Organizations: School of Mechanical, Medical and Process Engineering, Queensland University of Technology, Brisbane, 4000, QLD, Australia · Institute of Biomechanics and Medical Engineering, AML, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China · State Key Laboratory of Advanced Environmental Technology, Guangzhou Institute of Geochemistry, Chinese Academy of Sciences, Guangzhou, 510640, China · School of Civil and Environmental Engineering, Queensland University of Technology, Brisbane, 4000, QLD, Australia
Abstract
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.
Surrogate models for topology optimization (TO) exhibit highly variable out-of-distribution (OOD) generalization under distribution shifts such as changing loads or boundary conditions, yet the source of this variability remains unclear. We hypothesize that OOD performance is governed by how much information the conditioning signal preserves about the adjoint sensitivity (reduced gradient) that drives classical TO. Modeling the TO pipeline as a causal Markov chain, the Data Processing Inequality establishes that, under this abstraction, the sensitivity field is an information-theoretically optimal conditioning signal for topology prediction. However, computing exact adjoint sensitivities can be expensive or unavailable in practice; we observe that certain physical fields can approximate sensitivities through monotone transformations. To formalize this, we introduce \textbf{pseudo-sensitivities} to characterize which fields enable generalization versus those that are information-poor. We then show that a sensitivity-conditioned Bernoulli flow-matching generator empirically confirms these predictions: conditioning on sensitivities yields state-of-the-art OOD performance, while increasingly distant physical fields degrade toward raw parameter conditioning. Results hold across structural TO benchmarks under load shifts and our new CFD-TO dataset under boundary-condition shifts such as multi-outlet configurations. Code and datasets are available at https://tum-pbs.github.io/topotransformer/ .
Mohammad Rashed, Duarte F. Valoroso Madeira, Babak Gholami +3
Flow matching is a powerful generative modeling framework, valued for its simplicity and strong empirical performance. However, its standard formulation treats signals on structured spaces, such as fMRI data on brain graphs, as points in Euclidean space, overlooking the rich topological features of their domains. To address this, we introduce topological flow matching, a topology-aware generalization of flow matching. We interpret flow matching as a framework for solving a degenerate Schrödinger bridge problem and inject topological information by augmenting the reference process with a Laplacian-derived drift. This principled modification captures the structure of the underlying domain while preserving the desirable properties of flow matching: a stable, simulation-free objective and deterministic sample paths. As a result, our framework serves as a drop-in replacement for standard flow matching. We demonstrate its effectiveness on diverse structured datasets, including brain fMRIs, ocean currents, seismic events, and traffic flows.
Kacper Wyrwal, İsmail İlkan Ceylan, Alexander Tong
Natural load-bearing patterns such as leaf venation, trabecular bone, and spider webs achieve high stiffness per unit mass, yet classical topology optimizers rarely reach such geometries, and few let engineers express structural design intent through natural language. This work treats a frozen text-to-image diffusion model as a training-free source of design knowledge and distills it into the physics loop of density-based topology optimization via score distillation sampling, so that a text prompt becomes an explicit, machine-interpretable representation of engineer intent. The prompt-induced generative gradient and the finite element sensitivity are combined at every iteration, letting physics decide which prompt-induced features survive. In 245 primary SDS runs spanning four geometric domains and two physics regimes, 38 of 49 prompt--domain combinations achieved statistically significant compliance reductions (up to −31.5% mechanical and −23.0% thermoelastic), outperforming gradient-based baselines. Cross-domain morphological analysis identifies a recurring structural signature of improvement: in most domains the generative prior suppresses dead-end branches in the rib skeleton, with endpoint--compliance correlation r=+0.56 to +0.99. A Heaviside projection with β-continuation resolves a pronounced intermediate-density tendency in this diffusion--physics coupling (42.6% to <3%), and an automated skeleton-based pipeline converts optimized density fields into \rev{candidate geometry ready for computer-aided design. By retargeting the generative prior across domains, loading conditions, and physics objectives through a change of text prompt, with each new problem's physics setup specified separately, the framework uses a pretrained generative model as a reusable, training-free prior for engineering design.