Separable Operator Networks

• URL: http://arxiv.org/abs/2407.11253v2
• Date: Tue, 13 Aug 2024 07:08:49 GMT
• Title: Separable Operator Networks
• Authors: Xinling Yu, Sean Hooten, Ziyue Liu, Yequan Zhao, Marco Fiorentino, Thomas Van Vaerenbergh, Zheng Zhang,
• Abstract summary: Operator learning has become a powerful tool in machine learning for modeling complex physical systems governed by partial differential equations (PDEs) We introduce Separable Operator Networks (SepONet), a novel framework that significantly enhances the efficiency of physics-informed operator learning. SepONet uses independent trunk networks to learn basis functions separately for different coordinate axes, enabling faster and more memory-efficient training.
• Score: 4.688862638563124
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: Operator learning has become a powerful tool in machine learning for modeling complex physical systems governed by partial differential equations (PDEs). Although Deep Operator Networks (DeepONet) show promise, they require extensive data acquisition. Physics-informed DeepONets (PI-DeepONet) mitigate data scarcity but suffer from inefficient training processes. We introduce Separable Operator Networks (SepONet), a novel framework that significantly enhances the efficiency of physics-informed operator learning. SepONet uses independent trunk networks to learn basis functions separately for different coordinate axes, enabling faster and more memory-efficient training via forward-mode automatic differentiation. We provide a universal approximation theorem for SepONet proving that it generalizes to arbitrary operator learning problems, and then validate its performance through comprehensive benchmarking against PI-DeepONet. Our results demonstrate SepONet's superior performance across various nonlinear and inseparable PDEs, with SepONet's advantages increasing with problem complexity, dimension, and scale. For 1D time-dependent PDEs, SepONet achieves up to $112\times$ faster training and $82\times$ reduction in GPU memory usage compared to PI-DeepONet, while maintaining comparable accuracy. For the 2D time-dependent nonlinear diffusion equation, SepONet efficiently handles the complexity, achieving a 6.44\% mean relative $\ell_{2}$ test error, while PI-DeepONet fails due to memory constraints. This work paves the way for extreme-scale learning of continuous mappings between infinite-dimensional function spaces. Open source code is available at \url{https://github.com/HewlettPackard/separable-operator-networks}.

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