Physics-Informed Geometric Operators to Support Surrogate, Dimension Reduction and Generative Models for Engineering Design

TL;DR

This work introduces physics-informed geometric operators (GOs) that augment low-level shape representations with high-level intrinsic features derived from geometric moments, curvature, and 3D Fourier descriptors. By feeding GO-augmented descriptors into surrogate/discriminative models, dimension reduction methods, and generative models, the approach regularises learning, enhances latent-space quality, and improves the generation of valid, diverse designs for engineering shapes such as ship hulls and aerofoils. The authors demonstrate improved predictive accuracy for performance metrics like $C_w$ and $C_L/C_D$, more compact and diverse latent spaces in DRMs, and comparable quality/diversity in generative design when using PaDGAN-GO versus direct physics evaluations. This physics-informed representation accelerates convergence in shape optimisation, reduces computational costs, and offers a pathway to integrating physics more deeply into data-driven engineering design workflows.

Abstract

In this work, we propose a set of physics-informed geometric operators (GOs) to enrich the geometric data provided for training surrogate/discriminative models, dimension reduction, and generative models, typically employed for performance prediction, dimension reduction, and creating data-driven parameterisations, respectively. However, as both the input and output streams of these models consist of low-level shape representations, they often fail to capture shape characteristics essential for performance analyses. Therefore, the proposed GOs exploit the differential and integral properties of shapes--accessed through Fourier descriptors, curvature integrals, geometric moments, and their invariants--to infuse high-level intrinsic geometric information and physics into the feature vector used for training, even when employing simple model architectures or low-level parametric descriptions. We showed that for surrogate modelling, along with the inclusion of the notion of physics, GOs enact regularisation to reduce over-fitting and enhance generalisation to new, unseen designs. Furthermore, through extensive experimentation, we demonstrate that for dimension reduction and generative models, incorporating the proposed GOs enriches the training data with compact global and local geometric features. This significantly enhances the quality of the resulting latent space, thereby facilitating the generation of valid and diverse designs. Lastly, we also show that GOs can enable learning parametric sensitivities to a great extent. Consequently, these enhancements accelerate the convergence rate of shape optimisers towards optimal solutions.

Figures (14)

• Figure 1: Illustrative layout of typical surrogate/discriminative models (SMs), dimension reduction models (DRMs), and generative models (GMs). SMs are supervised models that require both design and performance data for training, which are later used to predict the performance of new designs. DRMs, on the other hand, are unsupervised; they only need the original designs to construct a lower-dimensional latent space. GMs are also unsupervised models where both input and output are designs. They are trained using a set of existing designs, adopting a probabilistic approach that allows the creation of new and unseen designs adhering to the learned patterns.
• Figure 2: Illustration of various geometric characteristics captured with geometric moments-, curvature integral- and Fourier descriptor-based geometric operators.
• Figure 3: (a) The 3D surface model, (b) its parameterisation, and (c) the resulting surface patches used in numerical evaluations.
• Figure 4: (a) Parameterisation of aerofoil profiles. (b) Randomly selected samples from the UIUC aerofoil database.
• Figure 5: Plots of $R^2$, MAPE, and RMSE for various combinations of GOs for the optimised NNs and GPRs trained for aerofoils.