A Method of Teaching a Neural Network to Generate Vector Fields for a Given Attractor

Consider a nite n dimensional space with an n dimensional vector S com pletely describes the state in which a nonlinear system resides to know the vector of state changes is a complete representation of the system dynamics provided it is deterministic and of rst order The totality of all state changes form a vector eld that contains the entire system dynamics A trajectory in state space is formed by pieceing together in nitesimal steps in the direction indicated by this vector eld The result is a curve whose tangent at each point is always aligned with the vector eld Thompson and Stewart If a neural network can be trained to learn this vector eld to associate the actual state vector S at the input with the vector of state change the network establishes a representation of the dynamic system equivalent to a set of nonlinear rst order di erential equations Various neural networks with generalization capabilities have been proposed in the literature recent Werntges for the task of vector eld approxima tion In this paper we present a method of teaching a multi layer perceptron MLP to generate a vector eld in state space that describes the dynamic of a nonlinear system when only knowledge of the attractor to which the system evolves is available Starting with an initial state the dynamic of the network governs the sequence of state vectors S t and generates a state space trajec tory The application of such a method of mapping desired trajectories onto embeddings in vector elds can bes considered for path planing modules with online obstacle avoidance features