Operator learning has become a powerful tool in machine learning for modeling complex physical systems. 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 theoretical guarantees for SepONet using the universal approximation theorem and validate its performance through comprehensive benchmarking against PI-DeepONet. Our results demonstrate that for the 1D time-dependent advection equation, when targeting a mean relative $\ell_{2}$ error of less than 6% on 100 unseen variable coefficients, SepONet provides up to $112 \times$ training speed-up and $82 \times$ GPU memory usage reduction compared to PI-DeepONet. Similar computational advantages are observed across various partial differential equations, with SepONet's efficiency gains scaling favorably as problem complexity increases. This work paves the way for extreme-scale learning of continuous mappings between infinite-dimensional function spaces.

翻译：算子学习已成为机器学习中建模复杂物理系统的有力工具。尽管深度算子网络（DeepONet）展现出潜力，但其需要大量数据采集。物理信息深度算子网络（PI-DeepONet）缓解了数据稀缺问题，但存在训练过程效率低下的缺陷。本文提出可分离算子网络（SepONet），一种显著提升物理信息算子学习效率的新型框架。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且更节省内存的训练。我们基于通用逼近定理为SepONet提供了理论保证，并通过与PI-DeepONet的全面基准测试验证了其性能。实验结果表明，针对一维时间依赖平流方程，当目标是在100个未见变量系数上实现低于6%的平均相对$\ell_{2}$误差时，与PI-DeepONet相比，SepONet可提供高达$112 \times$的训练加速和$82 \times$的GPU内存使用减少。在不同偏微分方程中均观察到类似的计算优势，且SepONet的效率增益随问题复杂度增加呈现良好的扩展性。此项工作为无限维函数空间之间连续映射的极端规模学习开辟了新途径。