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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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Systematic Sampling

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Dr. Rishabh Pathak

Systematic sampling is a method of sampling in which every nth element in a population is selected for inclusion in the sample. It is a systematic and structured approach to sampling that involves selecting elements at regular intervals from an ordered list or sequence. Here are some key points about systematic sampling:

1.    Process:

o    In systematic sampling, the researcher first determines the sampling interval (n) by dividing the population size by the desired sample size. Then, a random starting point is selected, and every nth element from that point is included in the sample until the desired sample size is reached.

2.    Example:

o    For example, if a researcher wants to select a systematic sample of 100 students from a population of 1000 students, they would calculate the sampling interval as 1000/100 = 10. Starting at a random point, every 10th student on the list would be included in the sample.

3.    Advantages:

o    Systematic sampling is relatively easy to implement and is more efficient than simple random sampling in terms of time and cost. It ensures a representative sample by covering the entire population at regular intervals, leading to a more evenly spread sample.

4.    Disadvantages:

o    One potential limitation of systematic sampling is the risk of periodicity in the population, which can introduce bias if the sampling interval coincides with a pattern or cycle in the data. If there is a systematic order or periodic trend in the population, systematic sampling may not provide a truly random sample.

5.    Applications:

o    Systematic sampling is commonly used in surveys, market research, quality control, and various scientific studies where a structured sampling approach is preferred. It is particularly useful when a complete list of the population is available and when researchers want to balance efficiency with randomness.

6.    Comparison with Random Sampling:

o    While systematic sampling is not a random sampling method in the strict sense, it is often treated as a reasonable approximation of random sampling. By starting at a random point and selecting elements at regular intervals, systematic sampling can achieve a level of randomness that is sufficient for many research purposes.

7.    Considerations:

o    To minimize bias in systematic sampling, researchers should ensure that the sampling interval is not related to any underlying patterns in the population. Randomizing the starting point and periodically changing the interval can help reduce the risk of systematic errors in the sample selection process.

Systematic sampling offers a practical and systematic approach to selecting samples from populations, balancing efficiency with randomness. By following established procedures and considering potential sources of bias, researchers can use systematic sampling effectively to obtain representative samples for their studies and surveys.

 

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