• Designable disorder creates novel isotropic architected materials. • Data-driven MBO framework with dual-acquisition strategy optimizes TAAMs. • Optimized TAAMs outperform SOTA benchmarks in stiffness-isotropy trade-off. • Framework boosts isotropy index to > 0.9 with minimal stiffness loss. • Experimental validation confirms superior performance of fabricated TAAMs. For their excellent stiffness-to-weight characteristics, triply periodic minimal surfaces (TPMS) are widely adopted in architected materials. However, their geometric regularity often leads to elastic anisotropy, limiting their effectiveness under complex loading. To address this, we propose TPMS-based aperiodicity-induced architected materials (TAAMs), which incorporate controllable geometric disorder as a tunable design variable. This concept of designable disorder broadens the geometric design space, enabling the simultaneous optimization of stiffness and isotropy. A data-driven framework integrating computational homogenization with multi-objective Bayesian optimization is employed to discover high-performance TAAMs. Selected designs were fabricated using fused deposition modeling and validated through uniaxial compression tests, showing strong agreement with numerical predictions. Compared to conventional TPMS, TAAMs exhibit significantly improved elastic isotropy while maintaining high stiffness across various relative densities. This approach offers a robust and scalable pathway for developing architected materials tailored to applications requiring isotropic performance, such as biomedical implants, protective systems, and aerospace components.