Solving complex optimization problems in engineering and the physical sciences requires repetitive computation of multi-dimensional function derivatives. Commonly, this requires computationally-demanding numerical differentiation such as perturbation techniques, which ultimately limits the use for time-sensitive applications. In particular, in nonlinear inverse problems Gauss-Newton methods are used that require iterative updates to be computed from the Jacobian. Computationally more efficient alternatives are Quasi-Newton methods, where the repeated computation of the Jacobian is replaced by an approximate update. Here we present a highly efficient data-driven Quasi-Newton method applicable to nonlinear inverse problems. We achieve this, by using the singular value decomposition and learning a mapping from model outputs to the singular values to compute the updated Jacobian. This enables a speed-up expected of Quasi-Newton methods without accumulating roundoff errors, enabling time-critical applications and allowing for flexible incorporation of prior knowledge necessary to solve ill-posed problems. We present results for the highly non-linear inverse problem of electrical impedance tomography with experimental data.