发表机构
ETH Zürich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对物理信息神经算子训练中损失函数病态性问题，提出预条件残差损失函数，利用多重网格实现网格无关条件数，在多个方程上达到监督训练精度且快四到二十五倍。
AI 中文摘要
神经算子通常以监督方式进行训练，这需要使用经典求解器生成数据集。以物理信息方式训练，即纯粹从控制方程出发，消除了这一巨大的离线成本，并允许在每个优化步骤中抽取新样本，但迄今为止仅限于简化问题，且在准确性上落后于监督训练。障碍在于物理信息损失函数的病态性，这种病态性由微分算子引起，并随着离散化细化而恶化。因此，我们提出了一种预条件残差损失函数，并展示了椭圆问题的网格无关条件数以及鞍点问题的显著改善条件数。通过几何和代数多重网格实现，该构造适用于线性和非线性方程，稳态或时变问题，在结构化与非结构化网格上，对神经算子架构不可知，且在推理时不增加成本。在泊松方程、Allen-Cahn方程和稳态Stokes方程上，所得的无标签训练与监督训练相匹配，且比先前的物理信息算子学习方法准确四到二十五倍。
英文摘要
Neural operators are typically trained in a supervised fashion, which requires a dataset to be generated with a classical solver. Training them physics-informed, i.e., purely from the governing equations, removes this large offline cost and allows fresh samples to be drawn at every optimization step, but has so far been limited to simplified problems and trails supervised training in accuracy. The obstacle is the ill-conditioning of physics-informed losses, which differential operators induce and which worsens as the discretization is refined. We therefore propose a preconditioned residual loss function and show mesh-independent conditioning for elliptic problems and greatly improved conditioning for saddle point problems. Realized through geometric and algebraic multigrid, the construction applies to linear and nonlinear equations, steady or time-dependent, on structured and unstructured meshes, is agnostic to the neural operator architecture, and adds no cost at inference. On the Poisson, Allen-Cahn and stationary Stokes equations, the resulting label-free training matches supervised training and is four to twenty-five times more accurate than previous physics-informed operator learning methods.
