arXiv:2605.00760cs.LG2026-05

用深度算子网络学习任意形状障碍物的散射场,比有限元法更快更省资源。

Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries

论文配图:Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries
图 1 · 摘自论文原文
  • 用符号距离函数编码任意几何,输入到DeepONet分支与支干部分。
  • 在未见过的几何上测试,预测结果与有限元法误差小于5%。
  • 无需重网格化,可快速扩展到新区域,适合复杂几何仿真场景。

本文研究在非参数化二维域上求解二维赫姆霍兹方程,采用物理信息神经算子网络DeepONet框架。考虑一个中心带有任意边界几何形状的包含物作为谐波入射波的散射体。目标是学习从散射体几何到散射场之间的算子映射。通过在域内多个点计算内含物边界的符号距离函数来编码其几何形状,并作为DeepONet架构中分支部分的输入;局部信息则作为主干部分的输入。该方法可有效表示任意几何(无论是否参数化)。模型在未见过的几何上的表现与有限元法(FEM)对比,验证了其泛化能力。训练后的网络权重隐式嵌入了局部物理规律及其与域几何的相互作用。若训练空间充分覆盖目标评估空间,模型可良好泛化。此外,模型可无需从头训练即可扩展至新区域。该框架避免了每种几何均需重新划分网格的步骤。相比传统FEM替代方案,所提方法计算成本更低,且不依赖于由FEM生成的训练数据。

原文摘要 · Abstract (English)

This paper deals with solving the 2D Helmholtz equation on non-parametric domains, leveraging a physics-informed neural operator network, the DeepONet framework. We consider a 2D square domain with an inclusion of arbitrary boundary geometry at its center. It acts as a scatterer for an incoming harmonic wave. The aim is to learn the operator linking the geometry of the scatterer to the resulting scattered field. A signed distance function to the boundary of the inner inclusion evaluated in several points on the domain is used to encode its geometry. It serves as input for the branch part of the DeepONet architecture and local information as the input for the trunk part. This approach enables the encoding of arbitrary geometries, whether they are parameterized or not. The evaluation of the model on unseen geometries was compared to its finite element method (FEM) equivalent to test its generalization capabilities. The trained network weights implicitly embed the local physics and their interaction with the domain geometry. If the training space sufficiently covers the target evaluation space, the model can generalize accordingly. Furthermore, it can be refined to extend to another region of interest without retraining from scratch. This framework also avoids the need to remesh the domain for each geometry. The proposed approach delivers a computationally lighter surrogate model than FEM alternatives and avoids relying on FEM generated training data.

算子网络赫姆霍兹方程非参数几何物理信息网络

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