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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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Basics Principles of Local Control

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Dr. Rishabh Pathak

The principle of local control, also known as blocking, is a fundamental concept in experimental design that involves controlling for known sources of variability by grouping experimental units into homogeneous blocks. Here are the basic principles of local control:


1.    Definition:

o    Principle: Local control, or blocking, is the process of grouping experimental units into blocks based on a known source of variability that may affect the outcomes of the study. By controlling for this source of variation within each block, researchers can reduce the impact of extraneous factors on the results.

2.    Homogeneous Blocks:

o    Principle: Blocks are created to be as similar as possible in terms of the known source of variability being controlled. By grouping experimental units into homogeneous blocks, researchers ensure that any differences in the outcomes can be attributed to the treatments or interventions rather than the blocking factor.

3.    Reducing Variability:

o    Principle: The primary goal of local control is to reduce the overall variability in the study by holding constant the known source of variation within each block. By minimizing the impact of extraneous factors through blocking, researchers can increase the precision and accuracy of their results.

4.    Two-Way Analysis of Variance:

o    Principle: Local control facilitates the use of two-way analysis of variance (ANOVA) by partitioning the total variability in the data into components attributed to treatments and the blocking factor. By incorporating blocking into the analysis, researchers can assess the main effects and interactions more effectively.

5.    Eliminating Extraneous Factors:

o   Principle: Through local control, researchers can eliminate the variability introduced by known extraneous factors from the experimental error. By holding these factors constant within blocks, researchers can isolate the effects of the treatments and improve the internal validity of the study.

6.    Enhancing Precision:

o   Principle: Local control increases the precision of estimates by reducing the impact of extraneous variability on the results. By minimizing the influence of known sources of variation through blocking, researchers can obtain more accurate and reliable estimates of treatment effects.

7.    Blocking Procedures:

o Principle: Researchers should carefully plan and implement blocking procedures to ensure that the blocks are truly homogeneous with respect to the blocking factor. Transparent reporting of the blocking strategy is essential for replicability and understanding the design of the study.

By applying the principles of local control and incorporating blocking into experimental designs, researchers can enhance the internal validity, precision, and reliability of their studies. Local control is a valuable technique for controlling known sources of variability and improving the accuracy of research findings in experimental settings.

 

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