DRM Revisited: A Complete Error Analysis
This work addresses a foundational theoretical gap for researchers in numerical PDEs and machine learning, offering incremental improvements in error bounds.
The authors tackled the problem of determining the necessary training samples, neural network architecture, step size, and iterations for the Deep Ritz Method to approximate PDE solutions with a target precision, providing a complete error analysis under over-parameterization.
In this work, we address a foundational question in the theoretical analysis of the Deep Ritz Method (DRM) under the over-parameteriztion regime: Given a target precision level, how can one determine the appropriate number of training samples, the key architectural parameters of the neural networks, the step size for the projected gradient descent optimization procedure, and the requisite number of iterations, such that the output of the gradient descent process closely approximates the true solution of the underlying partial differential equation to the specified precision?