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Andrej Karpathy summarizes how neural networks function, including the process of using a loss function in training and applying backpropagation for optimizing the network parameters through gradient descent.

tetsuo

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141,964 次观看 • 7 个月前 •via X (Twitter)

科学技术教育

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This video, created by my dear coauthor Mahdi E Kahou for our teaching and papers, shows how overparameterized neural networks produce smooth function approximations even in the context of the Runge phenomenon. Some background. Imagine you want to approximate the Runge function using polynomial interpolation at equally spaced points. It is well known that, despite targeting an infinitely differentiable function, such a polynomial approximation produces oscillatory behavior that worsens with the degree of the polynomial. In other words, higher-degree polynomial approximations might not improve accuracy. Instead, approximate the Runge function with a neural network (here, two layers are just to make the example concrete; nothing fundamental depends on it). As you increase the number of parameters well above the 11 training points (in our example, a two-layer neural network with 128 nodes each), you nicely converge to the target, without wild oscillations. Yes, this has much to do with double descent and benign overparameterization, but the main punchline of this post is that neural networks are really very different types of animals than polynomial approximations. And yes, Chebyshev nodes and splines exist, and in this case, they will prevent the oscillations. But that's not the point. Chebyshev nodes and splines still confront Faber’s theorem, which states that for any system of polynomial interpolation nodes, there exists a continuous function whose sequence of interpolating polynomials diverges as the number of nodes grows to infinity. Faber’s theorem does not apply to neural networks because they are not polynomials. The notebook, if you want to check the details, is here: Stay tuned for more on this 👀

This video, created by my dear coauthor Mahdi E Kahou for our teaching and papers, shows how overparameterized neural networks produce smooth function approximations even in the context of the Runge phenomenon. Some background. Imagine you want to approximate the Runge function using polynomial interpolation at equally spaced points. It is well known that, despite targeting an infinitely differentiable function, such a polynomial approximation produces oscillatory behavior that worsens with the degree of the polynomial. In other words, higher-degree polynomial approximations might not improve accuracy. Instead, approximate the Runge function with a neural network (here, two layers are just to make the example concrete; nothing fundamental depends on it). As you increase the number of parameters well above the 11 training points (in our example, a two-layer neural network with 128 nodes each), you nicely converge to the target, without wild oscillations. Yes, this has much to do with double descent and benign overparameterization, but the main punchline of this post is that neural networks are really very different types of animals than polynomial approximations. And yes, Chebyshev nodes and splines exist, and in this case, they will prevent the oscillations. But that's not the point. Chebyshev nodes and splines still confront Faber’s theorem, which states that for any system of polynomial interpolation nodes, there exists a continuous function whose sequence of interpolating polynomials diverges as the number of nodes grows to infinity. Faber’s theorem does not apply to neural networks because they are not polynomials. The notebook, if you want to check the details, is here: Stay tuned for more on this 👀

Jesús Fernández-Villaverde

46,527 次观看 • 25 天前

Individual Neuron: Neural Network A neuron in a neural network performs a weighted sum of inputs, adds a bias, and applies an activation function like sigmoid, ReLU, or tanh, introducing non-linearity. This output helps the neuron learn and represent patterns in the data.

Individual Neuron: Neural Network A neuron in a neural network performs a weighted sum of inputs, adds a bias, and applies an activation function like sigmoid, ReLU, or tanh, introducing non-linearity. This output helps the neuron learn and represent patterns in the data.

ₕₐₘₚₜₒₙ — e/acc

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gradient descent navigating the loss landscape

gradient descent navigating the loss landscape

tender

224,330 次观看 • 4 个月前

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

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

33,044 次观看 • 9 个月前

Ever wondered what neural networks are and how they work? Systems like ChatGPT use neural networks to work as well as they do. Neural networks are composed of "layers" of neurons, layers with different functions; connections between layers called "weights"; and mathematical functions called "activation functions". If you’re interested in learning about these systems, check the comments. Ultimately, the neural network structure of the model serves to visually demonstrate that it is, in fact, a complex mathematical equation. When companies release the model's weights, they are releasing a key component needed to run the model's complete equation. Without the weights, the equation is incomplete. For the math-minded: the weights of a model are the learned numbers (they are variables during training) that are then used as constants in the mathematical functions that make up the model. Neural networks are ultimately just one big, hyper-complex mathematical function, and when a model is trained, it learns the constants associated with the high-dimensional input.

