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...
The analytical model used to estimate
critical conditions at the onset of folding in the brain is based on the
Föppl–von Kármán theory. This theory is applied to approximate cortical folding
as the instability problem of a confined, layered medium subjected to
growth-induced compression. The model focuses on predicting the critical time,
pressure, and wavelength at the onset of folding in the brain's surface
morphology.
The analytical model adopts the classical fourth-order plate equation to
model the cortical deflection. This equation considers parameters such as
cortical thickness, stiffness, growth, and external loading to analyze the
behavior of the brain tissue during the folding process. By utilizing the
Föppl–von Kármán theory and the plate equation, researchers can derive
analytical estimates for the critical conditions that lead to the initiation of
folding in the brain.
Analytical modeling provides a quick initial insight into the critical
conditions at the onset of folding, allowing researchers to understand the
fundamental mechanisms driving cortical folding. However, it may not fully
capture the evolution of complex instability patterns in the post-critical
regime. Therefore, while the analytical model helps in estimating the critical
parameters for folding initiation, a computational model based on the continuum
theory of finite growth is often employed to predict more realistic surface
morphologies and complex folding patterns beyond the onset of folding.
In conclusion, the analytical model based on the Föppl–von Kármán theory
provides a foundational framework for understanding the critical conditions
that trigger folding in the brain's surface morphology. It serves as a valuable
tool for estimating key parameters at the onset of cortical folding and guiding
further computational modeling efforts to explore the evolution of brain
surface morphologies.
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