Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Method
This work addresses the need for rigorous theoretical guarantees in applying overparameterized neural networks to PDE solving, offering practical guidance for parameter settings without requiring additional assumptions on solutions, though it is incremental as it builds on existing deep Ritz method frameworks.
The authors tackled the problem of solving second-order elliptic PDEs with various boundary conditions using a three-layer tanh neural network trained with projected gradient descent within the deep Ritz method, establishing global convergence and providing a comprehensive error analysis that includes approximation, generalization, and optimization errors with bounds in terms of sample size.
Machine learning is a rapidly advancing field with diverse applications across various domains. One prominent area of research is the utilization of deep learning techniques for solving partial differential equations(PDEs). In this work, we specifically focus on employing a three-layer tanh neural network within the framework of the deep Ritz method(DRM) to solve second-order elliptic equations with three different types of boundary conditions. We perform projected gradient descent(PDG) to train the three-layer network and we establish its global convergence. To the best of our knowledge, we are the first to provide a comprehensive error analysis of using overparameterized networks to solve PDE problems, as our analysis simultaneously includes estimates for approximation error, generalization error, and optimization error. We present error bound in terms of the sample size $n$ and our work provides guidance on how to set the network depth, width, step size, and number of iterations for the projected gradient descent algorithm. Importantly, our assumptions in this work are classical and we do not require any additional assumptions on the solution of the equation. This ensures the broad applicability and generality of our results.