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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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Ex Post Facto Research Design

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Dr. Rishabh Pathak

Ex post facto research design, also known as causal-comparative research, is a type of non-experimental research design that examines the relationship between independent and dependent variables after they have occurred naturally. In ex post facto studies, the researcher does not manipulate the independent variable but rather observes and analyzes the existing differences or relationships between variables. Here are key characteristics and components of ex post facto research design:


1. No Manipulation of Variables: In ex post facto research, the researcher does not have control over the independent variable. The independent variable has already occurred or been naturally assigned, and the researcher observes the effects or relationships with the dependent variable after the fact.


2.    Retrospective Analysis: Ex post facto research involves a retrospective analysis of existing data or conditions. Researchers analyze data that have already been collected or events that have already taken place to investigate possible causal relationships between variables.


3.    Causal-Comparative Analysis: Ex post facto research aims to compare groups or conditions that differ on the independent variable to determine the effects on the dependent variable. The researcher seeks to establish causal relationships or associations based on observed differences or correlations.


4.    Identifying Causal Relationships: While ex post facto research cannot establish causation definitively due to the lack of experimental control, it can provide valuable insights into potential causal relationships between variables. Researchers may use statistical analyses to explore the relationships and draw inferences based on the observed patterns.


5.    Control of Extraneous Variables: Researchers in ex post facto studies must consider and control for extraneous variables that could influence the relationship between the independent and dependent variables. Statistical techniques such as regression analysis or analysis of covariance may be used to account for these variables.


6.    Cross-Sectional or Longitudinal Design: Ex post facto research can utilize cross-sectional or longitudinal designs to examine relationships between variables at a specific point in time or over a period. Longitudinal studies allow researchers to track changes and trends in variables over time.


7.    Applications: Ex post facto research is commonly used in educational research, social sciences, and healthcare to investigate the effects of variables that cannot be manipulated for ethical or practical reasons. For example, studying the impact of gender on academic achievement or the relationship between socioeconomic status and health outcomes.


8.    Limitations: One of the main limitations of ex post facto research is the inability to establish causation definitively due to the lack of experimental control. Researchers must be cautious in interpreting the results and consider alternative explanations for the observed relationships.


Ex post facto research design provides a valuable approach for exploring causal relationships between variables in situations where experimental manipulation is not feasible or ethical. By analyzing existing data and conditions, researchers can gain insights into potential causal links and contribute to the understanding of complex phenomena in various fields of study.

 

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