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}.

翻译：算子学习已成为机器学习中用于建模由偏微分方程（PDE）控制的复杂物理系统的强大工具。尽管深度算子网络（DeepONet）显示出潜力，但它们需要大量的数据采集。物理信息深度算子网络（PI-DeepONet）缓解了数据稀缺问题，但其训练过程效率低下。我们引入了可分离算子网络（SepONet），这是一个新颖的框架，能显著提升物理信息算子学习的效率。SepONet使用独立的骨干网络为不同坐标轴分别学习基函数，从而通过前向模式自动微分实现更快、更节省内存的训练。我们为SepONet提供了一个通用逼近定理，证明了对任意非线性连续算子存在可分离逼近。随后，我们全面评估了其表示能力和计算性能，并与PI-DeepONet进行了基准测试。我们的结果表明，SepONet在各种非线性和不可分离的PDE上均表现出优越性能，且SepONet的优势随着问题复杂性、维度和规模的增加而增加。对于一维时间相关PDE，与PI-DeepONet相比，SepONet实现了高达112倍的训练加速和82倍的GPU内存使用减少，同时保持了相当的精度。对于二维时间相关非线性扩散方程，SepONet高效地处理了其复杂性，实现了6.44%的平均相对$\ell_{2}$测试误差，而PI-DeepONet则因内存限制而失败。这项工作为无限维函数空间之间连续映射的极大规模学习铺平了道路。开源代码可在 \url{https://github.com/HewlettPackard/separable-operator-networks} 获取。