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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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Sequential Sampling

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Dr. Rishabh Pathak

Sequential sampling is a sampling method where the sample size is not fixed in advance but is determined based on the information gathered during the survey process. Here are some key points about sequential sampling:


1.    Process:

o    In sequential sampling, data collection and analysis occur in stages, with the sample size increasing or decreasing based on the information obtained at each stage.

o    The decision to continue sampling or stop the sampling process is based on predetermined criteria, such as reaching a certain level of precision or statistical significance.

2.    Purpose:

o    Sequential sampling is often used in quality control, acceptance sampling, and other situations where decisions need to be made progressively based on accumulating data.

o    It allows researchers to adapt the sample size and sampling process in real-time based on the results obtained during the survey.

3.    Advantages:

o    Provides flexibility in sample size determination, allowing researchers to optimize the sample size based on the information collected.

o    Can lead to more efficient data collection by focusing resources on areas where additional data are most needed.

o    Enables researchers to make decisions during the survey process, rather than waiting until the end of data collection.

4.    Disadvantages:

o    Requires clear criteria for stopping the sampling process to avoid bias or premature conclusions.

o    May introduce complexities in data analysis and interpretation due to the varying sample sizes at different stages.

o    Can be more resource-intensive and time-consuming compared to fixed sample size methods.

5.    Applications:

o    Sequential sampling is commonly used in quality control processes, where decisions about product acceptance or rejection are made based on sequential sampling results.

o    It is also used in clinical trials, market research, and other fields where data collection occurs in stages and decisions need to be made iteratively.

6.    Considerations:

o    Researchers must define stopping rules or criteria in advance to ensure the validity and reliability of the results obtained through sequential sampling.

o    Careful monitoring of the sampling process is essential to make informed decisions about sample size adjustments and data collection continuation.

7.    Advantages over Fixed Sample Size:

o    Sequential sampling allows for adaptive sampling, where the sample size can be adjusted based on the evolving information during data collection.

o    It can lead to more efficient use of resources by focusing on areas of interest or uncertainty, potentially reducing the overall sample size needed.

Sequential sampling offers a dynamic approach to data collection, allowing researchers to adjust the sample size based on the information gathered during the survey process. By making decisions iteratively and adaptively, researchers can optimize the sampling process and make informed conclusions based on evolving data.

 

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