A fundamental building block in characterizing and tackling scientific and industrial questions boils down to the ability of quickly solving mathematical equations. However, with the ever-growing volume of information and unsustainable integration growth in electronic processors, a radically new modality for solving equations is highly imminent. Here, we introduce an electromagnetic counterpart to solve multivariable complex equations, where two metamaterial kernels are connected in series to form a closed-loop electromagnetic system. Complex valued information is carried by electromagnetic fields, and the equation solution for arbitrary input signals can be recursively attained after a number of feedbacks. As an illustration, we present the capability of such a system in solving eight complex equations, and inversely design two 4×4 metamaterial kernels by topology optimization, whose average element error is reduced to smaller than 10 −4 . Having accomplished all unknown coefficients with high fidelity, our work represents a conspicuous apparatus for a myriad of enticing applications in ultracompact signal processing and neuromorphic computing.

Optical analog computing has recently sparked growing interest due to the appealing characteristics of low energy consumption parallel processing and ultrafast speed, spawning it complementary to conventional electronic computing. As the basic computing unit, optical logic operation plays a pivotal role for integrated photonics. However, the reported optical logic operations are volumetric and single-functional, which considerably hinders the practical cascadability and complex computing requirement. Here, we propose an on-chip combinational optical logic circuit using inverse design. By precisely engineering the scattering matrix of each small-footprint logic gate, all basic optical logic gates (OR, XOR, NOT, AND, XNOR, NAND, and NOR) are realized. On this foundation, we explore the assembly of these basic logic gates for general purpose combinational logic circuits, including optical half-adder and code converter. Our work provides a path for the development of integrated, miniaturized, and cascadable photonic processor for future optical computing technologies.