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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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What role do 'real' scientists and their scientific ideas such as uncertainty and complementarity play in the play?

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Dr. Rishabh Pathak

 


In Michael Frayn's play "Copenhagen," 'real' scientists and their scientific ideas, such as uncertainty and complementarity, play a central role in shaping the narrative and thematic depth of the story. The characters of Niels Bohr and Werner Heisenberg, based on the actual historical figures, are portrayed not just as scientists but as complex individuals grappling with profound scientific concepts and personal dilemmas.

1. **Uncertainty**: The concept of uncertainty, famously formulated by Heisenberg in his Uncertainty Principle, is a recurring theme in the play. Heisenberg's uncertainty principle, which states that the more precisely the position of a particle is known, the less precisely its momentum can be known, serves as a metaphor for the uncertainties and ambiguities in human relationships and moral decisions. The characters' interactions are fraught with uncertainty, mirroring the quantum indeterminacy at the heart of Heisenberg's principle.

2. **Complementarity**: Another key scientific idea explored in the play is complementarity, a concept developed by Bohr to explain the dual nature of light as both particles and waves. In the context of the play, complementarity symbolizes the interconnectedness of opposing perspectives and the coexistence of conflicting truths. Bohr and Heisenberg's differing viewpoints and interpretations of their past actions reflect the notion of complementarity, highlighting the complexity of human nature and the multifaceted nature of truth.

By incorporating these scientific ideas and the personas of real-life scientists into the fabric of the play, Frayn not only adds intellectual depth but also explores profound philosophical questions about knowledge, perception, and the limitations of human understanding. The characters' engagement with uncertainty and complementarity serves as a lens through which broader themes of morality, responsibility, and the nature of reality are examined, enriching the narrative with layers of complexity and intrigue.

 

Frayn, M. (2000). Copenhagen.


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