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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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Different types of Simple Random Sampling

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Dr. Rishabh Pathak

Simple random sampling is a fundamental sampling technique where each element in the population has an equal probability of being selected for the sample. It is considered one of the most straightforward and unbiased sampling methods. Here are some different types of simple random sampling techniques:


1.    Lottery Method:

§  In the lottery method, each element in the population is assigned a unique number, and a random selection process (e.g., drawing numbers from a hat or using a random number generator) is used to choose the sample. This method ensures that every element has an equal chance of being selected.

2.    Random Number Table:

§  Researchers can use random number tables to select a sample from a population. In this method, each element is assigned a number, and random numbers from the table are used to choose the sample. By following a systematic pattern in the random number table, researchers can ensure randomness in the selection process.

3.    Random Sampling Software:

§  With advancements in technology, researchers can use random sampling software or computer programs to generate random samples from a population. These software tools use algorithms to ensure randomness and eliminate human bias in the selection process. Researchers can input the population data and specify the sample size to obtain a simple random sample.

4.    Simple Random Sampling Without Replacement:

§  In simple random sampling without replacement, once an element is selected for the sample, it is not returned to the population. This ensures that each element is selected only once in the sample, maintaining the randomness of the selection process. This method is commonly used in research studies to avoid duplication of elements in the sample.

5.    Table of Random Digits:

§  A table of random digits is a tool used in simple random sampling to select elements from a population. Researchers can start at a random point in the table and follow a predetermined pattern to choose sample elements. By using a table of random digits, researchers can achieve randomness and avoid bias in the selection process.

6.    Physical Randomization:

§  Physical randomization techniques involve physically mixing or shuffling elements in the population to ensure randomness in sample selection. For example, placing numbered balls in a container and drawing them one by one can be a form of physical randomization to select a sample. This method is simple and transparent, ensuring fairness in the selection process.

Simple random sampling techniques provide researchers with a straightforward and unbiased way to select samples from populations. By employing different methods such as the lottery method, random number tables, and random sampling software, researchers can ensure that each element in the population has an equal chance of being included in the sample, leading to representative and reliable research outcomes.

 


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