Stream-Based Active Learning with Cooperative Neural Networks for Data-Efficient Partial Inverse Design: An Automotive Glass Run Channel Case Study

Agung Nugraha; Hyerin Kwon; Heungjun Im; Gian Antariksa; Jihwan Lee

cs.LGOct 7, 2026

Stream-Based Active Learning with Cooperative Neural Networks for Data-Efficient Partial Inverse Design: An Automotive Glass Run Channel Case Study

Authors: Agung Nugraha, Hyerin Kwon, Heungjun Im, Gian Antariksa, Jihwan Lee

Organizations: Department of Industrial and Data Engineering, Major in Industrial Data Science and Engineering, Pukyong National University, Busan, 48513, Republic of Korea · DRB Co., Ltd., Busan, 46329, Republic of Korea

Abstract

Inverse design in engineering often runs into a simple problem. Each labeled training sample must be produced through expensive simulation, so building a large dataset is slow and costly. This study addresses that problem for partial inverse design, where only some design variables are specified and the rest must be inferred to reach a target performance value. We propose CoNN-AL, a framework for data-efficient partial inverse design that adds stream-based active learning to the Cooperative Neural Network with Denoising Autoencoder (CoNN-DAE). The model estimates predictive uncertainty through Monte Carlo dropout and uses it to decide, in real time, which incoming candidate samples are worth labeling, so the limited labeling budget is spent on the most informative designs. We validate the framework on a real-world automotive glass run channel dataset of more than 900,000 unique simulated designs. With only 20,000 actively selected labels, about 2.3% of the training pool, CoNN-AL reaches R-squared values of 0.967 to 0.982 across all missing-variable levels, approaching the upper-bound models trained on far more data. It reaches R-squared of at least 0.95 with 30 to 40% fewer labels than random sampling at the more difficult missing-variable levels and, at the most challenging level, is the only strategy in this study to reach R-squared of 0.98. Together with this work, we publicly release the dataset to support future research on data-driven design.

Figures & tables

Figure 1: Architecture of the Cooperative Neural Network with Denoising Autoencoder (CoNN-DAE), adapted from Nugraha et al. [ 9 ] . The surrogate model (top) maps complete designs to the performance output. The imputation model (bottom) reconstructs missing variables via an autoencoder, jointly optimized through reconstruction and performance losses.

Figure 2: Stream-based active learning loop integrated with CoNN-DAE. Unlabeled samples stream in continuously, and the model estimates predictive uncertainty via MC-Dropout. High-uncertainty samples are queried from the oracle (FEA simulation) and added to the labeled pool, while low-uncertainty samples are discarded.

Figure 3: (a) Illustration of the GRC sealing profile location on the vehicle window frame. (b) The GRC profile during finite element analysis in MSC Marc.

Group	Variable	Unit
Lip Related Data	Seal gap	mm
	Lip tip radius	mm
	Lip radius at glass flat	mm
	Lip radius at top flat	mm
Notch Related Data	Notch lip thickness	mm
	Notch radius	mm

Table 1: Groups and units of the GRC design variables and output. Units refer to the original measurements; the shared dataset provides the design variables in standardized form.

Metric	Definition
Coefficient of Determination ( $R^{2}$ )	$1-\dfrac{\sum_{i=1}^{m}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{m}(y_{i}-\bar{y})^{2}}$
Mean Squared Error (MSE)	$\dfrac{1}{m}\sum_{i=1}^{m}(y_{i}-\hat{y}_{i})^{2}$
Mean Absolute Error (MAE)	$\dfrac{1}{m}\sum_{i=1}^{m}\left\|y_{i}-\hat{y}_{i}\right\|$
Root Mean Squared Error (RMSE)	$\sqrt{\dfrac{1}{m}\sum_{i=1}^{m}(y_{i}-\hat{y}_{i})^{2}}$
Symmetric Mean Absolute Percentage Error (sMAPE)	$\dfrac{2}{m}\displaystyle\sum_{i=1}^{m}\dfrac{\left\|y_{i}-\hat{y}_{i}\right\|}{\left\|y_{i}\right\|+\left\|\hat{y}_{i}\right\|}$

Table 2: Evaluation metrics and their definitions.

Figure 4: Radar chart comparison of upper-bound performance metrics (MSE, MAE, sMAPE, $R^{2}$ ) between 100,000 and 886,215 training sizes across MM levels. Performance is largely saturated at 100,000 samples.

