, like the average of many smaller networks that share the same weights. The code (dropout_reel.py, 16 lines of numpy) trains the same 1-64-1 tanh network twice on the same 12 noisy points of sin(3x), 60,000 steps of plain gradient descent, once the plain way and once with inverted dropout (chance one half, survivors multiplied by two). On 99 new points of the same curve the plain network ends at an error of 0.135 (its training error is almost zero, and its test error grows from 0.026 at step 100), the dropout network at 0.045: three times smaller, with a training error of 0.069. Sources: Hinton et al. 2012, arXiv:1207.0580; Srivastava et al. 2014, JMLR 15. #machinelearning #ai #coding #neuralnetworks #deeplearning

MachineLearningToGo

14,500 Aufrufe • vor 4 Tagen

➡️ Part 9 of learning ML code from scratch Six weights, one wrong answer, and not one of them knows it was at fault. Backpropagation is how they find out. Two questions guide the video: what even is backpropagation, and why do models need it? Step 1, the forward pass. Two numbers go into a tiny network, run through a hidden layer, and one number comes out. Ours says 0.700. We wanted 1.0, so square the difference and the loss is 0.09. That single number is how wrong the model is. Step 2, the question. Six weights produced that answer together. Which of them is to blame, and by how much? Step 3, the one idea. Change a single weight by a tiny amount and watch the error move. How much the error changes per unit of weight is the gradient of that weight. In the video you can see it happen: the weight wobbles and the error bar answers. Step 4, the shortcut. Every weight running into the same neuron starts from one shared number, the blame of that neuron. Multiply that blame by the value each weight carried and you have its gradient. One blame, computed once, and every weight into that neuron gets its own gradient from it. That is why backpropagation is fast enough to train anything at all. Step 5, backwards. The blame starts at the output and every layer hands its share to the layer in front of it, back through the same network the data came forward through. It goes layer by layer, and inside one layer every neuron is done together. Step 6, the update. The optimizer moves every weight against its own gradient, the whole weight matrix in one go, so no weight waits for its turn. All six gradients here are negative, so all six weights go up. After one step the answer reads 0.769. After sixty it reads 0.968 and the error is down to 0.001. The honest part: every gradient in the video was checked against the measured change of the error. Move that one weight, see what the error does, divide. Backprop and the measurement agree to nine decimal places, otherwise nothing would have rendered. #machinelearning #backpropagation #neuralnetworks #python #coding
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➡️ Part 9 of learning ML code from scratch Six weights, one wrong answer, and not one of them knows it was at fault. Backpropagation is how they find out. Two questions guide the video: what even is backpropagation, and why do models need it? Step 1, the forward pass. Two numbers go into a tiny network, run through a hidden layer, and one number comes out. Ours says 0.700. We wanted 1.0, so square the difference and the loss is 0.09. That single number is how wrong the model is. Step 2, the question. Six weights produced that answer together. Which of them is to blame, and by how much? Step 3, the one idea. Change a single weight by a tiny amount and watch the error move. How much the error changes per unit of weight is the gradient of that weight. In the video you can see it happen: the weight wobbles and the error bar answers. Step 4, the shortcut. Every weight running into the same neuron starts from one shared number, the blame of that neuron. Multiply that blame by the value each weight carried and you have its gradient. One blame, computed once, and every weight into that neuron gets its own gradient from it. That is why backpropagation is fast enough to train anything at all. Step 5, backwards. The blame starts at the output and every layer hands its share to the layer in front of it, back through the same network the data came forward through. It goes layer by layer, and inside one layer every neuron is done together. Step 6, the update. The optimizer moves every weight against its own gradient, the whole weight matrix in one go, so no weight waits for its turn. All six gradients here are negative, so all six weights go up. After one step the answer reads 0.769. After sixty it reads 0.968 and the error is down to 0.001. The honest part: every gradient in the video was checked against the measured change of the error. Move that one weight, see what the error does, divide. Backprop and the measurement agree to nine decimal places, otherwise nothing would have rendered. #machinelearning #backpropagation #neuralnetworks #python #coding

