An Iterative Deep Ritz Method for Monotone Elliptic Problems
This work addresses the problem of solving elliptic PDEs for researchers in computational mathematics, offering an incremental improvement over prior neural network-based methods.
The paper introduces an iterative deep Ritz method to solve monotone elliptic problems, improving accuracy over existing neural PDE solvers like physics-informed neural networks and the deep Ritz method, with demonstrated effectiveness in challenging examples.
In this work, we present a novel iterative deep Ritz method (IDRM) for solving a general class of elliptic problems. It is inspired by the iterative procedure for minimizing the loss during the training of the neural network, but at each step encodes the geometry of the underlying function space and incorporates a convex penalty to enhance the performance of the algorithm. The algorithm is applicable to elliptic problems involving a monotone operator (not necessarily of variational form) and does not impose any stringent regularity assumption on the solution. It improves several existing neural PDE solvers, e.g., physics informed neural network and deep Ritz method, in terms of the accuracy for the concerned class of elliptic problems. Further, we establish a convergence rate for the method using tools from geometry of Banach spaces and theory of monotone operators, and also analyze the learning error. To illustrate the effectiveness of the method, we present several challenging examples, including a comparative study with existing techniques.