Surrogate modelling of solutions of integral equations by neural networks is investigated theoretically. Estimates of speed of convergence of suboptimal surrogate solutions to solutions described by Fredholm theorem are derived.

In previous work the author found solutions to the Toda equations that were expressed in terms of determinants of integral operators. Here it is observed that a simple variant yields solutions to the matrix Toda equations. As an application another derivation is given of a differential equation of Sato, Miwa and Jimbo for a particular Fredholm determinant. During the last twenty years, beginnin...