Interpolation Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems

2025-03-03

AI Summary

To address the high computational cost, low accuracy, and poor interpretability of deep learning models in large-scale, high-dimensional partial differential equation (PDE) modeling, this paper proposes a sparse, interpretable neural network architecture that integrates finite element interpolation functions with tensor decomposition (CP/Tucker). The method innovatively embeds locally supported finite element basis functions into the network structure, achieving organic unification of machine learning and classical numerical methods. Tensor decomposition enables parameter-efficient sparsification, significantly reducing memory footprint and training overhead. The resulting framework delivers high accuracy, strong scalability, and physical interpretability for both forward and inverse problems involving high-dimensional parametric PDEs. It is particularly suited for industrial-grade physics-informed simulation and optimization tasks.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Deep learning has been extensively employed as a powerful function approximator for modeling physics-based problems described by partial differential equations (PDEs). Despite their popularity, standard deep learning models often demand prohibitively large computational resources and yield limited accuracy when scaling to large-scale, high-dimensional physical problems. Their black-box nature further hinders the application in industrial problems where interpretability and high precision are critical. To overcome these challenges, this paper introduces Interpolation Neural Network-Tensor Decomposition (INN-TD), a scalable and interpretable framework that has the merits of both machine learning and finite element methods for modeling large-scale physical systems. By integrating locally supported interpolation functions from finite element into the network architecture, INN-TD achieves a sparse learning structure with enhanced accuracy, faster training/solving speed, and reduced memory footprint. This makes it particularly effective for tackling large-scale high-dimensional parametric PDEs in training, solving, and inverse optimization tasks in physical problems where high precision is required.

Problem

• Addresses computational inefficiency in large-scale physics-based problems.
• Enhances accuracy and interpretability in high-dimensional PDE modeling.
• Reduces memory usage and speeds up training for physical systems.

Innovation

• Integrates interpolation functions into neural networks
• Enhances accuracy and reduces memory usage
• Effective for high-dimensional parametric PDEs