Convolutional Hierarchical Deep Learning Neural Networks-Tensor Decomposition (C-HiDeNN-TD): a scalable surrogate modeling approach for large-scale physical systems

2024-08-31

arXiv.org

AI Summary

Traditional numerical methods for ultra-large-scale physical system simulation suffer from high computational cost and memory overhead, while data-driven surrogate models critically depend on large volumes of labeled training data—constituting a dual bottleneck. To address this, we propose a PDE-driven surrogate modeling framework that requires no offline training data. Our method establishes an end-to-end differentiable solver, innovatively integrating convolutional hierarchical deep networks with tensor decomposition to directly parameterize and solve spatiotemporal partial differential equations, thereby unifying physical consistency with high-fidelity modeling. Compared to classical numerical solvers, it achieves significant reductions in both computation time and memory usage while preserving accuracy. Unlike data-driven approaches, it eliminates reliance on labeled datasets entirely. Experimental results demonstrate strong scalability, enabling real-time or near-real-time simulation of systems with up to hundreds of millions of degrees of freedom.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

A common trend in simulation-driven engineering applications is the ever-increasing size and complexity of the problem, where classical numerical methods typically suffer from significant computational time and huge memory cost. Methods based on artificial intelligence have been extensively investigated to accelerate partial differential equations (PDE) solvers using data-driven surrogates. However, most data-driven surrogates require an extremely large amount of training data. In this paper, we propose the Convolutional Hierarchical Deep Learning Neural Network-Tensor Decomposition (C-HiDeNN-TD) method, which can directly obtain surrogate models by solving large-scale space-time PDE without generating any offline training data. We compare the performance of the proposed method against classical numerical methods for extremely large-scale systems.

Problem

• Solving large-scale space-time PDEs without training data
• Reducing computational time and memory costs for simulations
• Providing scalable surrogate models for complex physical systems

Innovation

• Convolutional Hierarchical Deep Learning Neural Networks-Tensor Decomposition
• Directly obtains surrogate models without offline training data
• Solves large-scale space-time partial differential equations