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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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Lottery Methods

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Dr. Rishabh Pathak

Lottery methods are a type of simple random sampling technique where each element in the population is assigned a unique identifier (such as a number or a ticket), and a random selection process resembling a lottery is used to choose the sample. This method ensures that every element in the population has an equal probability of being selected for the sample. Here are some common lottery methods used in sampling:


1.    Traditional Lottery Draw:

o  In a traditional lottery draw, each element in the population is assigned a unique number or ticket. These numbers are placed in a container (e.g., a hat, a bowl, or a drum). A person then randomly selects numbers from the container without looking, ensuring randomness in the selection process. This method mimics a lottery draw where each element has an equal chance of being chosen.

2.    Random Number Generator:

o    With advancements in technology, random number generators are commonly used to conduct lottery-based sampling. Random number generators are algorithms or software programs that generate a sequence of random numbers. Researchers can assign numbers to elements in the population and use a random number generator to select sample elements. This method ensures unbiased and random selection.

3.    Electronic Raffle Systems:

o    Electronic raffle systems are modern tools that facilitate lottery-based sampling in a digital format. These systems use electronic devices or software to conduct random draws. Researchers input the population data into the system, and the electronic raffle system selects sample elements randomly. Electronic raffle systems offer efficiency and accuracy in the sampling process.

4.    Online Random Selection Tools:

o  Online random selection tools are web-based applications or websites that enable researchers to conduct lottery-based sampling online. These tools use algorithms to generate random selections from a given population. Researchers can input the population data, specify the sample size, and use the online tool to perform the random selection process. Online random selection tools are convenient for virtual research settings.

5.    Physical Lottery Machines:

o  Physical lottery machines are mechanical devices designed for conducting random draws. These machines are commonly used in lottery events and can also be employed for sampling purposes. Researchers can load the identifiers of population elements into the machine, operate it to select sample elements randomly, and ensure fairness in the selection process. Physical lottery machines provide a tangible and transparent way of conducting random sampling.

By utilizing lottery methods in sampling, researchers can achieve randomness and fairness in selecting samples from populations. Whether through traditional lottery draws, random number generators, electronic raffle systems, online tools, or physical machines, lottery methods offer a systematic approach to simple random sampling, ensuring that each element in the population has an equal opportunity to be included in the sample.

 

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