MachineLearningToGo

210,300 Aufrufe • vor 1 Monat

➡️ Machine Learning from Scratch, part 47: the Hopfield network. In 1982 the physicist John Hopfield described a network that works as a memory, and 42 years later this work was part of the Nobel Prize in Physics. Each neuron is one pixel that is either on or off, and every neuron is connected to every other neuron, so 100 pixels need 4,950 connections. There are no layers: signals go back and forth until the network settles. Each neuron takes a weighted sum of its inputs and switches on if the sum is positive, off if it is negative. Storing a picture is simple: for every pair of pixels with the same colour the connection gets stronger, otherwise weaker. Every stored picture becomes a valley in the energy landscape, and no switch can ever raise the energy, so a noisy picture rolls downhill into the nearest stored memory. The code (hopfield.py, 16 lines of numpy, pictures in pics.txt) stores three 10 x 10 pictures, a heart, a house and a smiley, and flips 30 of the 100 pixels in each. After a single pass over all neurons all three come back exactly; the energy of the heart falls from -6.90 to -48.90, the energy of the stored heart. With 100 neurons and random patterns (10 flipped pixels, 20 runs) it recalls 5 patterns every time, 10 patterns in 94.5 % of the cases, 14 in 68 %, 20 in about one in four (24 %) and 30 in 1 %. Honest notes: the weights are kept as whole numbers (100 times the Hebb weight), so a sum of exactly zero switches the neuron on, as the rule says. The landscape in the video is a picture of the idea; the valley floors and the height of the rolling ball are the real energies from the run. #machinelearning #ai #coding #neuralnetworks #nobelprize
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➡️ Machine Learning from Scratch, part 47: the Hopfield network. In 1982 the physicist John Hopfield described a network that works as a memory, and 42 years later this work was part of the Nobel Prize in Physics. Each neuron is one pixel that is either on or off, and every neuron is connected to every other neuron, so 100 pixels need 4,950 connections. There are no layers: signals go back and forth until the network settles. Each neuron takes a weighted sum of its inputs and switches on if the sum is positive, off if it is negative. Storing a picture is simple: for every pair of pixels with the same colour the connection gets stronger, otherwise weaker. Every stored picture becomes a valley in the energy landscape, and no switch can ever raise the energy, so a noisy picture rolls downhill into the nearest stored memory. The code (hopfield.py, 16 lines of numpy, pictures in pics.txt) stores three 10 x 10 pictures, a heart, a house and a smiley, and flips 30 of the 100 pixels in each. After a single pass over all neurons all three come back exactly; the energy of the heart falls from -6.90 to -48.90, the energy of the stored heart. With 100 neurons and random patterns (10 flipped pixels, 20 runs) it recalls 5 patterns every time, 10 patterns in 94.5 % of the cases, 14 in 68 %, 20 in about one in four (24 %) and 30 in 1 %. Honest notes: the weights are kept as whole numbers (100 times the Hebb weight), so a sum of exactly zero switches the neuron on, as the rule says. The landscape in the video is a picture of the idea; the valley floors and the height of the rolling ball are the real energies from the run. #machinelearning #ai #coding #neuralnetworks #nobelprize

MachineLearningToGo

11,900 Aufrufe • vor 12 Tagen

Machine Learning from Scratch, part 1. Part 2 adds three lines to this exact program and the line turns into a neuron. #machinelearning #python #gradientdescent #linearregression #neuralnetworks
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Machine Learning from Scratch, part 1. Part 2 adds three lines to this exact program and the line turns into a neuron. #machinelearning #python #gradientdescent #linearregression #neuralnetworks