Ever wondered what neural networks are and how they work? Systems like ChatGPT use neural networks to work as well as they do. Neural networks are composed of "layers" of neurons, layers with different functions; connections between layers called "weights"; and mathematical functions called "activation functions". If you’re interested in learning about these systems, check the comments. Ultimately, the neural network structure of the model serves to visually demonstrate that it is, in fact, a complex mathematical equation. When companies release the model's weights, they are releasing a key component needed to run the model's complete equation. Without the weights, the equation is incomplete. For the math-minded: the weights of a model are the learned numbers (they are variables during training) that are then used as constants in the mathematical functions that make up the model. Neural networks are ultimately just one big, hyper-complex mathematical function, and when a model is trained, it learns the constants associated with the high-dimensional input.

Harper Carroll

26,930 次观看 • 7 个月前

this is how machine learning actually works (gradient descent with bad starting parameters)

this is how machine learning actually works (gradient descent with bad starting parameters)

LaurieWired

38,694 次观看 • 1 年前

#SIGGRAPHAsia2024 🔥 Replace your Marching Cubes code with our Occupancy-Based Dual Contouring to reveal the "real" shape from either a signed distance function or an occupancy function. No neural networks involved. Web:

#SIGGRAPHAsia2024 🔥 Replace your Marching Cubes code with our Occupancy-Based Dual Contouring to reveal the "real" shape from either a signed distance function or an occupancy function. No neural networks involved. Web:

Minhyuk Sung

14,627 次观看 • 1 年前

I've added a function to propagate a gradient across the mesh (it follows the curvature of the hair), and then copy that gradient to the Windows clipboard as a transparent picture, for use as a selection mask in the image editor

I've added a function to propagate a gradient across the mesh (it follows the curvature of the hair), and then copy that gradient to the Windows clipboard as a transparent picture, for use as a selection mask in the image editor

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13,261 次观看 • 1 年前

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):

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):

ℏεsam

117,570 次观看 • 1 年前

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):

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

108,861 次观看 • 1 年前

we're turning University of Waterloo into a supercomputer! arceus is a cross-device distributed compute network for training large models, using model/tensor/pipeline parallelism. you can train anything, from deep neural networks to language models on the network. oss & deployment soon!

we're turning University of Waterloo into a supercomputer! arceus is a cross-device distributed compute network for training large models, using model/tensor/pipeline parallelism. you can train anything, from deep neural networks to language models on the network. oss & deployment soon!

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35,534 次观看 • 1 年前

JANE STREET PAYS $650,000 A YEAR FOR QUANTS WHO TRADE USING NEURAL NETWORKS. STANFORD JUST DROPPED THE COMPLETE NEURAL NETWORK FUNDAMENTALS LECTURE FOR FREE. BOOKMARK & WATCH THIS TODAY, THEN READ HOW QUANTS APPLY THIS IN THE ARTICLE BELOW BEFORE SOMEONE TAKES IT DOWN.

JANE STREET PAYS $650,000 A YEAR FOR QUANTS WHO TRADE USING NEURAL NETWORKS. STANFORD JUST DROPPED THE COMPLETE NEURAL NETWORK FUNDAMENTALS LECTURE FOR FREE. BOOKMARK & WATCH THIS TODAY, THEN READ HOW QUANTS APPLY THIS IN THE ARTICLE BELOW BEFORE SOMEONE TAKES IT DOWN.

Roan

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Does anyone know what the function of the black and hot liquid is in this process?

Does anyone know what the function of the black and hot liquid is in this process?

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How to combine first and last names using the TEXTJOIN function. 🤓

How to combine first and last names using the TEXTJOIN function. 🤓

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A 2-layer neural network goes from total chaos to perfectly separating left vs right classes in real time. Watch the decision boundary form live as gradient descent works its magic! Pure maths beauty in motion..

A 2-layer neural network goes from total chaos to perfectly separating left vs right classes in real time. Watch the decision boundary form live as gradient descent works its magic! Pure maths beauty in motion..

Mathematica

144,883 次观看 • 2 个月前

$If one remains only in ascent, the vessel stays uninhabited. Clarity without descent fractures the body, splits function, and leaves life unrealized. Only descent completes the vessel.$

If one remains only in ascent, the vessel stays uninhabited. Clarity without descent fractures the body, splits function, and leaves life unrealized. Only descent completes the vessel.

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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

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

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How to insert images from the web using the IMAGE function. 📸

How to insert images from the web using the IMAGE function. 📸

Excel Dictionary

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Implemented a visualization for the neural network today. This is a neural network being trained to learn the snake game in a terminal. It's built in Rust using Ratatui More input details in thread

Implemented a visualization for the neural network today. This is a neural network being trained to learn the snake game in a terminal. It's built in Rust using Ratatui More input details in thread

Bones

165,629 次观看 • 1 年前