可分离算子网络
Separable Operator Networks
AI总结：
提出 SepONet，通过按坐标轴分离的主干网络和前向自动微分提升物理信息算子学习效率，并以理论与 PDE 实验证明其显著省显存、加速且精度可靠。
摘要：
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 Deep Operator Networks（DeepONet，深度算子网络）展现出良好前景，但它们需要大量的数据采集。Physics-informed DeepONets（PI-DeepONet，物理信息深度算子网络）缓解了数据稀缺问题，却存在训练过程效率低下的缺陷。我们提出 Separable Operator Networks（SepONet，可分离算子网络），这是一种显著提升物理信息算子学习效率的新颖框架。SepONet 使用独立的主干网络分别为不同坐标轴学习基函数，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 给出了一个通用逼近定理，证明任何非线性连续算子都存在可分离逼近。随后，我们相对于 PI-DeepONet 全面基准测试了其表示能力和计算性能。结果表明，SepONet 在各种非线性且不可分离的 PDE 上均具有优越性能，并且其优势会随问题复杂度、维度和规模的增加而扩大。对于一维含时 PDE，与 PI-DeepONet 相比，SepONet 在保持相当精度的同时，实现了最高 112 倍的训练加速和 82 倍的 GPU 显存占用降低。对于二维含时非线性扩散方程，SepONet 高效处理了其复杂性，取得 6.44% 的平均相对 ℓ2 测试误差，而 PI-DeepONet 因显存限制而失败。这项工作为无限维函数空间之间连续映射的超大规模学习铺平了道路。开源代码可在 https://github.com/HewlettPackard/separable-operator-networks 获取。
英文摘要：
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}.
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