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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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Sample Survey

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Dr. Rishabh Pathak

A sample survey is a research method that involves collecting data from a subset or sample of individuals or units selected from a larger population or universe. Here are some key points related to sample surveys:


1.    Definition:

o    A sample survey is a research method that involves collecting data from a representative subset of the population rather than from every individual or unit in the population. The sample is selected using various sampling techniques to ensure that it is representative of the larger population.

2.    Purpose:

o    Sample surveys are conducted to gather information about a population by studying a smaller, manageable subset of individuals or units. The data collected from the sample are then used to make inferences and draw conclusions about the population as a whole.

3.    Representativeness:

o    The key principle of sample surveys is to ensure that the sample selected is representative of the population from which it is drawn. By using appropriate sampling techniques, researchers aim to minimize bias and ensure that the sample accurately reflects the characteristics of the larger population.

4.    Efficiency:

o    Sample surveys are often more efficient in terms of time, cost, and resources compared to conducting a census. By studying a sample rather than the entire population, researchers can obtain valuable insights and make informed decisions while saving time and resources.

5.    Sampling Techniques:

o    Various sampling techniques are used in sample surveys, including random sampling, stratified sampling, cluster sampling, systematic sampling, convenience sampling, and quota sampling. The choice of sampling method depends on the research objectives, population characteristics, and available resources.

6.    Data Collection:

o    Data in sample surveys are typically collected through methods such as interviews, questionnaires, observations, or a combination of these approaches. Researchers must ensure that data collection procedures are standardized, reliable, and consistent to maintain the quality of the survey results.

7.    Analysis and Inference:

o    Once data are collected from the sample, researchers analyze the information to draw conclusions about the population. Statistical techniques are used to make inferences and generalize the findings from the sample to the larger population with a certain level of confidence.

8.    Validity and Reliability:

o    The validity and reliability of the survey findings depend on the quality of the sample design, data collection methods, and analysis techniques. Researchers must ensure that the sample is representative, the data are accurate, and the conclusions drawn are valid and reliable.

In summary, sample surveys are valuable research tools that allow researchers to study a subset of the population in a systematic and efficient manner. By selecting a representative sample and using appropriate methods for data collection and analysis, sample surveys provide valuable insights into population characteristics and help inform decision-making in various fields of study.

 

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