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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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Table of Random Digits

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Dr. Rishabh Pathak

A table of random digits is a systematic arrangement of numbers that are generated in a random and unpredictable manner. These tables are commonly used in statistical sampling, experimental design, and simulations to simulate random events or select random samples from populations. Here is an overview of a typical table of random digits:


1.    Structure:

o    A table of random digits is usually organized in rows and columns, with each cell containing a single digit from 0 to 9. The digits are arranged in a random sequence to ensure that each digit has an equal probability of appearing in any position.

2.    Purpose:

o    The primary purpose of a table of random digits is to provide a source of randomness for various statistical and research applications. Researchers use these tables to select random samples, assign treatments in experiments, simulate random events, and conduct Monte Carlo simulations.

3.    Usage:

o    Researchers can use a table of random digits to select random samples by following a systematic procedure. They start at a random point in the table and read the digits sequentially to determine the elements to include in the sample. By using random digits, researchers can ensure an unbiased and representative selection process.

4.    Generation:

o    Random digit tables are typically generated using algorithms that produce sequences of digits with no discernible pattern or order. These tables are pre-generated and widely available in statistical textbooks, software packages, and online resources for researchers to use in their studies.

5.    Applications:

o    Random digit tables are used in various statistical sampling techniques, such as simple random sampling, systematic sampling, and stratified sampling. They are also employed in quality control, market research, opinion polls, and other fields where randomization is essential for obtaining reliable results.

6.    Advantages:

o    Using a table of random digits ensures that the selection process is truly random and free from bias. It provides a standardized method for generating randomness in research studies and helps researchers make unbiased inferences about populations based on sampled data.

7.    Limitations:

o    One limitation of random digit tables is the potential for human error in reading and interpreting the digits. Researchers must follow the instructions carefully to avoid introducing unintentional biases into the sampling process. Additionally, the size of the table may limit the range of random numbers that can be generated.

Overall, a table of random digits is a valuable tool for researchers seeking to introduce randomness into their sampling procedures and experimental designs. By using these tables effectively, researchers can enhance the validity and reliability of their research findings through unbiased and systematic randomization.

 

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