Learning Mappings in Mesh-based Simulations

Abstract

Many real-world physics and engineering problems arise in geometrically complex domains discretized by meshes for numerical simulations. The nodes of these potentially irregular meshes naturally form point clouds whose limited tractability poses significant challenges for learning mappings via machine learning models. To address this, we introduce a novel and parameter-free encoding scheme that aggregates footprints of points onto grid vertices and yields information-rich grid representations of the topology. Such structured representations are well-suited for standard convolution and FFT (Fast Fourier Transform) operations and enable efficient learning of mappings between encoded input-output pairs using Convolutional Neural Networks (CNNs). Specifically, we integrate our encoder with a uniquely designed UNet (E-UNet) and benchmark its performance against Fourier- and transformer-based models across diverse 2D and 3D problems where we analyze the performance in terms of predictive accuracy, data efficiency, and noise robustness. Furthermore, we highlight the versatility of our encoding scheme in various mapping tasks including recovering full point cloud responses from partial observations. Our proposed framework offers a practical alternative to both primitive and computationally intensive encoding schemes; supporting broad adoption in computational science applications involving mesh-based simulations.