Extended Data Fig. 3: Reduction of the CMOS Microwave Neural Network circuit to a generalized coupled mode model.

a, The integrated microwave neural network consists of interconnected linear and nonlinear resonators. The linear resonator is designed as a single waveguide with an adjustable length, implemented through a cascade of sub-segments, each referred to as L_in. These sub-segments can be grounded via switches (S₁, S₂, S₃, S₄ and S₅), which immediately terminate the microwave signal’s return path at the first switch that is shorted to ground. In contrast, the nonlinear resonator features a transmission line loaded with polynomially nonlinear capacitors. These capacitors form C-L-C π sections that are coupled by delays. Microwave power from input pads is distributed to these resonators through symmetrically arranged couplers (whose equivalent circuits are marked in purple). The left coupler divides power into two linear waveguides, while the right coupler feeds into a linear waveguide and into a nonlinear waveguide. Saturable gain elements, implemented as cross-coupled transistor pairs, connect the waveguides on opposite sides of the circuit, compensating for losses within the electromagnetic structures. Additionally, a pair of capacitor banks provides a small degree of tunability to the modes supported by the waveguides. Critically, there is parametric coupling between the circuit’s upper and lower halves through a pair of slow bitstream-driven switches. b, To simplify the complex circuit, we recognize that since the linear resonators support only a single natural frequency, they can be represented as tank circuits composed of L_j, and C_j. The symmetry in the bottom half of the circuit allows us to approximate the capacitor banks as two evenly split capacitors, contributing to the overall capacitance of the tank circuits. However, the asymmetry in the configuration of the resonators in the upper half does not permit such a simplification. c, To focus on the primary mechanism by which the system’s sensitivity to incoming signals is enhanced, we can largely ignore the left half of the circuit and concentrate on the interaction between the nonlinear distributed resonances and the linear resonator on the right half. These components interact only through the inductive path via a coupler and a coupling capacitor between the turns of the coupler. The source of regenerative gain through the cross-coupled pair is retained. For ease of analysis, we represent the parametrically driven switch as a tunable capacitor, which can be toggled between a very small value (open circuit) and a very large value (short circuit). d, The reduced circuit can be represented as an ensemble of coupled modes–a cascade of nonlinear resonators connected to a linear resonator via a parametrically varied switched coupling and a fixed phase delay (through the coupler). These modes interact with the incoming drive (fast Gigabit/sec microwave-speed data), with internal losses being compensated by saturable gain.