Shape-informed surrogate models based on signed distance function domain encoding

TL;DR

The results show that the proposed method achieves accuracy comparable to the best-case scenarios where an explicit parametrization of the computational domain is available, and can even be applied in cases where geometries undergo changes in their topology.

Abstract

We propose a non-intrusive method to build surrogate models that approximate the solution of parameterized partial differential equations (PDEs), capable of taking into account the dependence of the solution on the shape of the computational domain. Our approach is based on the combination of two neural networks (NNs). The first NN, conditioned on a latent code, provides an implicit representation of geometry variability through signed distance functions. This automated shape encoding technique generates compact, low-dimensional representations of geometries within a latent space, without requiring the explicit construction of an encoder. The second NN reconstructs the output physical fields independently for each spatial point, thus avoiding the computational burden typically associated with high-dimensional discretizations like computational meshes. Furthermore, we show that accuracy in geometrical characterization can be further enhanced by employing Fourier feature mapping as input feature of the NN. The meshless nature of the proposed method, combined with the dimensionality reduction achieved through automatic feature extraction in latent space, makes it highly flexible and computationally efficient. This strategy eliminates the need for manual intervention in extracting geometric parameters, and can even be applied in cases where geometries undergo changes in their topology. Numerical tests in the field of fluid dynamics and solid mechanics demonstrate the effectiveness of the proposed method in accurately predict the solution of PDEs in domains of arbitrary shape. Remarkably, the results show that it achieves accuracy comparable to the best-case scenarios where an explicit parametrization of the computational domain is available.

Figures (19)

• Figure 1: SDF-USM-Net architecture. The network $\mathcal{NN}_\text{SDF}$ receives the input spatial coordinates $\mathbf{x}$ and the shape code $\mathbf{z}_i$ and outputs the SDF value for the corresponding point. Within an auto-decoder framework, it learns the shape code distribution in a latent space by reconstructing the continuous SDF in the spatial domain. After the training of $\mathcal{NN}_\text{SDF}$, the network $\mathcal{NN}_\text{Phys}$ is trained in a supervised manner, taking as input the shape code $\mathbf{z}_i$, spatial coordinates $\mathbf{x}$, and a distance function $DF(\mathbf{x})$ value and returning the approximation of QoI $\mathbf{u}(\mathbf{x};\Omega)$.
• Figure 2: Working principle of $\mathcal{NN}_\text{SDF}$. (a) During training, each sample geometry is associated with a randomly initialized shape code. The auto-decoder receives shape codes and coordinates as input. The shape codes are optimized along with the decoder trainable parameters $\boldsymbol{\theta}_\text{SDF}$ through standard back-propagation. (b) During inference, the trainable parameters $\boldsymbol{\theta}_\text{SDF}$ are fixed, and an optimal shape code $\mathbf{z}_i$ is estimated for any geometry by minimizing the discrepancy between predicted and expected SDF values.
• Figure 3: Test Case 1: Representation of some of the geometries $\Omega\in \mathcal{G}$ included in the training dataset in Test Case 1.
• Figure 4: Test Case 1: (a) A sample of $SDF(\mathbf{x})$ on a uniform grid. (b) A sample of ${DF}(\mathbf{x})$ on an unstructured grid.
• Figure 5: Test Case 1: geometrical landmarks.