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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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Libraries and tools of Python

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Dr. Rishabh Pathak


1. Jupyter Notebook

  • Description: An interactive, browser-based programming environment that supports running and combining live code, narrative text, equations, and images in a single document.
  • Purpose: Makes it easy to perform exploratory data analysis, rapid prototyping, and to communicate results effectively.
  • Usage: Widely used in data science because it facilitates iterative development and visualizations in line with code.

2. NumPy

  • Description: The fundamental package for scientific computing in Python.
  • Core Feature: Provides the ndarray class for efficient, multidimensional arrays that hold elements of the same type.
  • Functionality:
  • High-level mathematical functions, including linear algebra operations and Fourier transforms.
  • Efficient vectorized operations on arrays, which are crucial for performance in numerical computations.
  • Base data structure for most other scientific Python libraries.
  • Importance: Almost all data used with scikit-learn must be converted to NumPy arrays as it forms the core data structure.

3. SciPy

  • Description: Builds on top of NumPy to provide additional functionalities.
  • Functionality:
  • Modules for optimization, integration, interpolation, eigenvalue problems, algebraic equations, and other advanced mathematical computations.
  • Importance: Essential for many scientific computations that require more specialized mathematical operations.

4. matplotlib

  • Description: The primary plotting and visualization library in Python.
  • Functionality:
  • Supports publication-quality static, interactive, and animated plots.
  • Common plot types include line charts, scatter plots, histograms, and many others.
  • Interaction: Integrates tightly with the Jupyter Notebook using magic commands like %matplotlib inline or %matplotlib notebook to display plots directly.
  • Example: You can generate plots with ease — e.g., plotting sine functions with markers — enabling visual exploration of data.

5. pandas

  • Description: A library providing data structures and operations for manipulating numerical tables and time series.
  • Core Constructs:
  • DataFrame: A two-dimensional labeled data structure with columns that can be of different data types, similar to spreadsheets or SQL tables.
  • Series: One-dimensional labeled array.
  • Usage: Widely used for data cleaning, transformation, and analysis, integrating well with NumPy and matplotlib.

6. mglearn

  • Description: A utility library created specifically for this book.
  • Purpose: It contains functions to simplify tasks such as plotting and loading datasets, so code examples remain clear and focused on machine learning concepts.
  • Note: While useful for learning and creating visual demonstrations, it’s not essential for practical machine learning applications outside the book’s context.

7. scikit-learn

  • Description: The most prominent and widely-used Python machine learning library.
  • Functionality:
  • Provides simple, efficient tools for data mining, machine learning, and statistical modeling.
  • Implements a wide range of algorithms, including classification, regression, clustering, dimensionality reduction, model selection, and preprocessing.
  • Integration: Built on NumPy and SciPy, and designed to work well with pandas and matplotlib.
  • Popularity and Support: Open source with extensive documentation and a large community; suitable for both academic and industrial usage.

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