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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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Random Sample from an Infinite Universe

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Dr. Rishabh Pathak

Random sampling from an infinite universe presents unique challenges due to the theoretical nature of an infinite population. While random sampling is more commonly discussed in the context of finite populations, it is possible to conceptualize random samples from infinite populations using certain principles. Here are some key points regarding random sampling from an infinite universe:

1.    Conceptual Understanding:

§  An infinite universe refers to a population with an unlimited number of elements, making it impossible to enumerate or list all possible units.

§  Random sampling from an infinite universe involves selecting sample units in a way that each unit has an equal probability of being chosen, despite the infinite nature of the population.

2.    Examples of Infinite Universes:

§  Consider scenarios such as the results of infinite dice throws, infinite decimal numbers between 0 and 1, or infinite sequences of events where each event represents a potential sample unit.

§  These examples illustrate the concept of an infinite universe where the population size is theoretically limitless.

3.    Theoretical Sampling Methods:

§  In practice, random sampling from an infinite universe is challenging due to the inability to list or enumerate all elements.

§  Theoretical sampling methods involve conceptualizing the selection process rather than physically listing or numbering elements.

4.    Principles of Random Sampling:

§  The fundamental principle of random sampling remains the same for infinite populations: each element should have an equal probability of being selected for the sample.

§  Randomness ensures that the sample is representative and unbiased, even in the absence of a finite population list.

5.    Application in Hypothetical Scenarios:

§  Researchers may use hypothetical scenarios, such as infinite sequences or theoretical distributions, to illustrate the concept of random sampling from an infinite universe.

§  These scenarios help demonstrate the principles of random sampling and the importance of equal probability for sample selection.

6.    Considerations for Analysis:

§  When analyzing data from a random sample of an infinite universe, researchers must account for the theoretical nature of the population and the implications of infinite possibilities.

§  Statistical methods and theoretical frameworks may be used to interpret results and draw inferences from samples taken from infinite populations.

While random sampling from an infinite universe is a theoretical concept, understanding the principles of random sampling and applying them to hypothetical scenarios can provide insights into the importance of randomness and equal probability in sample selection. Researchers can use theoretical sampling methods to explore the concept of random sampling in infinite populations and its implications for research and analysis.

 

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