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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 Designs

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Dr. Rishabh Pathak

Sample designs are an essential component of research methodology that involve the selection of a subset of individuals or items from a larger population for the purpose of study. Here are some key points related to sample designs:


1.    Definition of Universe:

o    The first step in developing a sample design is to clearly define the universe, which refers to the total set of objects or individuals that the researcher is interested in studying. The universe can be finite (with a known number of items) or infinite (with an unknown number of items).

2.    Types of Sample Designs:

o    Researchers can choose from various sample designs based on their research objectives and the characteristics of the population under study. Some common types of sample designs include random sampling, stratified sampling, cluster sampling, systematic sampling, convenience sampling, and quota sampling.

3.    Sample Size Determination:

o    Sample design also involves determining the size of the sample, which refers to the number of individuals or items to be included in the study. The sample size is determined before data collection begins and is influenced by factors such as the research objectives, population characteristics, and desired level of precision.

4.    Steps in Sample Design:

o  When developing a sample design, researchers need to consider several key steps, including:

§  Defining the universe to be studied.

§  Selecting the appropriate sampling method.

§  Determining the sample size.

§  Implementing the sampling plan.

§  Collecting and analyzing the data obtained from the sample.

5.    Characteristics of a Good Sample Design:

o  A good sample design should be reliable, appropriate for the research study, and capable of providing valid and generalizable results. It should also consider factors such as representativeness, sampling error, and the practicality of data collection.

6.    Selection of Sampling Procedure:

o  Researchers must carefully consider the criteria for selecting a sampling procedure based on the research objectives, the nature of the population, the available resources, and the desired level of precision. The choice of sampling method can significantly impact the validity and reliability of the study results.

7.    Importance of Sample Design:

o  The sample design is crucial in ensuring the accuracy and reliability of research findings. A well-designed sample helps researchers draw valid inferences about the population from which the sample was drawn. It also allows for the generalization of study results to the larger population.

In summary, sample designs play a critical role in research methodology by guiding the selection of a representative subset of the population for study. By following appropriate sample design principles and methods, researchers can enhance the validity, reliability, and generalizability of their research findings.

 

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