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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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Before-and-after with Control Designs

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Dr. Rishabh Pathak

Before-and-after with Control Designs are a type of informal experimental design where two areas or groups are selected, and the dependent variable is measured in both areas for an identical time period before the treatment is introduced. After the treatment is implemented in one area (the test area), the dependent variable is measured in both areas for an identical time period post-treatment. Here are the key characteristics of Before-and-after with Control Designs:


1.    Two Areas or Groups:

o    In this design, two areas or groups are involved: a test area/group where the treatment is applied and a control area/group where no treatment is applied. Data on the dependent variable are collected from both areas before and after the treatment.

2.    Pre- and Post-Treatment Measurements:

o    Researchers measure the dependent variable in both the test and control areas/groups for the same duration before the treatment is introduced. After the treatment is implemented in the test area/group, measurements are taken in both areas/groups for the same duration post-treatment.

3.    Comparison of Changes:

o    The treatment effect in Before-and-after with Control Designs is determined by comparing the change in the dependent variable in the test area/group with the change in the control area/group. This comparison helps assess the impact of the treatment while accounting for potential confounding factors.

4.    Control for Extraneous Variations:

o    By including a control group or area that does not receive the treatment, Before-and-after with Control Designs aim to control for extraneous variations that may influence the dependent variable. This design allows researchers to isolate the effects of the treatment from other factors.

5.    Avoidance of Extraneous Variation:

o    This design is considered superior to Before-and-after without Control Designs because it helps avoid extraneous variations resulting from the passage of time and non-comparability of the test and control areas. By comparing changes in both areas/groups, researchers can better attribute observed effects to the treatment.

6.    Enhanced Validity:

o    Before-and-after with Control Designs enhance the internal validity of the study by providing a basis for comparison between the effects of the treatment and the absence of treatment. This design allows for a more robust evaluation of the treatment's impact on the dependent variable.

7.    Practical Considerations:

o    Researchers may choose Before-and-after with Control Designs when historical data, time, or a comparable control area are available. This design offers a balance between simplicity and control over extraneous variables compared to other informal experimental designs.

Before-and-after with Control Designs offer a practical and comparative approach to studying the effects of interventions by including a control group or area for reference. By comparing changes in both the test and control groups, researchers can better assess the true impact of the treatment on the dependent variable while minimizing the influence of external factors.

 

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