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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 Select a Random Sample?

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Dr. Rishabh Pathak

Selecting a random sample is a crucial aspect of research methodology to ensure the representativeness and generalizability of study findings. Here are some common methods and considerations for selecting a random sample:


1.    Simple Random Sampling:

o    In simple random sampling, each element in the population has an equal chance of being selected for the sample.

o    One method is to assign a unique identifier (e.g., numbers) to each element in the population and then use a random number generator to select sample units.

o    Another approach is to use random sampling techniques such as lottery methods or random number tables to choose sample units.

2.    Systematic Sampling:

o    In systematic sampling, researchers select every nth element from a list of the population after randomly determining a starting point.

o    This method is efficient and easy to implement, especially when the population is ordered or arranged in a sequence.

3.    Stratified Sampling:

o Stratified sampling involves dividing the population into homogeneous subgroups (strata) based on certain characteristics and then randomly selecting samples from each stratum.

o    This method ensures representation from different subgroups and allows for comparisons between strata.

4.    Cluster Sampling:

o    Cluster sampling involves dividing the population into clusters or groups, randomly selecting clusters, and then sampling all elements within the chosen clusters.

o    This method is useful when it is impractical to sample individuals directly from the population.

5.    Considerations for Random Sampling:

o    Define the population of interest and clearly specify the sampling frame (list of all elements in the population).

o    Use randomization techniques to ensure that each element in the population has an equal chance of being selected.

o    Minimize bias by avoiding non-random selection methods and ensuring transparency in the sampling process.

o    Consider the sample size needed to achieve the desired level of precision and statistical power.

o    Document the sampling procedure and any deviations from the original plan to maintain transparency and replicability.

6.    Practical Implementation:

o Use random number generators, software tools, or statistical packages to facilitate random sample selection.

o    Ensure that the sampling process is conducted in a systematic and unbiased manner to uphold the principles of random sampling.

By following established random sampling techniques and considering key factors such as representativeness, transparency, and precision, researchers can select a random sample that accurately reflects the characteristics of the population of interest. Random sampling is essential for generating reliable and valid research results that can be generalized to the broader population.

 

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