Neural operators have been explored as surrogate models for simulating physical systems to overcome the limitations of traditional partial differential equation (PDE) solvers. However, most existing operator learning methods assume that the data originate from a single physical mechanism, limiting their applicability and performance in more realistic scenarios. To this end, we propose Physical Invariant Attention Neural Operator (PIANO) to decipher and integrate the physical invariants (PI) for operator learning from the PDE series with various physical mechanisms. PIANO employs self-supervised learning to extract physical knowledge and attention mechanisms to integrate them into dynamic convolutional layers. Compared to existing techniques, PIANO can reduce the relative error by 13.6\%-82.2\% on PDE forecasting tasks across varying coefficients, forces, or boundary conditions. Additionally, varied downstream tasks reveal that the PI embeddings deciphered by PIANO align well with the underlying invariants in the PDE systems, verifying the physical significance of PIANO. The source code will be publicly available at: https://github.com/optray/PIANO.

翻译：神经算子作为模拟物理系统的替代模型被广泛探索，以克服传统偏微分方程求解器的局限性。然而，现有算子学习方法大多假设数据源自单一物理机制，这限制了它们在更真实场景中的适用性和性能。为此，我们提出物理不变量注意力神经算子（PIANO），用于从具有多种物理机制的偏微分方程序列中解译并集成物理不变量。PIANO采用自监督学习提取物理知识，并通过注意力机制将其集成到动态卷积层中。与现有技术相比，PIANO可在不同系数、作用力或边界条件下的偏微分方程预测任务中，将相对误差降低13.6%-82.2%。此外，多样化下游任务表明，PIANO解译的物理不变量嵌入与偏微分方程系统中的潜在不变量高度吻合，验证了PIANO的物理意义。源代码将在https://github.com/optray/PIANO 公开。