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An insightful investigation into the mechanisms underlying the predictive functions of neural networks--and their ability to chart a new path for AI. Prediction is a cognitive advantage like few others, inherently linked to our ability to survive and thrive. Our brains are awash in signals that embody prediction. Can we extend this capability more explicitly into synthetic neural networks to improve the function of AI and enhance its place in our world Gradient Expectations is a bold effort by Keith L. Downing to map the origins and anatomy of natural and artificial neural networks to explore how, when designed as predictive modules, their components might serve as the basis for the simulated evolution of advanced neural network systems. Downing delves into the known neural architecture of the mammalian brain to illuminate the structure of predictive networks and determine more precisely how the ability to predict might have evolved from more primitive neural circuits. He then surveys past and present computational neural models that leverage predictive mechanisms with biological plausibility, identifying elements, such as gradients, that natural and artificial networks share. Behind well-founded predictions lie gradients, Downing finds, but of a different scope than those that belong to today's deep learning. Digging into the connections between predictions and gradients, and their manifestation in the brain and neural networks, is one compelling example of how Downing enriches both our understanding of such relationships and their role in strengthening AI tools. Synthesizing critical research in neuroscience, cognitive science, and connectionism, Gradient Expectations offers unique depth and breadth of perspective on predictive neural-network models, including a grasp of predictive neural circuits that enables the integration of computational models of prediction with evolutionary algorithms.

Neural networks (Computer science) --- Artificial neural networks --- Nets, Neural (Computer science) --- Networks, Neural (Computer science) --- Neural nets (Computer science) --- Artificial intelligence --- Natural computation --- Soft computing --- Deep learning (Machine learning) --- Conjugate gradient methods. --- Gradient methods, Conjugate --- Approximation theory --- Equations --- Iterative methods (Mathematics) --- Learning, Deep (Machine learning) --- Machine learning --- Numerical solutions

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It is very well known that differential equations are related with the rise of physical science in the last several decades and they are used successfully for models of real-world problems in a variety of fields from several disciplines. Additionally, difference equations represent the discrete analogues of differential equations. These types of equations started to be used intensively during the last several years for their multiple applications, particularly in complex chaotic behavior. A certain class of differential and related difference equations is represented by their respective fractional forms, which have been utilized to better describe non-local phenomena appearing in all branches of science and engineering. The purpose of this book is to present some common results given by mathematicians together with physicists, engineers, as well as other scientists, for whom differential and difference equations are valuable research tools. The reported results can be used by researchers and academics working in both pure and applied differential equations.

dynamic equations --- time scales --- classification --- existence --- necessary and sufficient conditions --- fractional calculus --- triangular fuzzy number --- double-parametric form --- FRDTM --- fractional dynamical model of marriage --- approximate controllability --- degenerate evolution equation --- fractional Caputo derivative --- sectorial operator --- fractional symmetric Hahn integral --- fractional symmetric Hahn difference operator --- Arrhenius activation energy --- rotating disk --- Darcy–Forchheimer flow --- binary chemical reaction --- nanoparticles --- numerical solution --- fractional differential equations --- two-dimensional wavelets --- finite differences --- fractional diffusion-wave equation --- fractional derivative --- ill-posed problem --- Tikhonov regularization method --- non-linear differential equation --- cubic B-spline --- central finite difference approximations --- absolute errors --- second order differential equations --- mild solution --- non-instantaneous impulses --- Kuratowski measure of noncompactness --- Darbo fixed point --- multi-stage method --- multi-step method --- Runge–Kutta method --- backward difference formula --- stiff system --- numerical solutions --- Riemann-Liouville fractional integral --- Caputo fractional derivative --- fractional Taylor vector --- kerosene oil-based fluid --- stagnation point --- carbon nanotubes --- variable thicker surface --- thermal radiation --- differential equations --- symmetric identities --- degenerate Hermite polynomials --- complex zeros --- oscillation --- third order --- mixed neutral differential equations --- powers of stochastic Gompertz diffusion models --- powers of stochastic lognormal diffusion models --- estimation in diffusion process --- stationary distribution and ergodicity --- trend function --- application to simulated data --- n-th order linear differential equation --- two-point boundary value problem --- Green function --- linear differential equation --- exponential stability --- linear output feedback --- stabilization --- uncertain system --- nonlocal effects --- linear control system --- Hilbert space --- state feedback control --- exact controllability --- upper Bohl exponent