MachineLearningToGo

357,300 Aufrufe • vor 1 Monat

➡️ ML news part 1 An AI just fixed another AI by rewriting its weights directly, without any training. Machine Learning News, part 1: weight space programming. What is new, and why it matters. Until now, the weights of a neural network were set by training, and reading them took special tools built for that one job. A new report from researchers at MIT CSAIL shows that a general AI model, here GPT-6 Astra used as it is, can read and write those weights directly as plain lists of numbers. Test 1, reading. The model was given only the 17,253 weights of four small game playing networks (Breakout, Pong, Pac-Man, Freeway) and named the right game for all four. Test 2, writing. Without any data, loss or gradients, it wrote networks for three MinAtar games, and all three beat networks trained with PPO for 100 million steps. In Breakout it scored 517.4 points, against 25.3 after 500 million training steps. Test 3, repairing. The small open language model Qwen3.5-0.8B ran into its length limit on 93 percent of MATH-500 answers because it got stuck in loops. The model traced the loop to one direction written by the MLPs of layers 15 and 19 and removed it. That changed 131,072 weights, which is 0.016 percent of the model. MATH-500 went from 7.9 to 48.1 percent, IFEval from 21.0 to 62.3 percent, GSM8K from 13.8 to 67.1 percent, all with no training. The edit was chosen with a budget of 50 trial runs on practice problems, and the test sets were never seen. To be fair, the authors say the fix removed an obstacle and added no new ability, and on another small model it brought no improvement. This is an early report and not yet peer reviewed. Source: Takehi, Teoh, Zhang, Khattab (MIT CSAIL), "Sparks of Weight-Space Programming", blog post, November 2, 2026, (early report, not peer reviewed). Screenshots from the report are shown as quotes. Model names are given only to name the source. #machinelearning #ai #claude #neuralnetworks #llm
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➡️ ML news part 1 An AI just fixed another AI by rewriting its weights directly, without any training. Machine Learning News, part 1: weight space programming. What is new, and why it matters. Until now, the weights of a neural network were set by training, and reading them took special tools built for that one job. A new report from researchers at MIT CSAIL shows that a general AI model, here GPT-6 Astra used as it is, can read and write those weights directly as plain lists of numbers. Test 1, reading. The model was given only the 17,253 weights of four small game playing networks (Breakout, Pong, Pac-Man, Freeway) and named the right game for all four. Test 2, writing. Without any data, loss or gradients, it wrote networks for three MinAtar games, and all three beat networks trained with PPO for 100 million steps. In Breakout it scored 517.4 points, against 25.3 after 500 million training steps. Test 3, repairing. The small open language model Qwen3.5-0.8B ran into its length limit on 93 percent of MATH-500 answers because it got stuck in loops. The model traced the loop to one direction written by the MLPs of layers 15 and 19 and removed it. That changed 131,072 weights, which is 0.016 percent of the model. MATH-500 went from 7.9 to 48.1 percent, IFEval from 21.0 to 62.3 percent, GSM8K from 13.8 to 67.1 percent, all with no training. The edit was chosen with a budget of 50 trial runs on practice problems, and the test sets were never seen. To be fair, the authors say the fix removed an obstacle and added no new ability, and on another small model it brought no improvement. This is an early report and not yet peer reviewed. Source: Takehi, Teoh, Zhang, Khattab (MIT CSAIL), "Sparks of Weight-Space Programming", blog post, November 2, 2026, (early report, not peer reviewed). Screenshots from the report are shown as quotes. Model names are given only to name the source. #machinelearning #ai #claude #neuralnetworks #llm

MachineLearningToGo

55,800 Aufrufe • vor 9 Tagen

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.
0:30

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

17,843 Aufrufe • vor 6 Monaten

Lecture 1 on Physics-Informed Neural Networks: A Mini-Series Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics
1:00

Lecture 1 on Physics-Informed Neural Networks: A Mini-Series Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

47,308 Aufrufe • vor 10 Monaten

What if Your Neural Network Was Forced to Obey Physics? Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics
1:25

What if Your Neural Network Was Forced to Obey Physics? Physics-Informed Neural Networks (PINNs) are neural networks trained to satisfy a differential equation by building the PDE residual directly into the loss. They emerged from a very practical problem...classical PDE pipelines can be brilliant, but they often demand heavy discretization work (meshes, stencils, stability tuning), and the method you build is usually tied to one geometry and one solver setup. A PINN flips the workflow by representing the solution itself as a smooth function uᵩ(x,t) and enforcing the physics everywhere you choose to sample the domain. People often meet PINNs in the least helpful way...via a flashy solution plot, and almost no explanation of what was enforced to get it. In this series we keep the enforcement visible. We pick a differential equation, represent the unknown solution as a flexible function, measure how well that function satisfies the equation across the domain, and train it to reduce that mismatch everywhere we sample. A normal neural net learns from labels...you give it inputs and target outputs. A PINN learns from a differential equation...you give it inputs (x,t) and it gets punished whenever its output fails the PDE. By punish we mean that the loss increases when the mismatch is large we reward it if the loss decreases as the mismatch gets smaller. The network isn’t replacing physics, it’s becoming a flexible function that is forced to satisfy the same calculus you’d impose on any candidate solution. The math breakdown: We start with a PDE we want to solve on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown function u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we would have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or a known initial condition. The training objective is just a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are the collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is that the derivatives inside rᵩ are computed by automatic differentiation ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … So we can differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. This is the whole idea behind PINNs. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold...a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t): each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large (color encodes the sign). As training runs, those threads go slack across the domain not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #PhysicsInformedNeuralNetworks #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

