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Energy Manifold Natural Gradient Descent: From Riemannian Optimization to Modern Neuroscience, NeuroAI and Quantum Physics

Dr. Rishabh Pathak
When Geometry, Energy, Artificial Intelligence and Neuroscience Converge Modern Artificial Intelligence is rapidly moving beyond the idea that learning simply means minimizing an error function. Increasingly, researchers are asking a deeper question: what is the structure of the space in which learning takes place? This question becomes particularly important when the system being modelled is constrained, nonlinear, dynamic, or governed by physical principles. A recent work titled “Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers” , by Zhangyong Liang and Huanhuan Gao, introduces Energy Manifold Natural Gradient Descent (EMNGD) , a mathematical framework that extends energy-based natural-gradient optimization from unconstrained Euclidean parameter spaces to constrained Riemannian parameter manifolds . At its core, the framework proposes a simple but powerful principle: An optimization algorithm should not only determine how to reduce error; it sh...
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Supervised Learning

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Dr. Rishabh Pathak

What is Supervised Learning?

·   Definition: Supervised learning involves training a model on a labeled dataset, where the input data (features) are paired with the correct output (labels). The model learns to map inputs to outputs and can predict labels for unseen input data.

·   Goal: To learn a function that generalizes well from training data to accurately predict labels for new data.

·         Types:

·         Classification: Predicting categorical labels (e.g., classifying iris flowers into species).

·         Regression: Predicting continuous values (e.g., predicting house prices).

Key Concepts:

·   Generalization: The ability of a model to perform well on previously unseen data, not just the training data.

·       Overfitting and Underfitting:

·    Overfitting: The model learns noise in the training data, performing very well on training data but poorly on new data.

·    Underfitting: The model is too simple to capture the underlying pattern, resulting in poor performance on both training and testing data.

·  Relation to Model Complexity: The model's complexity must be appropriate for the size and nature of the dataset to avoid overfitting or underfitting.

Popular Supervised Learning Algorithms Covered:

·         k-Nearest Neighbors (k-NN): Classifies data points based on the labels of their nearest neighbors in the feature space.

·         Linear Models: Includes linear regression and logistic regression, which make predictions based on a linear combination of input features.

·      Naive Bayes Classifier: Probabilistic classifiers based on Bayes’ theorem with strong independence assumptions between features.

·         Decision Trees: Models that split data into branches to make predictions based on feature thresholds.

·   Ensembles of Decision Trees: Methods like Random Forests and Gradient Boosting that combine multiple trees to improve performance.

·     Support Vector Machines (SVM): Effective for classification tasks by finding the hyperplane that best separates classes.

·   Neural Networks (Deep Learning): Models inspired by biological neural networks capable of learning complex patterns.

Practical Application Example:

  • Early in the book, supervised learning is illustrated with the classic problem of classifying iris flowers into several species based on physical measurements such as petal and sepal length.
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