Operator-based neural network architectures such as DeepONets have emerged as a promising tool for the surrogate modeling of physical systems. In general, towards operator surrogate modeling, the training data is generated by solving the PDEs using techniques such as Finite Element Method (FEM). The computationally intensive nature of data generation is one of the biggest bottleneck in deploying these surrogate models for practical applications. In this study, we propose a novel methodology to alleviate the computational burden associated with training data generation for DeepONets. Unlike existing literature, the proposed framework for data generation does not use any partial differential equation integration strategy, thereby significantly reducing the computational cost associated with generating training dataset for DeepONet. In the proposed strategy, first, the output field is generated randomly, satisfying the boundary conditions using Gaussian Process Regression (GPR). From the output field, the input source field can be calculated easily using finite difference techniques. The proposed methodology can be extended to other operator learning methods, making the approach widely applicable. To validate the proposed approach, we employ the heat equations as the model problem and develop the surrogate model for numerous boundary value problems.

翻译：算子型神经网络架构（如DeepONet）已成为物理系统代理建模的有力工具。通常，在算子代理建模中，训练数据需通过有限元法等偏微分方程求解技术生成。数据生成的高计算成本是制约此类代理模型在实际应用中部署的主要瓶颈之一。本研究提出了一种创新方法，以缓解DeepONet训练数据生成的计算负担。与现有文献不同，本文提出的数据生成框架无需使用任何偏微分方程积分策略，从而显著降低DeepONet训练数据集生成的计算成本。具体策略为：首先利用高斯过程回归随机生成满足边界条件的输出场，随后通过有限差分法从输出场轻松计算输入源场。该方案可推广至其他算子学习方法，具有广泛适用性。为验证所提方法，我们以热传导方程作为模型问题，针对多个边值问题开发了相应的代理模型。