Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometries

TL;DR

This paper tackles the challenge of predicting full-field solutions on 3D geometries that vary with design parameters. It introduces Geom-DeepONet, a DeepONet variant that augments the trunk with signed distance function inputs and uses SIREN activations, along with intermediate feature fusion, to handle irregular, parameterized 3D domains and vector fields. Through two geometry families (a beam with a circular hole and a cuboid with a random ellipsoidal void), Geom-DeepONet demonstrates superior accuracy and significantly lower memory usage compared with PointNet and vanilla DeepONet, plus strong generalization to unseen designs and massive speedups over finite element simulations. The work highlights practical potential for rapid design exploration and digital-twin workflows while noting limitations such as spectral bias and the need for parametric geometry descriptions, with proposed future directions including implicit geometry encoding and Fourier-based methods to better capture high-frequency features.

Abstract

Modern digital engineering design process commonly involves expensive repeated simulations on varying three-dimensional (3D) geometries. The efficient prediction capability of neural networks (NNs) makes them a suitable surrogate to provide design insights. Nevertheless, few available NNs can handle solution prediction on varying 3D shapes. We present a novel deep operator network (DeepONet) variant called Geom-DeepONet, which encodes parameterized 3D geometries and predicts full-field solutions on an arbitrary number of nodes. To the best of the authors' knowledge, this is the first attempt in the literature and is our primary novelty. In addition to expressing shapes using mesh coordinates, the signed distance function for each node is evaluated and used to augment the inputs to the trunk network of the Geom-DeepONet, thereby capturing both explicit and implicit representations of the 3D shapes. The powerful geometric encoding capability of a sinusoidal representation network (SIREN) is also exploited by replacing the classical feedforward neural networks in the trunk with SIREN. Additional data fusion between the branch and trunk networks is introduced by an element-wise product. A numerical benchmark was conducted to compare Geom-DeepONet to PointNet and vanilla DeepONet, where results show that our architecture trains fast with a small memory footprint and yields the most accurate results among the three with less than 2 MPa stress error. Results show a much lower generalization error of our architecture on unseen dissimilar designs than vanilla DeepONet. Once trained, the model can predict vector solutions, and speed can be over $10^5$ times faster than implicit finite element simulations for large meshes.

Figures (20)

• Figure 1: Schematic of the PointNet used for comparison with other networks. Input and output tensor shapes are shown in green and red, respectively. $?$ and $BN$ denote the flexible batch size dimension and batch normalization.
• Figure 2: Schematic of the vanilla DeepONet used for comparison with other networks. Input and output shapes are shown in green and red, respectively. $?$ and $*$ denote the flexible batch size and number of nodes in the data. $n$ and $N$ are the number of input parameters and the number of nodes included in the training.
• Figure 3: Schematic of the proposed Geom-DeepONet. Input and output shapes are shown in green and red. $?$ and $*$ denote the flexible batch size and number of nodes in the data. $n$, $N$, and $c$ are the number of input parameters and nodes included in training and output vector components, respectively.
• Figure 4: Randomly selected beam designs.
• Figure 5: Histograms of number of mesh nodes in different geometries for \ref{['beam_nodes']} beams and \ref{['cube_nodes']} cuboids with random void.