Figure 5: Ground truth vs. prediction scatter plots for CoNN-DAE trained on 100,000 samples (MM $=$ 1 to MM $=$ 6). Points align closely with the identity line across all MM levels.

Figure 6: Ground truth vs. prediction scatter plots for CoNN-DAE trained on 886,215 samples (MM $=$ 1 to MM $=$ 6). The pattern closely matches the 100,000 model, confirming performance saturation.

Figure 7: Learning curves for random sampling baseline across all MM levels. Performance metrics (MSE, MAE, RMSE, sMAPE, $R^{2}$ ) are shown as a function of training size (10,000–100,000). Insets report each metric at the 40,000-sample checkpoint (dashed line). The bottom-right panel shows the number of training samples required to reach $R^{2}\geq 0.95$ at each MM level. A performance plateau is reached around 40,000 samples.

Figure 8: Active learning convergence curves (MSE, MAE, RMSE, sMAPE, $R^{2}$ ) over 100 rounds, with insets reporting each metric at round 18 (20,000 labeled samples), marked by the dashed line. The bottom-right panel shows the number of labeled samples, and the corresponding round, required to reach $R^{2}\geq 0.95$ at each MM level. Performance is consistent across all MM levels at this early checkpoint.

Figure 9: Ground truth vs. prediction scatter plots at active learning round 18 (20,000 labels) for MM $=$ 1 to MM $=$ 6. Near upper-bound prediction accuracy is achieved with approximately 2.3% of the full labeling budget.

Figure 10: $R^{2}$ comparison at three active learning checkpoints (round 18, round 74, round 100) across MM levels. Round 74 (76,000 labeled samples) represents the optimal stopping point.

Figure 11: Performance comparison of active learning, random sampling, and the upper-bound models (100,000 and 886,215) at 20,000 labeled samples (MM $=$ 6). Active learning outperforms random sampling and approaches the upper-bound.

Figure 12: Performance comparison at the most efficient active learning checkpoint (76,000 labels) vs. random sampling (60,000) and upper-bound (100,000, 886,215) under MM $=$ 6. Active learning achieves lower MSE and slightly higher $R^{2}$ than the upper-bound models, while MAE is higher.

$R^{2}$ Threshold	Random Sampling	Active Learning	Efficiency Ratio
0.90	10,000	8,000	1.3 $\times$
0.92	10,000	10,000	1.0 $\times$
0.94	20,000	12,000	1.7 $\times$
0.95	20,000	14,000	1.4 $\times$
0.96	30,000	15,000	2.0 $\times$
0.97	40,000	20,000	2.0 $\times$

Table 3: Label efficiency comparison at

R^{2}

thresholds (MM

=

6), showing the number of labeled samples required by random sampling and active learning to reach each threshold.

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Generative Inverse Design with Abstention via Diagonal Flow Matching

Miguel de Campos, Werner Krebs, Hanno Gottschalk

Institute of Mathematics, TU Berlin · Siemens Energy

Flow Matching Uncertainty Quantification Flow-Based Generative Modeling Inverse Design

Open paper AskPDF arXiv

Inverse design aims to find design parameters

x

achieving target performance

y^*

. Generative approaches learn bidirectional mappings between designs and labels, enabling diverse solution sampling. However, standard conditional flow matching (CFM), when adapted to inverse problems by pairing labels with design parameters, exhibits strong sensitivity to their arbitrary ordering and scaling, leading to unstable training. We introduce Diagonal Flow Matching (Diag--CFM), which resolves this through a zero-anchoring strategy that pairs design coordinates with noise and labels with zero, making the learning problem provably invariant to coordinate permutations. This yields substantially lower round-trip error than CFM and invertible neural network baselines across design dimensions up to

P{=}784

, including order-of-magnitude gains on several benchmarks. We develop two architecture-intrinsic uncertainty metrics, Zero-Deviation and Self-Consistency, that enable three practical capabilities: selecting the best candidate among multiple generations, abstaining from unreliable predictions, and detecting out-of-distribution targets; consistently outperforming ensemble and general-purpose alternatives across all tasks. We validate on airfoil, gas turbine combustor, scalable analytical benchmarks, a photonics inverse-design task, and an image-statistics benchmark.

Abstract

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