A coarse space acceleration of deep-DDM

Abstract

The use of deep learning methods for solving PDEs is a field in full expansion. In particular, Physical Informed Neural Networks, that implement a sampling of the physical domain and use a loss function that penalizes the violation of the partial differential equation, have shown their great potential. Yet, to address large scale problems encountered in real applications and compete with existing numerical methods for PDEs, it is important to design parallel algorithms with good scalability properties. In the vein of traditional domain decomposition methods (DDM), we consider the recently proposed deep-ddm approach. We present an extension of this method that relies on the use of a coarse space correction, similarly to what is done in traditional DDM solvers. Our investigations shows that the coarse correction is able to alleviate the deterioration of the convergence of the solver when the number of subdomains is increased thanks to an instantaneous information exchange between subdomains at each iteration. Experimental results demonstrate that our approach induces a remarkable acceleration of the original deep-ddm method, at a reduced additional computational cost.

Domain Decomposition Learning Methods for Solving Elliptic Problems

“…Robin-Robin Algorithm via Solution-Oriented Learning Methods. Note that in addition to the deep learning analogue of overlapping Schwarz methods [29,35,30,42,44], the non-overlapping Robin-Robin algorithm [45,38] is also based on a direct exchange of solution value between neighbouring subdomains (see Figure 1 or algorithm 2.1). Moreover, as the decomposition leads to simpler and smoother functions to be learned on each subregion, it thus enables us to employ the standard solution-oriented learning methods [39,47,48,43] for the numerical solution of local problems.…”

“…Learning Methods interface exchange solution interface exchange solution and flux [ 29,30,35], namely,…”

“…On the other hand, as the classical domain decomposition methods [45] can be formulated at the continuous or the weak level, various works have been devoted to employing the learning approaches for solving the decomposed subproblems and therefore benefit from the mesh free nature of deep learning solvers. Under such circumstances, the machine learning analogue of the overlapping domain decomposition methods have emerged recently and have successfully handled many problems [32,29,35,42], however, the more general non-overlapping counterpart has not been thoroughly studied yet. A major difficulty is that the network solutions of local problems are prone to overfitting at and near the interface [9,1] since the interface conditions are enforced as soft penalty functions during training and the size of training data on interface is smaller than that of interior domains, which eventually propagates the errors to neighboring subdomains and hampers the convergence of outer iteration.…”

PFNN-2: A Domain Decomposed Penalty-Free Neural Network Method for Solving Partial Differential Equations

“…Inspired by the idea of classical domain decomposition methods [53], it was proposed to introduce domain decomposition strategies into neural network based PDE solvers [54][55][56][57][58][59][60][61][62] by dividing the learning task into training a series of sub-networks related to solutions on sub-domains. This tends to be more natural than the plain distributed training approaches because by introducing domain decomposition the training of each sub-network only requires a small part of dataset related to the corresponding sub-domain, therefore can significantly decrease the computational cost.…”

Learning Interface Conditions in Domain Decomposition Solvers

“…Using the continuous formulation of DDM, the so-called D3M [16] uses a variational deep learning solver, implementing local neural networks on physical subdomains in a parallel fashion. Likewise, Deep-DDM [17] utilizes PINNs to discretize and solve DDM problems, with coarse space corrections [18] being used to improve scalability.…”