We present initial results on Hessian-free force-gradient integrators for lattice field theories. Integrators of this framework promise to provide substantial performance enhancements, particularly
for larger lattice volumes where higher-order integrators demonstrate greater efficiency. Numerical
results demonstrate the superior efficiency of the proposed integrators compared to commonly
employed non-gradient schemes, particularly due to enhanced stability properties. It is anticipated
that the advantages of the Hessian-free framework will become even more pronounced in nested
integration approaches and for smaller fermion masses, where the numerical stability properties
of the integrators become increasingly important.