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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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Formal Experimental Designs

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Dr. Rishabh Pathak

Formal experimental designs are structured research designs that offer more control and precision compared to informal designs. These designs follow specific principles and procedures to minimize bias, control for extraneous variables, and enhance the validity of research findings. Here are some common formal experimental designs:


1.    Completely Randomized Design (C.R. Design):

o    Principle: Involves randomly assigning subjects to different experimental treatments or conditions. This design is simple and easy to analyze, with subjects allocated to treatments based on randomization principles.

o    Analysis: Typically analyzed using one-way analysis of variance (ANOVA) to compare the means of different treatment groups.

2.    Randomized Block Design (R.B. Design):

o    Principle: Involves grouping subjects into blocks based on a known source of variability, with each block containing subjects that are relatively homogeneous. Subjects within each block are then randomly assigned to different treatments.

o    Analysis: Analyzed using two-way ANOVA to assess the main effects of treatments and the blocking factor.

3.    Latin Square Design (L.S. Design):

o    Principle: Utilizes a Latin square arrangement to control for two sources of variability, typically used when there are two nuisance variables that need to be controlled. Treatments are assigned in a way that each treatment appears once in each row and column of the Latin square.

o    Analysis: Requires specialized analysis methods to account for the unique structure of the design.

4.    Simple and Complex Factorial Designs:

o   Principle: Involve manipulating two or more independent variables (factors) to study their main effects and interactions. Simple factorial designs involve two factors, while complex factorial designs involve more than two factors.

o    Analysis: Requires factorial ANOVA to analyze the main effects and interactions of the factors.

5.    Split-Plot Design:

o    Principle: Combines elements of completely randomized and randomized block designs, where one factor is applied to whole plots and another factor is applied to subplots within each whole plot. This design is useful when certain factors are more difficult or costly to change.

o    Analysis: Analyzed using specialized statistical techniques to account for the different levels of randomization.

Formal experimental designs provide researchers with a systematic framework for conducting controlled experiments, allowing for rigorous testing of hypotheses and drawing valid conclusions. By following established design principles and analysis methods, researchers can enhance the reliability and validity of their research findings in various fields of study.

 

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