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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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Random Number Table

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Dr. Rishabh Pathak

A random number table is a tool used in statistical sampling to generate random numbers for the purpose of selecting samples from a population. Random number tables consist of rows and columns of digits that have been arranged in a random order. Researchers can use these tables to ensure a systematic and unbiased selection process in simple random sampling. Here is an overview of how random number tables are used in sampling:


1.    Structure of Random Number Tables:

o    Random number tables typically consist of rows and columns of digits ranging from 0 to 9. The digits are arranged in a random sequence to eliminate any patterns or biases. Each digit in the table has an equal probability of being selected, ensuring randomness in the sampling process.

2.    Selection Process:

o    To use a random number table for sampling, researchers start by identifying a random starting point in the table. This starting point serves as the initial reference for selecting random numbers. Researchers then follow a predetermined pattern to read the digits from the table systematically.

3.    Assigning Numbers to Population Elements:

o    Before using the random number table, researchers assign numbers to each element in the population. These numbers serve as identifiers for the population elements and are used in the sampling process. Each element is associated with a unique number to facilitate random selection.

4.    Generating Random Samples:

o    Researchers use the random numbers obtained from the table to select sample elements from the population. By following a systematic pattern in reading the digits from the table, researchers can ensure that the sample selection process is random and unbiased. This method helps in avoiding human bias and ensures equal chances for all population elements to be included in the sample.

5.    Advantages of Random Number Tables:

o    Random number tables provide a structured and transparent way of conducting simple random sampling. They offer a systematic approach to selecting samples without introducing personal biases or preferences. By using random number tables, researchers can achieve randomness and representativeness in their samples, leading to reliable research outcomes.

6.    Limitations:

o    While random number tables are useful for generating random samples, they may be time-consuming when selecting large samples. Researchers need to ensure that the random number table is truly random and free from any patterns or biases. Additionally, the use of technology, such as random number generators, has become more common for generating random numbers efficiently.

By utilizing random number tables in sampling, researchers can ensure the fairness and randomness of sample selection processes. These tables provide a structured method for selecting samples from populations, contributing to the validity and reliability of research studies.

 

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