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

Mohammad Rashed1,2,4, Duarte F. Valoroso Madeira2,3, Babak Gholami2, Caglar Guerbuez2, Yunjia Yang1, Nils Thuerey1,4
1Technical University of Munich   2BMW AG   3Hamburg University of Technology   4Munich Center for Machine Learning
ICML 2026
Paper (coming soon) arXiv Code Dataset (coming soon)
Information-theoretic framework: causal Markov chain from physics parameters to optimal topology

(A) The topology optimization pipeline forms a causal Markov chain. The Data Processing Inequality implies that the sensitivity field is the information-theoretically optimal conditioning signal. (B) Fields approximating sensitivities via monotone transforms (pseudo-sensitivities) generalize OOD; information-poor fields do not.

Abstract

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 that drives classical TO. Modeling the TO pipeline as a causal Markov chain, the Data Processing Inequality establishes that 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 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.

Main Results

We evaluate on two physics domains: CFD topology optimization (turbulent channel flow, RANS k-ε) and structural topology optimization (2D compliance minimization). Both include in-distribution (ID) and out-of-distribution (OOD) test sets with boundary conditions never seen during training.

CFD — Simulation-Level Accuracy

Generated topologies evaluated through full CFD simulation in STAR-CCM+. Metrics: relative pressure-drop error (%) and 10%-accuracy.

Model ID Mean±Std ID Med. ID Acc. OOD-M Med. OOD-M Acc. OOD-H Med. OOD-H Acc.
DiT 2.59±4.21 1.45 96.4 3.66 79.4 4.85 70.6
UDiT 2.37±3.55 1.65 97.2 3.59 76.4 7.00 62.7
PDE-T 2.98±6.03 1.82 95.9 3.05 75.5 4.35 68.6
Ours 3.44±4.03 2.63 94.0 2.82 77.6 2.70 74.5

Our model achieves the best OOD-Hard accuracy (74.5%) and median error (2.70%), demonstrating the strongest generalization to unseen boundary conditions.

Structural — Compliance Error

Evaluated on 992 OOD samples with load configurations not seen during training.

Model Params (M) Mean ErrC (%) Med. ErrC (%)
TopoDiff 121 8.57 1.14
NITO 22 9.33 2.37
Ours 34 5.73 0.53

Conditioning Signal Matters

The choice of conditioning signal has a dramatic effect on OOD generalization. Sensitivity and pseudo-sensitivities (e.g., strain energy density) generalize well, while information-poor fields (e.g., raw displacement or pressure) degrade sharply under distribution shift.

Cross-entropy across conditioning signals

Topology cross-entropy across conditioning signals. Sensitivity and SED (pseudo-sensitivity) achieve nearly identical CE; physics parameters yield substantially higher uncertainty.

Generated topologies across conditioning signals

OOD structural topologies: Ground truth vs. predictions conditioned on Sensitivity, SED (pseudo-sensitivity), and Displacement.

Deployment Enhancements

Raw generative outputs contain salt-and-pepper noise that makes topologies unusable in simulation software. Engineers also need control over material budget and spatial constraints — all without retraining. We address this with inference-time strategies that work with the frozen, pre-trained model.

Interactive AI-Powered Design pipeline

The full deployment pipeline: an engineer defines a problem, runs one simulation to extract a sensitivity field, optionally edits the input, and the frozen model generates a clean, simulation-ready topology — no retraining required.

Greedy terminal sampling

Greedy Terminal Sampling

Greedy last-step decoding eliminates salt-and-pepper artifacts, producing clean binary topologies directly usable in simulation.

Volume fraction control

Volume Fraction Control

Confidence-based progressive pruning enforces material budget constraints during generation while preserving structural integrity.

Spatial blocking

Spatial Blocking

Engineers mark keep-out zones in the input; the model generates topologies that respect these spatial constraints.

Architecture

Model architecture

The model uses a transformer backbone with cross-attention conditioning — physical fields attend to topology tokens via cross-attention, keeping conditioning and generation in separate streams. Training uses Bernoulli flow matching, an iterative refinement process from uniform random bits to binary topology via learned transition probabilities. At test time, the model generates a topology in a single forward pass with 50 refinement steps.

BibTeX

@misc{rashed2026generalizationtopologyoptimizationsensitivityconditioned,
      title={On the Generalization in Topology Optimization via Sensitivity-Conditioned Bernoulli Flow Matching}, 
      author={Mohammad Rashed and Duarte F. Valoroso Madeira and Babak Gholami and Caglar Guerbuez and Yunjia Yang and Nils Thuerey},
      year={2026},
      eprint={2606.02179},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2606.02179}, 
}