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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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Steps in Sample Designs

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Dr. Rishabh Pathak

The steps involved in designing a sample for a research study are crucial for ensuring the representativeness and reliability of the data collected. Here is a detailed explanation of the steps in sample design:


1.    Define the Universe:

o    The first step in sample design is to clearly define the target population or universe from which the sample will be drawn. The universe can be finite (with a known number of elements) or infinite (with an unknown number of elements). Defining the universe helps in determining the scope and boundaries of the study.

2.    Select the Sampling Frame:

o    The sampling frame is a list of all the elements or units in the population from which the sample will be selected. It is essential to have a comprehensive and accurate sampling frame to ensure that all elements in the population have an equal chance of being included in the sample. The sampling frame serves as the basis for selecting the sample.

3.    Choose a Sampling Method:

o    There are various sampling methods available, such as 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. Each sampling method has its advantages and limitations in terms of representativeness and efficiency.

4.    Determine Sample Size:

o    The sample size refers to the number of elements or units to be included in the sample. Determining the appropriate sample size is crucial for achieving the desired level of precision and confidence in the study results. Factors such as population variability, desired level of confidence, and budget constraints influence the determination of sample size.

5.    Select the Sample:

o    Once the sampling method and sample size are determined, the actual selection of the sample takes place. The sample should be selected in a systematic and unbiased manner to ensure representativeness. Randomization techniques are often used to minimize selection bias and ensure that each element in the population has an equal chance of being included in the sample.

6.    Implement Quality Control Measures:

o    Quality control measures are essential to ensure the reliability and validity of the data collected from the sample. Researchers should implement protocols for data collection, data entry, and data verification to minimize errors and inconsistencies. Quality control measures help in maintaining the integrity of the study findings.

7.    Pilot Testing:

o  Before conducting the main data collection, researchers may conduct a pilot test of the sample design to identify any potential issues or challenges. Pilot testing helps in refining the sampling procedures, data collection instruments, and overall research methodology before implementing the study on a larger scale.

8.    Monitor and Adjust:

o    Throughout the data collection process, researchers should monitor the sampling procedures and data quality to ensure that the sample design is being implemented effectively. If any issues or deviations are identified, adjustments may be made to maintain the integrity and validity of the study results.

By following these steps in sample design, researchers can ensure that the sample selected is representative, reliable, and suitable for making valid inferences about the larger population. Careful planning and execution of the sample design are essential for the success of a research study and the credibility of its findings.

 

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