Separable Operator Networks
This work addresses computational bottlenecks in operator learning for physical systems, enabling extreme-scale applications, though it is incremental as it builds on existing DeepONet frameworks.
The paper tackles the inefficiency of physics-informed operator learning for PDEs by introducing Separable Operator Networks (SepONet), which achieves up to 112x faster training and 82x reduction in GPU memory usage while maintaining accuracy.
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 the existence of a separable approximation to any nonlinear continuous operator. Then, we comprehensively benchmark its representational capacity and computational performance 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 112x faster training and 82x 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}.