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}.

翻译：算子学习已成为机器学习中建模由偏微分方程（PDE）控制的复杂物理系统的强大工具。尽管深度算子网络（DeepONet）展现出潜力，但其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使用独立的干线网络来分别为不同的坐标轴学习基函数，从而通过前向模式自动微分实现更快、内存效率更高的训练。我们为SepONet提供了一个通用逼近定理，证明其可推广到任意的算子学习问题，然后通过与PI-DeepONet的全面基准测试验证其性能。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$112\times$的训练加速和$82\times$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。