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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 the Universe

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Dr. Rishabh Pathak

In research methodology, the universe refers to the total group of items or units that are under study or of interest to the researcher. Universes can vary in terms of their characteristics, boundaries, and nature. Here are different types of universes commonly encountered in research:


1.    Finite Universe:

o    A finite universe is one in which the total number of elements or units is known and definite. This type of universe has a fixed and countable number of items. Examples of finite universes include the population of a city, the number of employees in a company, or the students in a school. Researchers can theoretically enumerate all the elements in a finite universe.

2.    Infinite Universe:

o    An infinite universe is one in which the total number of elements or units is uncertain and potentially limitless. In an infinite universe, the number of items is infinite, and it is impossible to list or count all the elements. Examples of infinite universes include the number of stars in the sky, the listeners of a radio program, or the possible outcomes of a random event.

3.    Hypothetical Universe:

o A hypothetical universe consists of items or units that are conceptual or imaginary in nature. These universes may not have physical existence but are used for theoretical or experimental purposes. Examples of hypothetical universes include tossing a coin, rolling a dice, or hypothetical scenarios in simulation studies. Researchers may use hypothetical universes to study theoretical relationships or test hypotheses.

4.    Existent Universe:

o    An existent universe comprises concrete objects or entities that actually exist in reality. This type of universe includes tangible elements that can be observed, measured, or studied. Examples of existent universes include the population of a country, the customers of a business, or the plants in a botanical garden. Existent universes form the basis for empirical research and data collection.

5.    Hypothetical vs. Existent Universe:

o    The distinction between hypothetical and existent universes lies in the nature of the items or units they encompass. Hypothetical universes involve abstract or theoretical elements that may not physically exist but are used for modeling or simulation purposes. In contrast, existent universes consist of real-world entities that can be directly observed or studied.

Understanding the different types of universes is essential for researchers to define the scope of their studies, select appropriate sampling methods, and make valid inferences about the target population. By identifying the characteristics and boundaries of the universe, researchers can effectively design research studies, collect relevant data, and draw meaningful conclusions based on the study findings.

 

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