Optimal parameter selection in Weeks’ method for numerical Laplace transform inversion based on machine learning

The Weeks method for the numerical inversion of the Laplace transform utilizes a Möbius transformation which is parameterized by two real quantities, σ and b. Proper selection of these parameters depends highly on the Laplace space function F(s) and is generally a nontrivial task. In this paper, a convolutional neural network is trained to determine optimal values for these parameters for the specific case of the matrix exponential. The matrix exponential e^A is estimated by numerically inverting the corresponding resolvent matrix $(sI-A)-1$ via the Weeks method at $(\sigma ,b)$ pairs provided by the network. For illustration, classes of square real matrices of size three to six are studied. For these small matrices, the Cayley-Hamilton theorem and rational approximations can be utilized to obtain values to compare with the results from the network derived estimates. The network learned by minimizing the error of the matrix exponentials from the Weeks method over a large data set spanning $(\sigma ,b)$ pairs. Network training using the Jacobi identity as a metric was found to yield a self-contained approach that does not require a truth matrix exponential for comparison.