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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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How to find out " What will be the sample design"?

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Dr. Rishabh Pathak

Determining the sample design is a critical aspect of research methodology that involves selecting a subset of individuals or items from a larger population for study. Here are steps to help you decide on the appropriate sample design for your research:

1.    Define the Population:

o    Clearly define the target population or universe from which you intend to draw your sample. Identify the characteristics, demographics, and parameters that define the population of interest for your study.

2.    Sampling Frame:

o    Create a sampling frame, which is a list or representation of all the elements in the population from which the sample will be selected. Ensure that the sampling frame is comprehensive, up-to-date, and accurately represents the target population.

3.    Sampling Methods:

o Choose a sampling method that aligns with your research objectives, study design, and data collection techniques. Common sampling methods include probability sampling (e.g., simple random sampling, stratified sampling, cluster sampling) and non-probability sampling (e.g., convenience sampling, purposive sampling).

4.    Sample Size:

o    Determine the appropriate sample size based on factors such as the level of precision required, the variability in the population, the desired confidence level, and the resources available for data collection. Use statistical formulas or sampling calculators to estimate the sample size needed for your study.

5.    Sampling Technique:

o    Select a sampling technique that suits the characteristics of your population and the research objectives. Consider whether random sampling, systematic sampling, quota sampling, or other sampling techniques are most suitable for obtaining a representative sample.

6.    Sampling Units:

o    Define the sampling units, which are the individual elements or entities within the population that are eligible for selection in the sample. Determine whether individuals, households, organizations, geographic areas, or other units will form the basis of your sampling design.

7.    Sampling Bias:

o    Identify potential sources of sampling bias that could affect the representativeness of your sample. Take steps to minimize bias through proper sampling techniques, randomization, stratification, or weighting to ensure that the sample accurately reflects the population characteristics.

8.    Sampling Plan:

o    Develop a detailed sampling plan that outlines the procedures for selecting the sample, contacting participants, obtaining consent, and collecting data. Specify the sampling method, sample size, sampling units, sampling frame, and any stratification or clustering strategies to be used.

9.    Pilot Testing:

o    Conduct a pilot test or pretest of your sampling design to assess the feasibility, effectiveness, and practicality of the sampling procedures. Use the pilot test to identify any issues or challenges that may arise during actual data collection and make necessary adjustments.

10.Ethical Considerations:

o  Ensure that your sample design adheres to ethical guidelines, respects participant rights, maintains confidentiality, and obtains informed consent from participants. Consider the ethical implications of sampling methods, data collection procedures, and participant recruitment strategies.

By following these steps and considering factors such as defining the population, creating a sampling frame, choosing sampling methods, determining sample size, selecting sampling techniques, defining sampling units, addressing sampling bias, developing a sampling plan, conducting pilot testing, and addressing ethical considerations, you can effectively determine the sample design for your research study.

 

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