We ask if analog computers can solve NP-complete problems efficiently. Regarding this as unlikely, we formulate a strong version of Church’s Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer. From this assumption and the assumption that P ≠ NP we can draw conclusions about the operation of physical devices used for computation. An NP-complete problem, 3-SAT, is reduced to the problem of checking whether a feasible point is a local optimum of an optimization problem. A mechanical device is proposed for the solution of this problem. It encodes variables as shaft angles and uses gears and smooth cams. If we grant Strong Church’s Thesis, that P ≠ NP, and a certain ‘‘Downhill Principle’ ’ governing the physical behavior of the machine, we conclude that it cannot operate successfully while using only polynomial resources. We next prove Strong Church’s Thesis for a class of analog computers described by well-behaved ordinary differential equations, which we can take as representing part of classical mechanics. We conclude with a comment on the recently discovered connection between spin glasses and combinatorial optimization. 1.

This research begins with reaction-diffusion, first proposed by Alan Turing in 1952 to account for morphogenesis -- the formation of hydranth tentacles, leopard spots, zebra stripes, etc. Reaction-diffusion systems have been researched primarily by biologists working on theories of natural pattern formation and by chemists modeling dynamics of oscillating reactions. The past few years have seen a new interest in reaction-diffusion spring up within the computer graphics and image processing communities. However, reaction-diffusion systems are generally unbounded, making them impractical for many applications. In this thesis we introduce a bounded and more flexible non-linear system, the "M-lattice", which preserves the natural pattern-formation properties of reaction-diffusion. On the theoretical front, we establish relationships between reaction-diffusion systems and paradigms in linear systems theory and certain types of artificial "neurally-inspired" systems. The M-lattice is closel...

Sliding modes are used to analyze a class of dynamical systems that solve convex programming problems. The analysis is carried out using concepts from the theory of differential equations with discontinuous right-hand sides and Lyapunov stability theory. It is shown that the equilibrium points of the system coincide with the minimizers of the convex programming problem, and that irrespective of the initial state of the system the state trajectory converges to the solution set of the problem. The dynamic behavior of the systems is illustrated by two numerical examples.

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The paper introduces a new approach to analyze the stability of neural network models without using any Lyapunov function. With the new method, we investigate the stability properties of the general gradient-based neural network model for optimization problems. Our discussion includes both isolated equilibrium points and connected equilibrium sets which could be unbounded. For a general optimization problem, if the objective function is bounded below and its gradient is Lipschitz continuous, we prove that a) any trajectory of the gradient-based neural network converges to an equilibrium point, and b) the Lyapunov stability is equivalent to the asymptotical stability in the gradient-based neural networks. For a convex optimization problem, under the same assumptions, we show that any trajectory of gradient-based neural networks will converge to an asymptotically stable equilibrium point of the neural networks. For a general nonlinear objective function, we propose a refined gradientbase...

This thesis develops an engineering practice and design methodology to enable us to use CMOS analog VLSI chips to perform more accurate and precise computation. These techniques form the basis of an approach that permits us to build computer graphics and neural network applications using analog VLSI. The nature of the design methodology focuses on defining goals for circuit behavior to be met as part of the design process. To increase the accuracy of analog computation, we develop techniques for creating compensated circuit building blocks, where compensation implies the cancellation of device variations, offsets, and nonlinearities. These compensated building blocks can be used as components in larger and more complex circuits, which can then also be compensated. To this end, we develop techniques for automatically determining appropriate parameters for circuits, using constrained optimization. We also fabricate circuits that implement multi-dimensional gradient estimation for a grad...

Some neural network models have been suggested to solve linear and quadratic programming problems. The Kennedy and Chua model[5] is one of those networks. In this paper results about the convergence of the model are obtained. Another related problem is how to choose a parameter value ~ s so that the equilibrium point of the network immediately and properly approximates the original solution. Such an estimation for the parameter is given in a closed form when the network is used to solve linear programming.