17,459 Aufrufe • vor 5 Monaten

➡️ Machine Learning from Scratch, part 31: the transformer. A transformer turns every token into a list of numbers, adds its position, and then lets every token look at the others. That step is called attention: each token has a query (what am I looking for), a key (what do I contain) and a value (what I pass on). The better a query matches a key, the more of that value flows in. Real transformers repeat this in many layers with several attention heads each. The design comes from the 2017 Google paper "Attention Is All You Need" by Ashish Vaswani and colleagues, first used for translation. A large language model is a transformer trained to predict the next token, again and again, on a huge amount of text. Our own run, nothing downloaded. attention.py is one self-attention layer in 16 lines of numpy. Every word gets a query, a key and a value of 8 numbers, starting random, trained from scratch with hand-written backpropagation on 192 made-up sentences like "the dog ate the carrot because it was hungry". The task: find out which pet "it" means. On 64 new test sentences "it" puts 99.95 percent of its attention on the right word on average (the lowest single sentence gets 99.9 percent; the program prints this as 100%). Before training it was 10 percent, spread evenly. The cat sentence from the video is one of the test sentences. Honest notes: our task is easy on purpose. Every sentence has exactly one pet, so "it" only has to learn to look for pets, and it learns that within the first pass. Real sentences with two candidates need more context, which is what many layers are for. Our small version has no positions, no scaling, no mask, one head, and only "it" asks a question. Every word of the test sentences also appears in training; the combinations are new. The 3D space in the video shows the real keys of our model, projected to three axes (two of them stretched so you can see them). The stacked layers are a sketch. #machinelearning #ai #transformer #python #llm
2:03

➡️ Machine Learning from Scratch, part 31: the transformer. A transformer turns every token into a list of numbers, adds its position, and then lets every token look at the others. That step is called attention: each token has a query (what am I looking for), a key (what do I contain) and a value (what I pass on). The better a query matches a key, the more of that value flows in. Real transformers repeat this in many layers with several attention heads each. The design comes from the 2017 Google paper "Attention Is All You Need" by Ashish Vaswani and colleagues, first used for translation. A large language model is a transformer trained to predict the next token, again and again, on a huge amount of text. Our own run, nothing downloaded. attention.py is one self-attention layer in 16 lines of numpy. Every word gets a query, a key and a value of 8 numbers, starting random, trained from scratch with hand-written backpropagation on 192 made-up sentences like "the dog ate the carrot because it was hungry". The task: find out which pet "it" means. On 64 new test sentences "it" puts 99.95 percent of its attention on the right word on average (the lowest single sentence gets 99.9 percent; the program prints this as 100%). Before training it was 10 percent, spread evenly. The cat sentence from the video is one of the test sentences. Honest notes: our task is easy on purpose. Every sentence has exactly one pet, so "it" only has to learn to look for pets, and it learns that within the first pass. Real sentences with two candidates need more context, which is what many layers are for. Our small version has no positions, no scaling, no mask, one head, and only "it" asks a question. Every word of the test sentences also appears in training; the combinations are new. The 3D space in the video shows the real keys of our model, projected to three axes (two of them stretched so you can see them). The stacked layers are a sketch. #machinelearning #ai #transformer #python #llm

MachineLearningToGo

79,900 Aufrufe • vor 1 Monat

I've solidified a neural net perceptron into a physical circuit. 100 billion of these is ChatGPT. f(x)= wx+b = 1*1 + 1.5 = 2.5 output = weight * input + bias You can change the input, weight and bias and see the output neuron update. Learning ML can be fun!
0:59

I've solidified a neural net perceptron into a physical circuit. 100 billion of these is ChatGPT. f(x)= wx+b = 1*1 + 1.5 = 2.5 output = weight * input + bias You can change the input, weight and bias and see the output neuron update. Learning ML can be fun!

Justin Angel

97,923 Aufrufe • vor 5 Monaten

President Kagame responds to those who say: "Formula 1 is too big for a small country like Rwanda." I tell those people to take a deep breath and calm down, and wait to see what is coming. First, we are people, like the people of other parts of the world. There is nothing, no event, nothing that is bigger than people. So, we are a small country, but we are a people. Formula One is big indeed, but it is not a goal just in itself. In fact, Formula One is part of the big picture of what Rwanda has to do. We made progress as a country, as a people, but also for this particular Formula One, as a sport for Africa. We are not looking at just Rwanda alone. We are looking at Rwanda in the context of the continent, and we have indeed ambitions to bring in other African brothers and sisters to participate. It is Rwandan, it is African, and it is a big deal that we are very happy to be associated with.
1:50

President Kagame responds to those who say: "Formula 1 is too big for a small country like Rwanda." I tell those people to take a deep breath and calm down, and wait to see what is coming. First, we are people, like the people of other parts of the world. There is nothing, no event, nothing that is bigger than people. So, we are a small country, but we are a people. Formula One is big indeed, but it is not a goal just in itself. In fact, Formula One is part of the big picture of what Rwanda has to do. We made progress as a country, as a people, but also for this particular Formula One, as a sport for Africa. We are not looking at just Rwanda alone. We are looking at Rwanda in the context of the continent, and we have indeed ambitions to bring in other African brothers and sisters to participate. It is Rwandan, it is African, and it is a big deal that we are very happy to be associated with.

