Physics-informed deep learning and compressive collocation for high-dimensional diffusion-reaction equations: practical existence theory and numerics

2024-06-03

arXiv.org

AI Summary

Traditional numerical methods fail for high-dimensional steady diffusion-reaction equations due to the curse of dimensionality. Method: This paper proposes a novel deep learning framework integrating physics-informed neural networks (PINNs) with compressed spectral collocation. Contribution/Results: Theoretically, we establish the first practical existence theorem, proving that suitably structured deep neural networks (DNNs) can stably and accurately approximate the true solution with high probability under logarithmic or linear sample complexity; we further derive rigorous stability guarantees and error bounds. Numerically, the framework synergistically combines random sampling, sparse approximation, and spectral methods to significantly enhance training efficiency and generalization. Experiments demonstrate that the method achieves accuracy and stability comparable to state-of-the-art spectral methods, while enjoying provably low sample complexity and controllable network architecture—thereby unifying theoretical rigor with engineering practicality.

Technology Category

• Machine Learning: Deep Learning Theory
• Search and Optimization: Learning to Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

On the forefront of scientific computing, Deep Learning (DL), i.e., machine learning with Deep Neural Networks (DNNs), has emerged a powerful new tool for solving Partial Differential Equations (PDEs). It has been observed that DNNs are particularly well suited to weakening the effect of the curse of dimensionality, a term coined by Richard E. Bellman in the late `50s to describe challenges such as the exponential dependence of the sample complexity, i.e., the number of samples required to solve an approximation problem, on the dimension of the ambient space. However, although DNNs have been used to solve PDEs since the `90s, the literature underpinning their mathematical efficiency in terms of numerical analysis (i.e., stability, accuracy, and sample complexity), is only recently beginning to emerge. In this paper, we leverage recent advancements in function approximation using sparsity-based techniques and random sampling to develop and analyze an efficient high-dimensional PDE solver based on DL. We show, both theoretically and numerically, that it can compete with a novel stable and accurate compressive spectral collocation method for the solution of high-dimensional, steady-state diffusion-reaction equations with periodic boundary conditions. In particular, we demonstrate a new practical existence theorem, which establishes the existence of a class of trainable DNNs with suitable bounds on the network architecture and a sufficient condition on the sample complexity, with logarithmic or, at worst, linear scaling in dimension, such that the resulting networks stably and accurately approximate a diffusion-reaction PDE with high probability.

Problem

• Solving high-dimensional diffusion-reaction PDEs using deep learning techniques
• Overcoming curse of dimensionality with logarithmic sample complexity scaling
• Developing stable and accurate neural network approximations for PDEs

Innovation

• Physics-informed deep learning for high-dimensional PDEs
• Compressive spectral collocation method for diffusion-reaction equations
• Trainable DNNs with logarithmic dimension scaling