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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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Sampling Procedures

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Dr. Rishabh Pathak

Sampling procedures are essential components of research methodology that involve selecting a subset of individuals or items from a larger population to represent the whole. Sampling procedures play a crucial role in ensuring the validity and reliability of research findings. Here is an overview of sampling procedures in research:


1.    Types of Sampling Procedures:

o    Probability Sampling:

§  Simple Random Sampling: Involves selecting individuals from the population at random, where each member has an equal chance of being chosen.

§  Stratified Sampling: Divides the population into subgroups (strata) based on certain characteristics and then samples from each stratum.

§  Systematic Sampling: Selects every nth individual from a list of the population after a random start.

§  Cluster Sampling: Divides the population into clusters and then randomly selects clusters to sample.

o    Non-Probability Sampling:

§  Convenience Sampling: Involves selecting individuals who are readily available and accessible.

§  Purposive Sampling: Involves selecting individuals based on specific criteria or characteristics relevant to the research study.

§  Snowball Sampling: Involves initial participants recruiting additional participants, often used in studies where the population is hard to reach.

2.    Sampling Design Considerations:

o  Population: Understanding the characteristics and size of the population under study is crucial for selecting an appropriate sampling procedure.

o    Research Objectives: The research questions and objectives guide the selection of the sampling procedure to ensure that the sample represents the population of interest.

o  Resource Constraints: Budgetary limitations, time constraints, and availability of resources influence the choice of sampling procedure.

o    Sampling Error: Researchers aim to minimize sampling error by selecting a sampling procedure that provides a representative sample of the population.

3.    Sampling Procedure Selection:

o  Researchers must carefully consider the characteristics of the population, the research objectives, and the constraints of the study when selecting a sampling procedure.

o   The chosen sampling procedure should align with the research design, data collection methods, and analytical techniques to ensure the validity and reliability of the study findings.

4.    Sampling Error:

o  Sampling error refers to the discrepancy between the characteristics of the sample and the population. Minimizing sampling error is crucial for ensuring the generalizability of research findings.

o    Proper sampling procedures, such as random sampling techniques, can help reduce sampling error and improve the accuracy of research results.

5.    Sample Size Determination:

o  Sampling procedures also involve determining the appropriate sample size to achieve the desired level of precision and confidence in research findings.

o    Sample size calculations consider factors such as the variability of the population, the level of confidence required, and the margin of error acceptable in the study.

In conclusion, sampling procedures are fundamental in research methodology for selecting representative samples from populations of interest. Researchers must carefully choose appropriate sampling procedures based on the research objectives, population characteristics, resource constraints, and desired level of sampling error to ensure the validity and reliability of their research findings. Proper sampling procedures contribute to the credibility and generalizability of research results.

 

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