Trajectory-Aware Flow Matching for Topology Optimisation

Topology optimisation (TO) often requires repeated finite element analysis and sensitivity-based material updates, which can be costly when multiple candidate designs are needed under varying physical and design conditions.

Generative TO offers a route to rapid design exploration, but existing models may rely on adversarial training, long reverse-diffusion sampling, or external guidance to maintain structural feasibility and physical consistency.

This study develops a flow matching-based topology optimisation (FMTO) framework for conditional topology generation.

Linear FMTO is first formulated as an endpoint-based baseline by interpolating between a Gaussian source field and the BESO reference topology.

To introduce mechanically meaningful intermediate states, a trajectory-aware FMTO formulation is proposed, where volume-fraction-indexed BESO states are used to construct the probability path and target velocity field.

This incorporates physics-guided optimisation history into generative flow learning without adding inference-time optimisation.

A path--velocity mismatch analysis explains why moderate trajectory weighting can improve generation stability, whereas excessive guidance may over-constrain the learned transport.

Numerical examples show that FMTO generates diverse topology candidates with improved compliance-related performance, volume-fraction satisfaction, topology fidelity, and substantially fewer sampling steps than a diffusion-based baseline.

Under limited training data, trajectory-aware FMTO achieves the best overall performance with a moderate trajectory weight.

Studies on trajectory-anchor density and three-dimensional topology generation further demonstrate the influence of path design and the applicability of the proposed framework beyond two-dimensional problems.