We consider a square non linear parametric equations system which is constituted of non differential equations in the unknowns that are the components of while is a set of parameters that play a role in the definition of the equations . We assume that is restricted to lie in a bounded region and we are interested in developing a solver for obtaining real solutions (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 called the . Starting from this set we show that we can create multiple pairs (parameters, solution) and that these pairs may be structured into coherent 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 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 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 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.

中文翻译：

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混合神经网络、延拓和符号计算来求解非线性方程的参数系统

我们考虑一个方形非线性参数方程组，它由未知数中的非微分方程组成，这些未知数是 的组成部分，而 是一组在方程定义中起作用的参数。我们假设 被限制在有界区域内，并且我们有兴趣开发一个求解器来获得有界区域内任何参数值的实数解（论文中定义的概念）。所提出方法的出发点是，我们假设数值方法使我们能够确定非常有限数量的固定点的真实解（但不一定是全部），称为 。从这个集合开始，我们证明我们可以创建多个对（参数、解决方案），并且这些对可以被构造成一致的，用于训练多层感知器（MLP）。训练过程是特定的：尽管它仍然使用损失函数的减少，但其主要目标是最大化出现次数，以训练集样本数的百分比表示，为此牛顿方案用MLP 预测，收敛到预期的解决方案。然后我们证明，对于足够多的 MLP，我们可以获得所有学习集 100% 的成功率。求解器是通过运行一组求解器获得的，其中每个求解器都基于特定的 MLP，其预测可能会导致系统的精确解。该求解器在由参数值（均不同于学习集中的样本）构成的样本集以及相应系统的所有解上进行测试。我们证明这些集合可以自动生成，并且它们也可以用于自学习过程中，以提高从初始解集建立的求解器的性能。