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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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Types of Sleep

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Dr. Rishabh Pathak

Sleep is generally categorized into two main types: Non-Rapid Eye Movement (NREM) sleep and Rapid Eye Movement (REM) sleep. Each type has distinct characteristics and plays different roles in overall health and well-being. 

1. Non-Rapid Eye Movement (NREM) Sleep

NREM sleep is further divided into three stages, each characterized by different brain wave patterns and physiological changes:

    • Stage 1 (N1):
      • This is the lightest stage of sleep, often considered the transition between wakefulness and sleep. It typically lasts a few minutes.
      • During this stage, muscle activity decreases, and the person can be easily awakened.
      • Brain waves begin to slow down, transitioning from alpha waves (associated with relaxed wakefulness) to theta waves.
    • Stage 2 (N2):
      • This stage constitutes about 50% of total sleep time in adults. It is characterized by a further slowing of brain waves, with the appearance of sleep spindles and K-complexes.
      • Heart rate slows, and body temperature drops. This stage is also considered a period of light sleep, but it is harder to awaken someone compared to Stage 1.
    • Stage 3 (N3):
      • Also known as deep sleep or slow-wave sleep (SWS), this stage is crucial for physical recovery and growth.
      • It is characterized by delta waves, which are slow brain waves. During this stage, it is very difficult to awaken someone.
      • Deep sleep is associated with restorative processes, including tissue repair, immune function, and energy restoration.

2. Rapid Eye Movement (REM) Sleep

    • REM sleep typically occurs about 90 minutes after falling asleep and recurs multiple times throughout the night, with each episode getting longer.
    • During REM sleep, brain activity increases, resembling wakefulness, and most dreaming occurs during this stage.
    • The eyes move rapidly under the eyelids, and muscle tone is significantly reduced (atonia), preventing the sleeper from acting out dreams.
    • REM sleep is important for cognitive functions such as memory consolidation, learning, and emotional regulation.

Sleep Cycle

    • A complete sleep cycle lasts about 90 to 120 minutes and includes both NREM and REM sleep. Throughout a typical night, a person will go through several cycles, with NREM sleep dominating the first half of the night and REM sleep becoming more prominent in the latter half.

Importance of Sleep Types

    • Each type of sleep plays a vital role in overall health. NREM sleep is essential for physical restoration and growth, while REM sleep is crucial for cognitive functions and emotional health. Disruptions in either type can lead to various health issues, including cognitive impairments, mood disorders, and weakened immune function.

Understanding these sleep types and their functions can help in recognizing the importance of good sleep hygiene and addressing sleep-related issues effectively.

 

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