Transfer Learning on Multi-Dimensional Data: A Novel Approach to Neural Network-Based Surrogate Modeling
This work addresses the data efficiency problem for researchers and engineers in computational science and engineering, offering an incremental improvement over existing CNN-based surrogate modeling methods.
The paper tackles the high cost of generating training data for neural network-based surrogate models of PDEs by proposing a transfer learning approach that mixes data from the full-dimensional problem and its lower-dimensional approximation, demonstrating on a multiphase flow test that it outperforms Monte Carlo methods with a fraction of the data generation budget.
The development of efficient surrogates for partial differential equations (PDEs) is a critical step towards scalable modeling of complex, multiscale systems-of-systems. Convolutional neural networks (CNNs) have gained popularity as the basis for such surrogate models due to their success in capturing high-dimensional input-output mappings and the negligible cost of a forward pass. However, the high cost of generating training data -- typically via classical numerical solvers -- raises the question of whether these models are worth pursuing over more straightforward alternatives with well-established theoretical foundations, such as Monte Carlo methods. To reduce the cost of data generation, we propose training a CNN surrogate model on a mixture of numerical solutions to both the $d$-dimensional problem and its ($d-1$)-dimensional approximation, taking advantage of the efficiency savings guaranteed by the curse of dimensionality. We demonstrate our approach on a multiphase flow test problem, using transfer learning to train a dense fully-convolutional encoder-decoder CNN on the two classes of data. Numerical results from a sample uncertainty quantification task demonstrate that our surrogate model outperforms Monte Carlo with several times the data generation budget.