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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 Sample Designs

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Dr. Rishabh Pathak

There are different types of sample designs that researchers can employ based on various factors such as the representation basis and the element selection technique. Here are the main categories of sample designs:


1.    Probability Sampling:

§  Probability sampling involves random selection of elements from the population, where each element has a known and non-zero chance of being included in the sample. Common types of probability sampling include:

§  Simple Random Sampling: Every member of the population has an equal chance of being selected.

§ Stratified Sampling: The population is divided into homogeneous subgroups (strata), and samples are randomly selected from each stratum.

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

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

2.    Non-Probability Sampling:

§  Non-probability sampling does not involve random selection of elements, and the likelihood of any element being included in the sample is unknown. Some common types of non-probability sampling include:

§  Convenience Sampling: Elements are selected based on their availability and accessibility.

§  Purposive Sampling: Researchers deliberately choose specific elements based on predefined criteria.

§ Snowball Sampling: Existing participants recruit new participants to form the sample.

§  Quota Sampling: Researchers select participants based on pre-defined quotas to ensure representation.

3.    Unrestricted and Restricted Sampling:

§  Based on the element selection technique, samples can be classified as unrestricted or restricted:

§  Unrestricted Sampling: Each sample element is drawn individually from the population at large, without any restrictions.

§  Restricted Sampling: In restricted sampling, there are limitations or conditions imposed on the selection of sample elements.

4.    Mixed Sampling Methods:

§  Researchers may also use a combination of different sampling methods to enhance the representativeness and efficiency of the sample design. For example, a study may employ a combination of stratified sampling and cluster sampling to achieve a more comprehensive sample representation.

5.    Complex Sampling Designs:

§  In some research studies, complex sampling designs may be necessary to address specific research questions or population characteristics. These designs may involve multiple stages of sampling, stratification, weighting, and clustering to ensure the validity and reliability of the results.

By selecting an appropriate sample design that aligns with the research objectives, population characteristics, and data collection methods, researchers can enhance the quality and generalizability of their findings. Understanding the different types of sample designs and their implications can help researchers make informed decisions when designing and implementing sampling strategies in research studies.

 

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