With an increase in the demand for smart wearable systems, artificial synapse arrays for flexible neural networks have received considerable attention. A synaptic device with a two-terminal configuration is promising for complex neural networks because of its ability to scale to a crossbar array architecture. To realize practical crossbar arrays with a high density, it is essential to achieve reliable electrode lines that act as signal terminals. However, an effective method to develop intrinsically flexible signal lines in artificial neural networks has not been developed. In this study, we achieved reliable polymer signal lines for flexible neural networks using coffee ring-free micromolding in capillaries (MIMIC). In a typical MIMIC, the outward convective flow of the polymer solution inherently deteriorates the pattern fidelity. To achieve reliable conducting polymer (CP) lines, we precisely controlled the flow of the polymer solution in the MIMIC by inducing the Marangoni force. When the convective and Marangoni flows for the solution were balanced in the MIMIC, the CP line patterns were reliably produced with high fidelity. The developed CP lines exhibited superior conductivity and high mechanical flexibility. Moreover, flexible memristor arrays consisting of CP signal lines demonstrated a high potential for realizing practical neuromorphic systems linked to artificial intelligence.