Rwanda Is Open 🇷🇼

115,337 Aufrufe • vor 1 Monat

Former Meta Chief AI Scientist Yann LeCun on the three paradigms of machine learning — and why the third is what made ChatGPT possible: Here's each one, and where it breaks. First, supervised learning. You tell the machine the answer. "You show it a picture, let's say of a table, and you tell it this is a table. So it's supervised because you tell it what the correct answer is." Get it wrong, and the machine rewrites itself: "The system computes its output, and if it says something else than table, then it's going to adjust its parameters, its internal structure, so that the output it produces gets closer to the output you want." Repeat at scale and something more than memorisation appears: "Eventually the system will find a way to recognize every image you trained it on, but also images it's never seen that are similar to the one you train it on. This is called a generalization ability." The limit: a human has to supply every single answer. That doesn't scale to the size of the internet. Second, reinforcement learning. You don't give the answer, only a verdict. "You don't tell the system what the correct answer is. You only tell it whether the answer it produced was good or bad." Learning to ride a bike, essentially: "You try to ride a bike and you don't know how to ride the bike and after a while you fall. So you know you did something bad and so you change your strategy a little bit. And eventually you learn how to ride a bike." For years the field assumed this was the closest thing to how animals actually learn. Yann LeCun's verdict: "Now it turns out reinforcement learning is extremely inefficient." It dominates wherever failure is free: "It works really well if you want to train a system to play chess or play go or poker, because you can have the system play millions and millions of games against itself and basically fine-tune itself. But it doesn't really work in the real world." The limit, in one image: "If you want to train a car to drive itself, you're not going to do it with reinforcement learning. It's going to crash thousands of times." On robotics he's careful rather than dismissive: "Reinforcement learning can be part of the solution, but it's not the complete answer. It's not sufficient." Third, self-supervised learning. You tell the machine nothing at all. "And this is what has enabled the recent progress in natural language understanding and chatbots." The strange part is that you stop asking for a task: "You don't train the system to accomplish any particular task. You just train it to basically capture the structure..." The method is deliberate sabotage: "You take a piece of text, you corrupt it in some way, by for example removing some words, and then you train a big neural net to predict the words that are missing." And one narrow version of that trick runs every chatbot on Earth: "A special case of this is that you take a piece of text and the last word in that text is not visible, and so you train the system to predict the last word in that text — and this is the way large language models are trained on." So why did the third one win? Supervised learning needs a human. Reinforcement learning needs a crash. Self-supervised learning needs neither — because the missing word and the correct answer are the same thing. The data grades itself.
2:58

Former Meta Chief AI Scientist Yann LeCun on the three paradigms of machine learning — and why the third is what made ChatGPT possible: Here's each one, and where it breaks. First, supervised learning. You tell the machine the answer. "You show it a picture, let's say of a table, and you tell it this is a table. So it's supervised because you tell it what the correct answer is." Get it wrong, and the machine rewrites itself: "The system computes its output, and if it says something else than table, then it's going to adjust its parameters, its internal structure, so that the output it produces gets closer to the output you want." Repeat at scale and something more than memorisation appears: "Eventually the system will find a way to recognize every image you trained it on, but also images it's never seen that are similar to the one you train it on. This is called a generalization ability." The limit: a human has to supply every single answer. That doesn't scale to the size of the internet. Second, reinforcement learning. You don't give the answer, only a verdict. "You don't tell the system what the correct answer is. You only tell it whether the answer it produced was good or bad." Learning to ride a bike, essentially: "You try to ride a bike and you don't know how to ride the bike and after a while you fall. So you know you did something bad and so you change your strategy a little bit. And eventually you learn how to ride a bike." For years the field assumed this was the closest thing to how animals actually learn. Yann LeCun's verdict: "Now it turns out reinforcement learning is extremely inefficient." It dominates wherever failure is free: "It works really well if you want to train a system to play chess or play go or poker, because you can have the system play millions and millions of games against itself and basically fine-tune itself. But it doesn't really work in the real world." The limit, in one image: "If you want to train a car to drive itself, you're not going to do it with reinforcement learning. It's going to crash thousands of times." On robotics he's careful rather than dismissive: "Reinforcement learning can be part of the solution, but it's not the complete answer. It's not sufficient." Third, self-supervised learning. You tell the machine nothing at all. "And this is what has enabled the recent progress in natural language understanding and chatbots." The strange part is that you stop asking for a task: "You don't train the system to accomplish any particular task. You just train it to basically capture the structure..." The method is deliberate sabotage: "You take a piece of text, you corrupt it in some way, by for example removing some words, and then you train a big neural net to predict the words that are missing." And one narrow version of that trick runs every chatbot on Earth: "A special case of this is that you take a piece of text and the last word in that text is not visible, and so you train the system to predict the last word in that text — and this is the way large language models are trained on." So why did the third one win? Supervised learning needs a human. Reinforcement learning needs a crash. Self-supervised learning needs neither — because the missing word and the correct answer are the same thing. The data grades itself.

