Finding the stiffness map of biological tissues is of great importance in evaluating their healthy or pathological conditions. However, due to the heterogeneity and anisotropy of biological fibrous tissues, this task presents challenges and significant uncertainty when characterized only by single-mode loading experiments. In this study, we propose a new method to accurately map the stiffness landscape of fibrous tissues, specifically focusing on brain white matter tissue. Initially, a finite element model of the fibrous tissue was subjected to six loading modes, and their corresponding stress-strain curves were characterized. By employing multiobjective optimization, an equivalent anisotropic material model was inversely extracted to best fit all six loading modes simultaneously. Subsequently, large-scale finite element simulations were conducted, incorporating various fiber volume fractions and orientations, to train a convolutional neural network capable of predicting the equivalent anisotropic material model solely based on the fibrous architecture of any given tissue. The method was applied to imaging data of brain white matter tissue, demonstrating its effectiveness in precisely mapping the anisotropic behavior of fibrous tissue. The findings of this study have direct applications in traumatic brain injury, brain folding studies, and neurodegenerative diseases, where accurately capturing the material behavior of the tissue is crucial for simulations and experiments.