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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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Criteria of Selecting a Sample Procedures

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Dr. Rishabh Pathak

Criteria for selecting sampling procedures are essential in research methodology to ensure that the chosen method is appropriate for the study objectives, population characteristics, and resource constraints. Here are some key criteria for selecting sampling procedures:


1.    Representativeness:

o The sampling procedure should result in a sample that is representative of the population under study. It should accurately reflect the characteristics, diversity, and variability of the population to allow for generalization of findings.

2.    Accuracy:

o    The sampling procedure should yield results that are accurate and reliable. It should minimize sampling error and bias to ensure that the findings are valid and trustworthy.

3.    Efficiency:

o    The selected sampling procedure should be efficient in terms of time, cost, and resources. It should provide the required level of precision and confidence without unnecessary expenditure of resources.

4.    Feasibility:

o    The sampling procedure should be feasible within the constraints of the research study, including budgetary limitations, time constraints, availability of resources, and logistical considerations.

5.    Generalizability:

o The sampling procedure should allow for generalizability of findings beyond the sample to the larger population. It should enable researchers to draw valid inferences and make meaningful conclusions based on the sample data.

6.    Bias Reduction:

o    The sampling procedure should minimize bias in sample selection to ensure that the sample is not skewed or unrepresentative of the population. Researchers should aim to reduce systematic bias and increase the reliability of results.

7.    Precision:

o    The sampling procedure should be designed to achieve the desired level of precision in estimating population parameters. It should provide accurate estimates with a known level of confidence to support robust statistical analysis.

8.    Compatibility:

o    The sampling procedure should be compatible with the research design, data collection methods, and analytical techniques employed in the study. It should align with the overall research framework and objectives.

9.    Ethical Considerations:

o    Researchers should consider ethical implications when selecting sampling procedures, ensuring that the rights and privacy of participants are respected. Ethical sampling practices contribute to the credibility and integrity of the research.

By considering these criteria when selecting sampling procedures, researchers can ensure that the chosen method is appropriate, reliable, and valid for their research study. Adhering to sound sampling principles enhances the quality and rigor of research findings, ultimately contributing to the credibility and impact of the research outcomes.

 

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