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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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Plasticity

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Dr. Rishabh Pathak

Plasticity refers to the brain's ability to reorganize itself by forming new neural connections throughout life in response to experiences, learning, and environmental stimuli. This adaptive capacity allows the brain to change its structure and function in order to optimize performance, recover from injury, and adapt to new challenges. Plasticity is a fundamental property of the nervous system that underlies learning, memory, and various cognitive processes.


There are two main types of plasticity in the brain:


1. Structural Plasticity: Structural plasticity involves changes in the physical structure of the brain, such as the formation of new synapses (connections between neurons), the growth of dendrites (branch-like extensions of neurons), and the reorganization of neural circuits. Structural changes in the brain occur in response to learning, environmental enrichment, and sensory experiences. For example, practicing a new skill can lead to the formation of new neural connections and the strengthening of existing ones, enhancing the brain's ability to perform that skill.


2.  Functional Plasticity: Functional plasticity refers to changes in the functional organization of the brain, including alterations in neural activity patterns and the recruitment of different brain regions for specific tasks. Functional plasticity allows the brain to adapt its processing strategies in response to changing demands and experiences. For instance, after a brain injury, other areas of the brain may compensate for the damaged region by taking on new functions, demonstrating the brain's ability to reorganize and adapt to maintain cognitive abilities.


Plasticity is most pronounced during critical periods of development, such as early childhood, when the brain is highly malleable and responsive to environmental influences. However, plasticity continues throughout life to a certain extent, allowing for ongoing learning, memory formation, and adaptation to new experiences.


Factors that influence brain plasticity include sensory stimulation, motor activities, social interactions, cognitive challenges, and environmental enrichment. By understanding and harnessing the principles of plasticity, researchers and clinicians can develop interventions to promote healthy brain development, enhance cognitive function, and facilitate recovery from brain injuries or neurological disorders.

 

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