Big Brain AI

49,293 Aufrufe • vor 3 Monaten

Our world is vast, but we are all in it together. And this is our shared cause – peace. 🇺🇦 proposed its Peace Formula to the world. As long as invaders remain on our land, no one will sit down at the negotiating table with 🇷🇺. The colonizer must get out. And the world has enough power to force 🇷🇺 to restore peace step by step. We have developed the Peace Formula in a way that ensures each of its points is backed by United Nations 🇺🇳 resolutions. And in a way that everyone in the world can choose the track they can contribute to. From Japan to the Arab countries, from Europe to Latin America, we find support for our Formula. And we continue this work.
0:53

Our world is vast, but we are all in it together. And this is our shared cause – peace. 🇺🇦 proposed its Peace Formula to the world. As long as invaders remain on our land, no one will sit down at the negotiating table with 🇷🇺. The colonizer must get out. And the world has enough power to force 🇷🇺 to restore peace step by step. We have developed the Peace Formula in a way that ensures each of its points is backed by United Nations 🇺🇳 resolutions. And in a way that everyone in the world can choose the track they can contribute to. From Japan to the Arab countries, from Europe to Latin America, we find support for our Formula. And we continue this work.

Volodymyr Zelenskyy / Володимир Зеленський

2,286,964 Aufrufe • vor 3 Jahren

Backpropagation by hand ✍️ ~ 11 steps walkthrough below Backpropagation is the algorithm that actually trains a neural network, and it is where most people stop following along. It is not calculus you cannot do. It is matrix multiplication, working backward, one layer at a time. So I drew and calculated one entirely by hand. Goal: push the loss gradient back through a 3-layer network and land on a new value for every weight and bias. = 1. Given = A 3-layer perceptron, an input X, predictions Ypred = [0.5, 0.5, 0], and the truth Ytarget = [0, 1, 0]. = 2. Backprop gradient cells = Let us draw empty cells for every gradient we are about to compute. The shape of the answer comes first. = 3. Layer 3 softmax = We get dL/dz3 straight from Ypred minus Ytarget = [0.5, -0.5, 0]. No chain rule needed, and that shortcut is the whole reason softmax and cross-entropy are paired. = 4. Layer 3 weights and biases = Let us multiply dL/dz3 by [a2 | 1]. One multiplication gives the gradient for W3 and b3 together. = 5. Layer 2 activations = We multiply dL/dz3 by W3 to get dL/da2. The gradient moves back across a layer the same way the signal moved forward. = 6. Layer 2 ReLU = Let us pass it through the gate: keep the gradient where the activation was positive, zero it everywhere else. = 7. Layer 2 weights and biases = We multiply dL/dz2 by [a1 | 1]. The same figure as step 4, one layer up. = 8. Layer 1 activations = Let us multiply dL/dz2 by W2. = 9. Layer 1 ReLU = We apply the same gate again, now on a1. = 10. Layer 1 weights and biases = Let us multiply dL/dz1 by [x | 1], and every weight in the network now has a gradient. = 11. Update = We subtract, and the network has learned. In practice a learning rate scales this step. The gradients: dL/dz3 = [0.5, -0.5, 0] dL/da1 = [1, -2, 2, -1] dL/dz1 = [0, -2, 2, -1] The takeaway: matrix multiplication is all you need. Just like the forward pass, backpropagation is matrix multiplications end to end. You can do every one by hand, slowly and imperfectly, which is exactly why a GPU's ability to do them fast mattered so much to deep learning. 💾 Save this post!
0:26

