Publisher Summary This chapter provides an account of different neural network architectures for pattern recognition. A neural network consists of several simple processing elements called neurons. Each neuron is connected to some other neurons and possibly to the input nodes. Neural networks provide a simple computing paradigm to perform complex recognition tasks in real time. The chapter categorizes neural networks into three types: single-layer networks, multilayer feedforward networks, and feedback networks. It discusses the gradient descent and the relaxation method as the two underlying mathematical themes for deriving learning algorithms. A lot of research activity is centered on learning algorithms because of their fundamental importance in neural networks. The chapter discusses two important directions of research to improve learning algorithms: the dynamic node generation, which is used by the cascade correlation algorithm; and designing learning algorithms where the choice of parameters is not an issue. It closes with the discussion of performance and implementation issues.

Neural Networks for Pattern Recognition

The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of $^{4}\mathrm{He}$.

Theory of Dynamic Critical Phenomena

The Renormalization group and the epsilon expansion

http://cds.cern.ch/record/147992/files/198312192.pdf

Chiral perturbation theory to one loop

Monte Carlo renormalization-group calculations of critical behavior in the simple-cubic Ising model

Feynman-Graph Expansion for the Equation of State near the Critical Point

The critical behavior of a classical Heisenberg ferromagnet is studied in the limit where the spin dimensionality $N$ is large. Corrections of order $\frac{1}{N}$ to the spherical model are obtained as functions of a continuous dimension $d$, $2<d<4$. Particular attention is given to the behavior near the coexistence curve. The divergence of the magnetic susceptibility below ${T}_{c}$ as the external field vanishes is discussed through a nonlinear realization of the $O(N)$ symmetry, as well as in the $\frac{1}{N}$ and $4\ensuremath{-}d$ expansions.

Critical Behavior of a Classical Heisenberg Ferromagnet with Many Degrees of Freedom

The continuous n-component Potts model is studied in the framework of Wilson's multiplicative renormalization group. Apart from the isotropic phi 4 interaction, there is another one inherent in the model for n>2. For n>or=4, it is distinct from any phi 4 interaction that has been investigated in detail. Critical exponents are calculated. The n dependence of the fixed points shows new behaviour; in particular for n in the neighbourhood of five, the epsilon expansion must be reformulated as a power series in epsilon 12/.

https://iopscience.iop.org/article/10.1088/0305-4470/8/9/019/pdf

Critical behaviour of the continuous n-component Potts model

The authors introduce effective Hamiltonians for Goldstone modes of the Euclidean group, representing fluctuations in the surface of a critical droplet or in the interface between two phases. The Euclidean invariance is non-linearly realised on the Goldstone fields. The Hamiltonians are non-renormalisable in more than one dimension, showing that the disappearance of a phase transition in one dimension for systems with a discrete symmetry may be interpreted in terms of the infrared instabilities induced by these modes. The existence and form of these Hamiltonians indicates the universality of the essential singularity at a first-order phase transition in models with Euclidean invariance.

/pdf/goldstone-modes-in-vacuum-decay-and-first-order-phase-26upyn2x50.pdf

D. J. Wallace

Papers

Monte Carlo renormalization-group calculations of critical behavior in the simple-cubic Ising model

Feynman-Graph Expansion for the Equation of State near the Critical Point

Critical Behavior of a Classical Heisenberg Ferromagnet with Many Degrees of Freedom

Critical behaviour of the continuous n-component Potts model

Goldstone modes in vacuum decay and first-order phase transitions