AutoTandemML: Active Learning Enhanced Tandem Neural Networks for Inverse Design Problems

TL;DR

Inverse design in high-dimensional spaces is computationally expensive and often ill-posed. The authors present AutoTandemML, a hybrid framework that combines active learning with Tandem Neural Networks to efficiently learn forward and inverse mappings, using a forward model $\mathbf{F_{DNN}}$ and an inverse model $\mathbf{I_{DNN}}$ trained on actively acquired data and regularized by the forward mapping, with a high-fidelity surrogate $\mathbf{H}$ guiding data generation. The framework is validated on three benchmarks—Airfoil Inverse Design, Photonic Surfaces Inverse Design, and Scalar Boundary Reconstruction—demonstrating that active-learning–driven data generation yields better accuracy with fewer samples and reduced variability compared to standard samplers. Across the benchmarks, AutoTandemML shows strong performance in both forward and inverse tasks, highlighting its potential as a data-efficient tool for complex inverse design problems. The work suggests avenues for extending the approach with advanced uncertainty quantification and graph-based architectures to broaden applicability to discrete and graph-structured design spaces.

Abstract

Inverse design in science and engineering involves determining optimal design parameters that achieve desired performance outcomes, a process often hindered by the complexity and high dimensionality of design spaces, leading to significant computational costs. To tackle this challenge, we propose a novel hybrid approach that combines active learning with Tandem Neural Networks to enhance the efficiency and effectiveness of solving inverse design problems. Active learning allows to selectively sample the most informative data points, reducing the required dataset size without compromising accuracy. We investigate this approach using three benchmark problems: airfoil inverse design, photonic surface inverse design, and scalar boundary condition reconstruction in diffusion partial differential equations. We demonstrate that integrating active learning with Tandem Neural Networks outperforms standard approaches across the benchmark suite, achieving better accuracy with fewer training samples.

Figures (12)

• Figure 1: AutoTandemML framework segments: (a) Active learning to generate a dataset ($\mathbf{x}$, $\mathit{f}(\mathbf{x})$). (b) Forward deep neural network $\mathbf{F_{DNN}}$ training with the active learning generated dataset ($\mathbf{x}$, $\mathit{f}(\mathbf{x})$). (c) Inverse deep neural network $\mathbf{I_{DNN}}$ training with the active learning generated dataset ($\mathit{f}(\mathbf{x})$, $\mathbf{x}$), and a modified loss function that utilizes the $\mathbf{F_{DNN}}$ predictions.
• Figure 2: Inverse design benchmark problems: (a) Airfoil inverse design (AID). (b) Photonic surfaces inverse design (PSID). (c) Scalar boundary reconstruction (SBR).
• Figure 3: Inverse design validation procedure aiming to assess the accuracy of the trained $\mathbf{I_{DNN}}$ using the test data from each inverse design benchmark problem. In step (1), we define the test dataset comprising input-output pairs (T$_x$, T$_y$). In step (2), we utilize the output values T$_y$ as inputs to the inverse model $\mathbf{I_{DNN}}$ to obtain the predicted inputs P$_{\text{IDNN}}$. In step (3), we reconstruct the outputs P$_y$ by feeding the predicted inputs P$_{\text{IDNN}}$ into the HF surrogate $\mathbf{H}$. Finally, in step (4), we compare the original output values T$_y$ from the test dataset with the reconstructed outputs P$_y$ to evaluate the inverse model's accuracy using the specified metrics.
• Figure 4: Inverse DNN ($\mathbf{I_{DNN}}$) performance on AID benchmark problem using different dataset generation methods: (a) R$^2$ (higher is better), (b) RMSE (lower is better), and (c) NMAE (lower is better). Subscripts in $\mathbf{I_{DNN}}$ denote the sampling method (e.g., $\mathbf{I_{DNN_R}}$ for random sampling).
• Figure 5: $\mathbf{I_{DNN}}$ performance on PSID benchmark problem using different dataset generation methods: (a) R$^2$ (higher is better), (b) RMSE (lower is better), and (c) NMAE (lower is better). Subscripts in $\mathbf{I_{DNN}}$ denote the sampling method (e.g., $\mathbf{I_{DNN_R}}$ for random sampling).