Backpropagation by hand ✍️ ~ 11 steps walkthrough below Backpropagation is the algorithm that actually trains a neural network, and it is where most people stop following along. It is not calculus you cannot do. It is matrix multiplication, working backward, one layer at a time. So I drew and calculated one entirely by hand. Goal: push the loss gradient back through a 3-layer network and land on a new value for every weight and bias. = 1. Given = A 3-layer perceptron, an input X, predictions Ypred = [0.5, 0.5, 0], and the truth Ytarget = [0, 1, 0]. = 2. Backprop gradient cells = Let us draw empty cells for every gradient we are about to compute. The shape of the answer comes first. = 3. Layer 3 softmax = We get dL/dz3 straight from Ypred minus Ytarget = [0.5, -0.5, 0]. No chain rule needed, and that shortcut is the whole reason softmax and cross-entropy are paired. = 4. Layer 3 weights and biases = Let us multiply dL/dz3 by [a2 | 1]. One multiplication gives the gradient for W3 and b3 together. = 5. Layer 2 activations = We multiply dL/dz3 by W3 to get dL/da2. The gradient moves back across a layer the same way the signal moved forward. = 6. Layer 2 ReLU = Let us pass it through the gate: keep the gradient where the activation was positive, zero it everywhere else. = 7. Layer 2 weights and biases = We multiply dL/dz2 by [a1 | 1]. The same figure as step 4, one layer up. = 8. Layer 1 activations = Let us multiply dL/dz2 by W2. = 9. Layer 1 ReLU = We apply the same gate again, now on a1. = 10. Layer 1 weights and biases = Let us multiply dL/dz1 by [x | 1], and every weight in the network now has a gradient. = 11. Update = We subtract, and the network has learned. In practice a learning rate scales this step. The gradients: dL/dz3 = [0.5, -0.5, 0] dL/da1 = [1, -2, 2, -1] dL/dz1 = [0, -2, 2, -1] The takeaway: matrix multiplication is all you need. Just like the forward pass, backpropagation is matrix multiplications end to end. You can do every one by hand, slowly and imperfectly, which is exactly why a GPU's ability to do them fast mattered so much to deep learning. 💾 Save this post!

Tom Yeh

964,224 Aufrufe • vor 3 Monaten

What are Physics-Informed Neural Networks (PINNs) Physics-Informed Neural Networks (PINNs) are neural nets trained to satisfy a differential equation. The trick is simple. You bake the PDE residual straight into the loss. They came out of a very practical pain point. Classical PDE pipelines can be amazing, but they often demand a lot of setup work. Meshes. Stencils. Stability tuning. And once you build a solver, it’s usually tied to one geometry and one discretization choice. A PINN flips the workflow. You represent the solution itself as a smooth function uᵩ(x,t) and you enforce the physics wherever you choose to sample the domain. Most people first meet PINNs in the least helpful way. A pretty solution surface, almost no clarity on what was enforced to make it appear. In this series we keep the enforcement visible. We pick a PDE, represent the unknown solution as a flexible function, measure how badly that function violates the equation across the domain, and train it to reduce that mismatch at the points we sample. A normal neural net learns from labels. You give it inputs and target outputs. A PINN learns from an equation. You give it inputs (x,t), and it gets penalized whenever its output fails the PDE. Smaller mismatch means smaller loss. Bigger mismatch means bigger loss. That’s all “punish” and “reward” mean here. The network isn’t replacing physics. It’s just a flexible function that we force to obey the same calculus you’d demand from any candidate solution. The math breakdown: We start with a PDE on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we’d have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or from an initial condition. The training objective is a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is how we get the derivatives inside rᵩ. We don’t approximate them with finite differences. We compute them with automatic differentiation: ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … Then we differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. That’s the whole idea. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold, a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t). Each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large, with color showing the sign. As training runs, those threads go slack across the domain, not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics
1:00

