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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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Elements Selection Techniques

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Dr. Rishabh Pathak

Element selection techniques play a crucial role in determining how individual elements or units are chosen from the population to form a sample. Here are some common element selection techniques used in sampling:


1.    Unrestricted Sampling:

§  In unrestricted sampling, each element in the population has an equal chance of being selected for the sample. This approach is commonly used in simple random sampling, where every element is selected independently of other elements.

2.    Restricted Sampling:

§  Restricted sampling involves imposing certain restrictions or conditions on the selection of sample elements. This can include stratification, clustering, or other criteria that guide the selection process. Restricted sampling techniques include:

§  Stratified Sampling: The population is divided into homogeneous subgroups (strata), and samples are selected from each stratum to ensure representation of different characteristics.

§  Cluster Sampling: The population is divided into clusters, and a random sample of clusters is selected for inclusion in the study.

§  Systematic Sampling: Elements are selected at regular intervals from a list or sequence, following a predetermined pattern.

3.    Judgement Sampling:

§  In judgement sampling, the researcher's judgment or expertise is used to select sample elements that are deemed representative of the population. This technique is subjective and relies on the researcher's knowledge and experience to identify relevant elements for inclusion in the sample.

4.    Quota Sampling:

§  Quota sampling involves setting quotas for different subgroups of the population based on certain characteristics. Interviewers are then tasked with filling these quotas by selecting individuals who meet the specified criteria. Quota sampling is a non-probability sampling technique that allows for control over the composition of the sample.

5.    Convenience Sampling:

§  Convenience sampling involves selecting sample elements based on their ease of access or availability to the researcher. This technique is often used when time and resources are limited, but it may introduce bias if the selected elements do not adequately represent the population.

6.    Snowball Sampling:

§  Snowball sampling is a technique where existing participants in the study recruit new participants from their social networks. This method is commonly used in studies where the target population is hard to reach or identify initially, such as in studies of marginalized or hidden populations.

By understanding and selecting appropriate element selection techniques based on the research objectives, population characteristics, and sampling constraints, researchers can ensure the validity, representativeness, and reliability of their sample designs. Each technique has its advantages and limitations, and researchers should carefully consider the implications of their choices on the quality of the study results.

 

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