将物理约束融入深度算子网络，高效建模弯曲后向台阶流场。
Physics-constrained DeepONet for Surrogate CFD models: a curved backward-facing step case
- 在深度算子网络中嵌入连续性方程的散度约束。
- 仅用50个样本和50次迭代即收敛，稀疏数据下精度更高。
- 适合需要快速模拟复杂流场的工程场景。
本文提出物理约束深度算子网络（PC-DeepONet），将基础物理知识融入数据驱动的DeepONet模型。以计算流体力学中的典型问题——弯曲后向台阶流动为案例，利用参数化几何形状生成的CFD数据进行训练。该模型学习从几何参数到速度场与压力场的映射关系。与纯数据驱动的DeepONet相比，PC-DeepONet引入了连续性方程的散度约束。在仅使用50个样本的小数据集下，两者均能在50次迭代内实现收敛，且PC-DeepONet在稀疏数据条件下表现出更高的精度，凸显神经算子对偏微分方程所描述动力学的高效学习能力。
原文摘要 · Abstract (English)
The Physics-Constrained DeepONet (PC-DeepONet), an architecture that incorporates fundamental physics knowledge into the data-driven DeepONet model, is presented in this study. This methodology is exemplified through surrogate modeling of fluid dynamics over a curved backward-facing step, a benchmark problem in computational fluid dynamics. The model was trained on computational fluid dynamics data generated for a range of parameterized geometries. The PC-DeepONet was able to learn the mapping from the parameters describing the geometry to the velocity and pressure fields. While the DeepONet is solely data-driven, the PC-DeepONet imposes the divergence constraint from the continuity equation onto the network. The PC-DeepONet demonstrates higher accuracy than the data-driven baseline, especially when trained on sparse data. Both models attain convergence with a small dataset of 50 samples and require only 50 iterations for convergence, highlighting the efficiency of neural operators in learning the dynamics governed by partial differential equations.
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