On the Generalization in Topology Optimization via Sensitivity-Conditioned Bernoulli Flow Matching

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://github.com/tum-pbs/topotransformer.

Topology optimization is a method for automatically designing efficient structures, such as lightweight mechanical parts or fluid systems. Modern AI models can speed up this process, but they often fail when conditions change. For example, when forces are applied in new ways or fluid boundaries differ from the training data. It has been unclear why some models generalize well while others do not. We studied this problem from an information perspective. In classical topology optimization, a quantity called the “sensitivity field” describes how design changes affect performance. We show mathematically that this field contains the most useful information for predicting good designs under changing conditions. We also introduce the idea of “pseudo-sensitivities”: physical fields that approximate sensitivities and may still support good generalization when exact sensitivities are too expensive to compute. To test this idea, we built a generative AI model for topology optimization and evaluated it on structural and fluid-design benchmarks with changing loads and boundary conditions. Models conditioned on sensitivities achieved the best out-of-distribution performance, while weaker conditioning signals performed substantially worse. Our findings help explain why some AI approaches for engineering design are more reliable than others and provide practical guidance for building more robust optimization systems.