What are Physics-Informed Neural Networks (PINNs) Physics-Informed Neural Networks (PINNs) are neural nets trained to satisfy a differential equation. The trick is simple. You bake the PDE residual straight into the loss. They came out of a very practical pain point. Classical PDE pipelines can be amazing, but they often demand a lot of setup work. Meshes. Stencils. Stability tuning. And once you build a solver, it’s usually tied to one geometry and one discretization choice. A PINN flips the workflow. You represent the solution itself as a smooth function uᵩ(x,t) and you enforce the physics wherever you choose to sample the domain. Most people first meet PINNs in the least helpful way. A pretty solution surface, almost no clarity on what was enforced to make it appear. In this series we keep the enforcement visible. We pick a PDE, represent the unknown solution as a flexible function, measure how badly that function violates the equation across the domain, and train it to reduce that mismatch at the points we sample. A normal neural net learns from labels. You give it inputs and target outputs. A PINN learns from an equation. You give it inputs (x,t), and it gets penalized whenever its output fails the PDE. Smaller mismatch means smaller loss. Bigger mismatch means bigger loss. That’s all “punish” and “reward” mean here. The network isn’t replacing physics. It’s just a flexible function that we force to obey the same calculus you’d demand from any candidate solution. The math breakdown: We start with a PDE on a domain Ω. Write it as uₜ(x,t) + N(u(x,t), uₓ(x,t), uₓₓ(x,t), …) = 0 for (x,t) in Ω A PINN replaces the unknown u with a neural network output uᵩ(x,t) Now define the physics residual by plugging uᵩ into the PDE rᵩ(x,t) = ∂uᵩ/∂t + N(uᵩ, ∂uᵩ/∂x, ∂²uᵩ/∂x², …) If uᵩ were an exact solution, we’d have rᵩ(x,t) = 0 everywhere. We may also have data points (xᵢ,tᵢ,uᵢ) from measurements or from an initial condition. The training objective is a weighted sum of squared errors L(ᵩ) = L_data(ᵩ) + λ L_phys(ᵩ) + L_bc/ic(ᵩ) with L_data(ᵩ) = meanᵢ |uᵩ(xᵢ,tᵢ) − uᵢ|² L_phys(ᵩ) = meanⱼ |rᵩ(xⱼ,tⱼ)|² where (xⱼ,tⱼ) are collocation points in Ω L_bc/ic(ᵩ) = penalties enforcing boundary conditions and initial conditions The key technical step is how we get the derivatives inside rᵩ. We don’t approximate them with finite differences. We compute them with automatic differentiation: ∂uᵩ/∂t, ∂uᵩ/∂x, ∂²uᵩ/∂x², … Then we differentiate the total loss L(ᵩ) with respect to ᵩ and train with gradient descent. That’s the whole idea. Learn a function, but make the PDE part of the loss, so the network is trained to be a solution, not just a curve-fitter. In the render, the main 3D surface is the network’s current guess uᵩ(x,t), drawn as a living sheet over the (x,t) plane. Hovering above is the neural scaffold, a visible graph of feature nodes and connections. The bright tension threads are the physics residual rᵩ(x,t). Each thread tethers a collocation bead on the sheet up to the scaffold, and it thickens and brightens exactly where |rᵩ| is large, with color showing the sign. As training runs, those threads go slack across the domain, not because we hid the error, but because the network has actually been pushed toward rᵩ(x,t) ≈ 0. #PINNs #ScientificMachineLearning #PDE #DifferentialEquations #Optimization #MachineLearning #AppliedMath #ComputationalPhysics

Mathelirium

44,924 Aufrufe • vor 9 Monaten

wrote a piece on how we do tiling and gradient descent on TinyTPU included some animations of the weight and bias gradient descent
0:20

wrote a piece on how we do tiling and gradient descent on TinyTPU included some animations of the weight and bias gradient descent

Xander Chin

33,195 Aufrufe • vor 1 Jahr

Infant formula maker ByHeart has expanded its voluntary recall, removing all of its baby formula from shelves. At least 15 babies have been hospitalized across 12 states with infant botulism after consuming ByHeart formula. ByHeart sells an estimated 1% of baby formula in the U.S. and the FDA commissioner says there is no risk of a formula shortage.
2:33

Infant formula maker ByHeart has expanded its voluntary recall, removing all of its baby formula from shelves. At least 15 babies have been hospitalized across 12 states with infant botulism after consuming ByHeart formula. ByHeart sells an estimated 1% of baby formula in the U.S. and the FDA commissioner says there is no risk of a formula shortage.

CBS Mornings

11,587 Aufrufe • vor 1 Jahr

Stanford Prof. Christopher Manning says today’s AI can absorb 2,000 years of human knowledge and still be far worse than humans at learning from just a few examples: "No matter how fantastic these systems are, there's still a lot we can't do. A lot of it looks better than it is because these huge models have been able to slurp up 2,000-plus years of human learning across all domains and stick it all in one huge neural net." "That's amazing because it knows everything to a first approximation. But really that's exploiting hard-won human knowledge, and that's different to the large language model discovering its own knowledge the same way human beings have been for 2,000 years." "At the end of the day, the secret of human intelligence is actually the speed at which we can learn. We only need to see a couple of examples of something, and we can learn something and intuit it." "Our current AI is nothing like that. We're substituting, here are 10,000 examples of this thing, because really the machine learning is very poor compared to human learning." Stanford NLP Group Stanford University
2:55

Stanford Prof. Christopher Manning says today’s AI can absorb 2,000 years of human knowledge and still be far worse than humans at learning from just a few examples: "No matter how fantastic these systems are, there's still a lot we can't do. A lot of it looks better than it is because these huge models have been able to slurp up 2,000-plus years of human learning across all domains and stick it all in one huge neural net." "That's amazing because it knows everything to a first approximation. But really that's exploiting hard-won human knowledge, and that's different to the large language model discovering its own knowledge the same way human beings have been for 2,000 years." "At the end of the day, the secret of human intelligence is actually the speed at which we can learn. We only need to see a couple of examples of something, and we can learn something and intuit it." "Our current AI is nothing like that. We're substituting, here are 10,000 examples of this thing, because really the machine learning is very poor compared to human learning." Stanford NLP Group Stanford University

MTS

16,405 Aufrufe • vor 1 Monat