Material properties, such as elastic modulus, Poisson’s ratio, mass density, electric permittivity, magnetic permeability, thermal conductivity, heat capacity, or others, are so defined that they are in general greater than zero. Thermodynamic bounds must be satisfied in conventional materials. However, when materials are allowed to have internal microstructures and processes, they may exhibit negative characteristics, such as negative stiffness due to postbuckling processes. Several physical considerations, such as violation of conservation laws in the non-Hermitian, non-reciprocal systems, or odd elasticity, are adopted to examine their stability or exceptional points, where eigenvalues change from real to complex numbers. Possible applications of materials with negative characteristics are abundant, e.g. vibration mitigation, noise reduction, cloaking in electromagnetic, or unbounded effective material properties in composites. In this minisymposium, all aspects from numerical viewpoints in understanding materials with negative characteristics are welcome, including, but not limited to, machine learning techniques to generate material microstructures or properties, and novel analytical or computational methods in calculating physical properties for materials in solid, liquid, or other states. Experimental or theoretical studies to correlate numerical results are also welcome.