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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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Restricted Sampling

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Dr. Rishabh Pathak

Restricted sampling involves the application of specific criteria or restrictions during the selection of sample elements from the population. This approach allows researchers to tailor their sampling methods to account for certain characteristics or conditions within the population. Here are some key points about restricted sampling techniques:


1.    Stratified Sampling:

§  In this technique, the population is divided into homogeneous subgroups or strata based on certain characteristics (e.g., age, gender, income level). Samples are then selected independently from each stratum to ensure representation of different strata in the final sample. This helps in capturing the variability within the population and can lead to more precise estimates for subgroups.

2.    Cluster Sampling:

§  Cluster sampling involves dividing the population into clusters or groups (e.g., geographical areas, classrooms) and then randomly selecting entire clusters to be included in the sample. This method is useful when it is more practical to sample clusters rather than individual elements, especially in large and geographically dispersed populations.

3.    Systematic Sampling:

§  Systematic sampling involves selecting sample elements at regular intervals from a list or sequence after a random start. For example, every 5th person on a list may be selected for inclusion in the sample. This method introduces an element of randomness through the initial random start point, while still maintaining a systematic selection process.

4.    Quota Sampling:

§  Quota sampling involves setting quotas for different subgroups of the population based on specific characteristics. Interviewers then select sample elements to fill these quotas, ensuring that the final sample reflects the distribution of these characteristics in the population. Quota sampling is a non-probability sampling technique that allows for control over the composition of the sample.

5.    Advantages:

§  Restricted sampling techniques can help researchers ensure that certain subgroups or characteristics of interest are adequately represented in the sample. By stratifying or clustering the population, researchers can improve the precision of their estimates for specific groups within the population.

6.    Challenges:

§  Implementing restricted sampling techniques may require additional resources and planning compared to simple random sampling. Researchers need to carefully define the strata, clusters, or quotas to avoid bias and ensure the representativeness of the sample.

By incorporating restricted sampling techniques into their research designs, researchers can enhance the precision and relevance of their study findings by accounting for specific characteristics or conditions within the population. Each technique offers unique advantages and considerations, and the choice of method should align with the research objectives and the nature of the population under study.

 

 

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