Lossless Cellular Neural Networks

Since their introduction, Cellular Neural Networks 4] have turned out to be useful architectures for the solution of many problems, e. g. in image processing or in the simulation of partial diierential equations. Therefore, there have been several attempts to introduce cell circuits suitable for large-scale integration 3]. Up to now, all of these cells need energy and therefore power supply. Just recently attempts have been made to build up circuitry being able to work without an external energy supply by using the energy stored in the initial state 1]. This principle can provide two major advantages. First, since no or at least not much energy is dissipated during computation, the circuit does not produce much heat. Therefore , there are no \hot spots" in integrated circuits, which limit integration density and operation speed. Furthermore, since there is no need for a power supply, the absence of voltage supply lines supports a high integration density. In this work an architecture for the realisation of a lossless CNN is proposed. Further on, since standard learning algorithms turn out to fail for lossless systems, a way to amend these is introduced. 1 Mathematical outline and circuit model 1.1 Mathematical outline As described in 2], lossless systems can be described by the following diierential equation: _ x = A(x)grad x H(x); (1) This diierential equation with a coupling matrix A, which depends on the state vector, leaves the Hamilto-nian H, an energy function of the state vector, unchanged, if A is skew-symmetric for all x. Normally the additional condition of Jacobi-identity has to be fulllled, too, but for the kind of Hamiltonian proposed later on it does not need to be taken into account. Since the Hamiltonian H is preserved, the trajectory of a system with a given initial energy H 0 will never leave a surface deened by H(x) = H 0 ; (2) This equation deenes an n-dimensional surface in the state space, which has the dimension n + 1. If we introduce coordinates on this reduced surface, we can map the given lossless diierential equation with the state vector x 2 R n+1 into a normal diierential equation with the state vector y 2 R n. The mapping is carried out by using a (nonlinear) mapping function r: y = r(x); (3) By using the inverse of this mapping for a speciic given energy H, here denoted by r ?1