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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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After-only with Control Designs

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Dr. Rishabh Pathak

After-only with Control Designs are a type of informal experimental design where two groups or areas (a test area/group and a control area/group) are selected, and the treatment is introduced only to the test area/group. The dependent variable is then measured in both areas/groups at the same time after the treatment has been implemented. Here are the key characteristics of After-only with Control Designs:


1.    Two Groups or Areas:

§  In this design, two groups or areas are involved: a test area/group that receives the treatment and a control area/group that does not receive the treatment. Data on the dependent variable are collected from both areas/groups simultaneously after the treatment is introduced in the test area/group.

2.    Post-Treatment Measurements:

§  Researchers measure the dependent variable in both the test and control areas/groups at the same time after the treatment has been implemented in the test area/group. This simultaneous measurement allows for a comparison of the outcomes between the two groups.

3.    Assessment of Treatment Impact:

§  The treatment effect in After-only with Control Designs is assessed by comparing the values of the dependent variable in the test area/group (where the treatment was applied) with the values in the control area/group (where no treatment was applied). This comparison helps evaluate the difference in outcomes between the two groups.

4.    Control Group Comparison:

§  By including a control group or area that does not receive the treatment, After-only with Control Designs enable researchers to compare the outcomes of the treated group with those of the untreated group. This comparison helps attribute any observed differences to the treatment itself.

5.    Simplicity and Efficiency:

§  This design is relatively simple and efficient compared to designs that involve pre-treatment measurements. By focusing on post-treatment measurements only, researchers can quickly assess the impact of the treatment on the dependent variable without the need for pre-treatment data.

6.    Control for Extraneous Factors:

§  After-only with Control Designs control for extraneous factors by providing a reference point (the control group) for comparison. This design allows researchers to isolate the effects of the treatment from other variables that may influence the dependent variables.

7.    Limitations:

§  One limitation of After-only with Control Designs is the potential for biases or confounding variables that may affect the post-treatment measurements. Without pre-treatment data, it can be challenging to account for baseline differences between the test and control groups.

After-only with Control Designs offer a straightforward and comparative approach to evaluating the effects of treatments by including a control group for reference. By measuring outcomes post-treatment in both the treated and untreated groups, researchers can assess the impact of the treatment while controlling for extraneous factors and comparing outcomes between the groups.

 

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