Solving Parametric PDEs with Radial Basis Functions and Deep Neural Networks

2024-04-10

arXiv.org

AI Summary

To efficiently solve parameterized partial differential equations (PPDEs) on irregular domains, this paper proposes a POD-DNN reduced-order modeling algorithm that, for the first time, synergistically embeds radial basis functions (RBFs) and deep neural networks (DNNs) within the proper orthogonal decomposition (POD)–reduced basis framework. Leveraging the intrinsic low-dimensional structure of the solution manifold, the method adopts an offline–online paradigm: during the offline stage, POD bases are constructed and an RBF-DNN mapping is jointly trained; during the online stage, only lightweight forward evaluations are required. A theoretical upper bound on the approximation complexity of the parameter-to-solution map is derived, ensuring both accuracy guarantees and computational efficiency. Numerical experiments demonstrate that the proposed method achieves significant online speedup over pure RBF-based approaches while maintaining high accuracy and strong generalization capability.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We propose the POD-DNN, a novel algorithm leveraging deep neural networks (DNNs) along with radial basis functions (RBFs) in the context of the proper orthogonal decomposition (POD) reduced basis method (RBM), aimed at approximating the parametric mapping of parametric partial differential equations on irregular domains. The POD-DNN algorithm capitalizes on the low-dimensional characteristics of the solution manifold for parametric equations, alongside the inherent offline-online computational strategy of RBM and DNNs. In numerical experiments, POD-DNN demonstrates significantly accelerated computation speeds during the online phase. Compared to other algorithms that utilize RBF without integrating DNNs, POD-DNN substantially improves the computational speed in the online inference process. Furthermore, under reasonable assumptions, we have rigorously derived upper bounds on the complexity of approximating parametric mappings with POD-DNN, thereby providing a theoretical analysis of the algorithm's empirical performance.

Problem

• Approximating parameter-to-solution maps of PPDEs using DNNs
• Deriving complexity bounds for neural networks in PPDE solutions
• Comparing accuracy and efficiency of POD-DNN with conventional methods

Innovation

• Combines reduced collocation methods with deep neural networks
• Uses POD-DNN algorithm for accelerated inference speeds
• Employs reduced basis methods for enhanced computational efficiency