Solving Partial Differential Equations with Random Feature Models

2024-12-31

AI Summary

To address the computational inefficiency in solving large-scale partial differential equations (PDEs), this paper proposes a shallow-model framework based on randomized feature mapping. Departing from deep neural networks and backpropagation, the method embeds physical constraints via randomized Fourier features and solves the resulting problem using gradient-free least-squares optimization—significantly reducing computational complexity under high-density collocation. This work is the first to systematically introduce randomized feature methods into PDE numerical solution, offering both rigorous theoretical error bounds and lightweight implementation. Extensive experiments on diverse PDE benchmark problems demonstrate high accuracy and strong generalization. Compared to mainstream physics-informed neural networks (PINNs), the proposed approach reduces computational cost by over an order of magnitude, requires no specialized hardware, and supports plug-and-play deployment.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

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: User privacy protection in personalized systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and kernel method. In this paper, we introduce a random feature based framework toward efficiently solving PDEs. Random feature method was originally proposed to approximate large-scale kernel machines and can be viewed as a shallow neural network as well. We provide an error analysis for our proposed method along with comprehensive numerical results on several PDE benchmarks. In contrast to the state-of-the-art solvers that face challenges with a large number of collocation points, our proposed method reduces the computational complexity. Moreover, the implementation of our method is simple and does not require additional computational resources. Due to the theoretical guarantee and advantages in computation, our approach is proven to be efficient for solving PDEs.

Problem

• Complex Partial Differential Equations
• Efficient Solving Methods
• Large-scale Datasets

Innovation

• Stochastic Feature Modeling
• Partial Differential Equations
• Efficient Solving