ICMS 2024 Session

Machine Learning within
Computer Algebra Systems

ICMS 2024 Home Page — Session Home Page

Yves Papegay and Jean-Pierre Merlet — Mixing computer algebra methods and neural networks for solving systems of parametric equations

Abstract

We define a system of parametric equations as a set of n equations F(p, x) = 0 where x is a vector of unknowns of dimension n and p a set of parameters that play a role in the definition of the elements of F - which are least C1 functions. For a given p the instance of F may admit several solutions.

We assume that CAS, interval analysis, continuation or algebraic geometry methods may provide all or part of these solutions but with a prohibitive computation time. Our aim is to develop a framework that will allow to obtain possibly all solutions for any instance of p within a bounded region in a very fast way, under the assumption that a significant number of instances of F will have to be solved.

The starting point is to obtain a set S of solutions (not necessary all of them) for a limited number instances for p (typically about 8 cases). We then show how different structured learning sets – i.e. pairs (pj, sj) where pj is an instance of p and sj is one of the solution of F for this instance – may be deduced from S using continuation and CAS. These learning sets will be used to train several multi-layer perceptrons (MLPs) that will provide various predictions of the solutions that will be used as guess for the Newton method and will possibly provide exact solutions – in the sense that they can be refined to obtain a solution whose distance to the exact solution may be as small as desired. The training of a given MLPs is based not only on the decrease of a loss function but also on the number of occurrences for which the prediction of the MLP for a given p in the learning set, used as initial guess for the Newton scheme, will lead to the corresponding solution in the learning set.

This approach will be illustrated on a difficult robotics problem for which alternate methods such as continuation and interval analysis require about 20 hours to solve F for a given instance of p. We will show that building the set of MLPs require about 70 hours but as soon as the MLPs have been obtained the solving time for any instance of P is less than 0.1 seconds. We cannot guarantee to obtain all solutions for a given P but we will exhibit a self-learning procedure that may allow to obtain additional MLPs that may provide missed solution(s).