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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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Different types of Complex Random Sampling

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Dr. Rishabh Pathak

Complex random sampling designs encompass various advanced sampling techniques that go beyond simple random sampling. These designs involve a combination of probability and non-probability sampling methods to address specific research requirements. Here are some different types of complex random sampling designs:


1.    Systematic Sampling:

§  Systematic sampling is a method where researchers select every ith element from a list after a random start. This approach combines elements of randomness (random start) with a systematic selection process. Systematic sampling is efficient and easier to implement compared to simple random sampling, especially when the population is ordered.

2.    Stratified Sampling:

§  Stratified sampling involves dividing the population into homogeneous subgroups or strata based on certain characteristics (e.g., age, gender, income) and then independently selecting samples from each stratum using a probability sampling method. This technique ensures representation from each subgroup and allows for comparisons between strata.

3.    Cluster Sampling:

§  Cluster sampling involves dividing the population into clusters or groups, randomly selecting some clusters, and then sampling all elements within the chosen clusters. This method is useful when it is more practical to sample clusters rather than individual elements, especially in large and geographically dispersed populations.

4.    Multi-stage Sampling:

§  Multi-stage sampling is an extension of cluster sampling where the sampling process occurs in multiple stages. Researchers first select large primary sampling units (e.g., states, districts), then further subdivide these units into smaller units for sampling. This method is suitable for large-scale inquiries covering extensive geographical areas.

5.    Sequential Sampling:

§  Sequential sampling is a complex design where the sample size is not predetermined but is determined based on information gathered during the survey. This method is often used in acceptance sampling plans for statistical quality control purposes. Sequential sampling allows for adjustments in sample size based on evolving data.

6.    Mixed Sampling:

§  Mixed sampling involves using a combination of different sampling methods within the same study. Researchers may employ various sampling techniques such as simple random sampling, systematic sampling, and stratified sampling to achieve specific research objectives. Mixed sampling allows for flexibility and customization in sample selection.

7.    Complex Random Sampling Designs:

§  Complex random sampling designs may involve a combination of the above techniques or other advanced sampling methods to meet the unique requirements of a research study. These designs aim to balance the need for randomness, representativeness, efficiency, and statistical validity in the sampling process.

By utilizing different types of complex random sampling designs, researchers can tailor their sampling strategies to suit the characteristics of the population, the research objectives, and practical considerations. Each type of complex random sampling design offers specific advantages and is suitable for different research contexts.

 

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