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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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Physical Randomization

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Dr. Rishabh Pathak

Physical randomization refers to the process of introducing randomness into an experiment or study using physical means or mechanisms. Unlike computer-generated randomization or random digit tables, physical randomization involves tangible objects or actions to ensure a random allocation of treatments, subjects, or samples. Here are some common methods of physical randomization:


1.    Randomization Cards:

o    Randomization cards are physical cards or slips with treatment assignments or group allocations written on them. Researchers shuffle the cards and draw them one by one to assign treatments or interventions to participants in a randomized manner. This method ensures that each participant has an equal chance of receiving any particular treatment.

2.    Randomization Barrels:

o    Randomization barrels are containers filled with balls or tokens representing different treatment groups or allocations. Researchers draw balls or tokens from the barrel to determine the assignment for each participant. By mixing the balls thoroughly and selecting them randomly, researchers achieve a randomized allocation process.

3.    Coin Flipping:

o    Coin flipping is a simple yet effective method of physical randomization. Researchers assign treatments or group allocations based on the outcome of a coin toss. For example, heads may represent one treatment group, while tails represent another. By flipping a coin for each participant, researchers can ensure a random assignment process.

4.    Dice Rolling:

o    Dice rolling involves using dice to determine treatment assignments or group allocations. Each face of the dice can correspond to a different treatment group or allocation. By rolling the dice for each participant, researchers introduce randomness into the assignment process based on the dice outcome.

5.    Drawing Lots:

o    Drawing lots is a traditional method of physical randomization where participants draw slips of paper or tokens from a container. Each slip corresponds to a treatment group or allocation, and participants are assigned based on the slip they draw. This method ensures a random and unbiased allocation process.

6.    Shuffling and Selection:

o    Researchers can also use physical objects like cards, tokens, or slips with participant IDs to conduct random selection. By shuffling the objects and selecting them without looking, researchers can achieve a random sample selection process for studies or experiments.

Physical randomization methods are particularly useful in situations where researchers prefer a hands-on approach to randomization or where access to electronic devices or computers is limited. By employing physical randomization techniques, researchers can ensure the fairness and impartiality of treatment assignments, group allocations, or sample selections in their studies. It is important to follow standardized procedures and protocols to maintain the integrity of the randomization process and minimize biases in research outcomes.

 

 

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