🤖 AI Summary
Convolutional neural networks (CNNs) for surrogate modeling of high-dimensional partial differential equations (PDEs) suffer from prohibitive computational costs due to reliance on large-scale, high-fidelity numerical simulations.
Method: We propose a cross-dimensional transfer learning framework featuring a novel hybrid-dimensional (d- and (d−1)-dimensional) joint training paradigm. It leverages approximate solutions of lower-dimensional PDEs to guide training of high-dimensional CNN surrogates, enabling knowledge transfer and error compensation. The architecture employs a fully convolutional encoder–decoder, multi-scale transfer mechanisms, and PDE-informed dimensionality-reduced data generation, augmented with uncertainty quantification.
Contribution/Results: On multiphase flow benchmark problems, our method achieves higher accuracy than Monte Carlo methods using only a few times fewer simulation budgets. Forward inference is negligible in cost, dramatically improving the cost-effectiveness and practicality of PDE surrogates.
📝 Abstract
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.