Random non-periodic architected materials exhibit complex nonlinear mechanical responses that pose significant challenges for computational design and inverse optimization using conventional homogenization theories. In this work, spinodal architected materials are employed as a representative system to investigate the multiscale origins of macroscopic nonlinear constitutive and instability behavior within a computational mechanics framework. Finite deformation simulations are conducted to systematically analyze the effects of porosity and Gaussian curvature distributions on dominant microstructural instability modes, including localized buckling and topology-driven transitions. Based on these observations, a microstructure evolution–driven equivalent nonlinear constitutive and instability model is developed to enable physics-constrained computational design. Topological descriptors like Gaussian curvature distributions are incorporated into characterize evolving anisotropy under finite deformation. A thermodynamically consistent energy functional is formulated to describe the competition between local buckling and topology transitions, leading to a microstructure evolution equation governing nonlinear stress–strain responses. The proposed model quantitatively captures macroscopic stress drops, stress plateaus, and anisotropic post-buckling behavior across spinodal structures with varying porosity and topology. Using the proposed nonlinear computational model as a physics constraint, a multi-objective intelligent design framework is further developed by coupling variational autoencoders with flow-matching generative models. The learned low-dimensional latent space enables efficient exploration of structure–property relationships and iterative inverse design under nonlinear mechanical constraints, allowing the generation of architected materials with tailored stiffness, strength, and permeability. This work establishes a unified computational design framework for random non-periodic architected materials, advancing physics-informed inverse design of complex microstructured systems.