A novel data generation scheme for surrogate modelling with deep operator networks

TL;DR

The paper tackles the data-generation bottleneck in operator-learning surrogates like DeepONet by introducing a Gaussian Process Regression (GPR) based method to synthesize training data without solving the governing PDEs. By first sampling a physically consistent temperature field that satisfies Dirichlet boundaries and then deriving the corresponding heat source via finite differences, the approach avoids expensive FEM-based data generation while producing physically meaningful input-output pairs. The method is validated on multiple 2-D heat-conduction problems (square, triangular, annular domains) with varying boundary conditions and conductivities, achieving near-perfect predictive accuracy (R^2 close to 1) and very small Normalized L2 errors. The framework is domain- and boundary-agnostic and extensible to other operator-learning methods, offering substantial computational savings and broad practical impact for rapid surrogate modeling of PDE systems.

Abstract

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.

Figures (15)

• Figure 1: DeepONet Architecture consists of two sub neural networks namely branch and trunk network.
• Figure 2: Flowchart of data generation for steady state heat conduction problem : (\ref{['5_1a']}) existing approach in literature, (\ref{['5_1b']}) proposed framework. Unlike existing work the proposed framework does not require the use of FEM technique.
• Figure 3: Example for grid points in a domain: Grid points for an annular domain are shown. The grid points can be used as sensor as well as output points.
• Figure 4: A sample representation prior and posterior temperature field for a) 1-D b) 2-D heat conduction. Figure shows that temperature generated from posterior distribution using GPR is able to satisfy the boundary conditions.
• Figure 5: Generation of the temperature data for heterogeneous boundary conditions [$T_{x=0,y}=200y(1-y), T_{x=1,y}=400y(1-y), T_{x,y=0}=200x(1-x), T_{y=1}=400x (1-x)$ : (\ref{['6_2ta']}) profile satisfying non-homogeneous boundary condition, (\ref{['6_2tb']}) profile satisfying homogeneous boundary condition, (\ref{['6_2tc']}) final profile obtained by addition of profile (a) and (b).