Additive manufacturing enables compact heat exchangers based on triply periodic minimal surfaces (TPMS) such as the gyroid, offering very high surface-to-volume ratios and promising thermohydraulic performance. However, the flow regime inside periodically curved and cross-flowed TPMS passages cannot be adequately described by the classical laminar–turbulent dichotomy of smooth ducts. By analogy with flow past a circular cylinder—where a steady laminar wake becomes unstable at Reynolds numbers of approximately 47–50—we hypothesize that the spatially interlaced gyroid ligaments induce repeated flow separation and reorientation. As a result, a geometry-driven transitional flow state can persist at moderate Reynolds numbers (Re ≈ 100–400) that would remain laminar in straight tubes. This contribution presents a conjugate heat-transfer CFD study of an additively manufactured Inconel gyroid heat exchanger under experimentally defined boundary conditions (hot inlet temperature ≈ 45.6 °C, cold inlet temperature ≈ 14.7 °C, volumetric flow rate ≈ 130 L/h). Simulations are performed using the finite-volume method in ANSYS Fluent on a second-order polyhedral (poly-hexcore) mesh with refined near-wall resolution. Mixing and secondary-flow intensity are quantified using helicity and normalized helicity based on velocity and vorticity fields. To enable automated, feature-conditioned design, the CFD field data are transformed into structured grids and used to train an operator-learning surrogate based on a Fourier Neural Operator. As a proof of concept, temperature fields on y–z slices are predicted from inlet features and normalized spatial coordinates, yielding a fast surrogate for rapid screening and optimisation of TPMS and inlet configurations. Validation against calorimetric measurements yields an overall cold-side heat-transfer coefficient of 23.9 kW m⁻² K⁻¹, within 3% of the experimental value. Helicity distributions reveal alternating helical vortical structures acting as passive vortex generators that disrupt thermal boundary layers while avoiding pressure losses typical of fully developed turbulence. The results support a topology-based interpretation of TPMS heat exchangers as multi-directional cross-flow obstacle arrays and provide guidance for performance-oriented, machine-learning-accelerated TPMS design.