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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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How does Michael Frayn's Copenhagen portray the meeting between Niels Bohr and Werner Heisenberg in 1941?

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Dr. Rishabh Pathak

 


In Michael Frayn's play "Copenhagen," the meeting between Niels Bohr and Werner Heisenberg in 1941 is portrayed as a pivotal and tense encounter. The play delves into the complex relationship between these two renowned physicists, who were once colleagues and friends but found themselves on opposite sides during World War II.

The meeting takes place in Nazi-occupied Copenhagen, were Heisenberg, working for Hitler's war effort, visits Bohr under Gestapo surveillance. The play captures the intense dialogue and emotional dynamics between the characters as they grapple with personal and ethical dilemmas. The tension escalates as Bohr becomes angry and their friendship ultimately dissolves, reflecting the larger historical context of scientific collaboration and moral responsibility during wartime.

Frayn's portrayal of the meeting goes beyond a simple historical reenactment; it delves into the characters' inner thoughts, memories, and conflicting perspectives. The play explores themes of uncertainty, memory, and the blurred boundaries between reality and fiction, inviting the audience to contemplate the complexities of human relationships and moral choices in times of crisis.

Overall, "Copenhagen" offers a nuanced and thought-provoking depiction of the meeting between Bohr and Heisenberg, highlighting the ethical and philosophical implications of their encounter within the context of wartime politics and scientific responsibility.

 

Frayn, M. (2000). Copenhagen.


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