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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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Standard Error of Mean (SEM)

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Dr. Rishabh Pathak

The Standard Error of the Mean (SEM) is a statistical measure that indicates the precision with which the sample mean estimates the population mean. Here is an explanation of the Standard Error of the Mean (SEM):


1.      Definition:

oThe Standard Error of the Mean (SEM) is a measure of the variability of sample means around the true population mean. It quantifies the accuracy of the sample mean as an estimate of the population mean.

2.     Calculation:

oThe SEM is calculated as the standard deviation of the sample divided by the square root of the sample size. Mathematically, SEM = SD / √(n), where SD is the standard deviation of the sample and n is the sample size.

3.     Interpretation:

oA smaller SEM indicates that the sample mean is likely to be close to the population mean, while a larger SEM suggests that the sample mean may be less precise in estimating the population mean.

4.    Confidence Interval:

oThe SEM is often used to calculate the confidence interval around the sample mean. The confidence interval provides a range within which the true population mean is likely to fall.

5.     Significance:

oResearchers use the SEM to assess the reliability of the sample mean and to determine the level of uncertainty associated with the estimate of the population mean. A smaller SEM indicates more precise estimates.

6.    Comparison with Standard Deviation:

oWhile the standard deviation measures the dispersion of data points around the sample mean, the SEM specifically quantifies the precision of the sample mean as an estimate of the population mean.

7.     Application:

oThe SEM is commonly reported in research studies, especially in scientific publications, to provide information about the reliability and accuracy of the reported sample means.

In summary, the Standard Error of the Mean (SEM) is a statistical measure that reflects the precision of the sample mean as an estimate of the population mean. It is calculated based on the standard deviation of the sample and the sample size, providing valuable information about the variability and reliability of the sample mean in representing the true population mean.

 

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