Mixing neural networks, continuation and symbolic computation to solve parametric systems of non linear equations

We consider a square non linear parametric equations system F(P,X) = 0 which is constituted of n non differential equations in the n unknowns {x₁,…,x_n} that are the components of X while P={p₁,…,p_m} is a set of m parameters that play a role in the definition of the equations F. We assume that P is restricted to lie in a bounded region and we are interested in developing a solver for obtaining all real solutions exactly (a notion that is defined in the paper) for any parameter values within the bounded region. The starting point of the proposed approach is that we assume that a numerical methods has allowed us to determine the real solutions (but not necessarily all of them) for a very limited number of fixed P called the initial solution set. Starting from this set we show that we can create multiple pairs (parameters, solution) and that these pairs may be structured into coherent learning sets that will be used to train multi-layer perceptrons (MLP). The training process is specific: although it still uses a decrease of a loss function its main objective is to maximize the success rate i.e. the number of occurrences, expressed in percentage of number of samples of the training set, for which the Newton scheme, initialized with the MLP prediction, converges toward the expected solution. We then show that for a sufficiently large number of MLPs we may obtain a 100% success rate for all learning sets. The solver is obtained by running a set of local solvers each of which is based on a specific MLP whose prediction may lead to an exact solution of the system. This solver is tested on verification sets i.e. set of samples constituted of parameter values (all different from the samples in the learning set) and all the solutions of the corresponding system. We show that these sets may be automatically generated and that they may also be used in a self-learning process for improving the performance of the solver established from the initial solution set. This approach is illustrated on two engineering problems in robotics and chemistry and it is shown that the solver provides all solutions for any instance of P with a high probability although we cannot guarantee it. The time required to design the final solver is large but the solving time is extremely low so that this approach should be used when the system has to be solved for a sufficient number of occurrences of the P. Furthermore we will show that the computation time required for establishing the solver may be drastically reduced by using a distributed implementation.