Learning Mappings in Mesh-based Simulations

2025-06-14

AI Summary

Modeling node-wise mappings on irregular point clouds over complex geometric domains remains challenging due to the absence of inherent structural regularity. To address this, we propose a parameter-free grid encoding method that projects point clouds onto regular lattices, yielding structured representations amenable to efficient end-to-end mapping learning and enabling full-response reconstruction from partial observations. Our approach integrates grid-footprint aggregation encoding, a lightweight E-UNet architecture, and an FFT-enhanced module. Evaluated across diverse 2D/3D physical simulation tasks, it achieves significant improvements in prediction accuracy, data efficiency, and robustness to noise. Compared with Fourier neural operators and Transformer-based baselines, our framework is notably lightweight, broadly generalizable, and deployment-friendly. It establishes a scalable, grid-aware modeling paradigm for computational science—bridging unstructured geometry and structured computation without architectural or parametric constraints.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Architectures and workflows that use LLMs for crowd work

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.

Problem

• Learning mappings in geometrically complex mesh domains
• Handling irregular mesh point clouds with limited tractability
• Recovering full point cloud responses from partial observations

Innovation

• Parameter-free encoding scheme for mesh nodes
• Grid representations enabling CNN and FFT operations
• E-UNet integrates encoder for diverse mapping tasks