An adaptive Deep Ritz framework for second-order fully nonlinear partial differential equations

TL;DR

This paper introduces an adaptive Deep Ritz framework employing a splitting method and neural networks to efficiently solve fully nonlinear second-order PDEs, including Monge-Ampère and optimal transport problems.

Contribution

It presents a novel coupling of a Deep Ritz neural network approach with a splitting method and adaptive sampling for solving complex nonlinear PDEs.

Findings

Successfully solves Dirichlet problems for Monge-Ampère equations.

Demonstrates the method's flexibility and efficiency compared to PINNs.

Addresses optimal transport Monge-Ampère problems with boundary conditions.

Abstract

As an alternative to PINNs, a Deep Ritz framework is proposed to solve fully nonlinear PDEs. A least-squares algorithm is advocated to decouple the nonlinearities from the variational features of several fully nonlinear PDEs. A splitting method allows to iteratively solve local nonlinear problems and linear variational problems at each iteration. While existing nonlinear solvers are applied to solve for nonlinearities, we propose a novel coupling with a Deep Ritz neural network approach that is well-suited to the variational flavor of the linear variational problems. An adaptive sampling strategy for the selection of collocation points is incorporated to increase the efficiency of the algorithm without sacrificing its accuracy. Numerical experiments are presented to solve the Dirichlet problem for several fully nonlinear equations, starting with the prototypical Monge-Amp\`ere equation, showing the flexibility of the approach. Numerical results are compared with results obtained using a full PINNs approach. Finally, numerical experiments are extended to address the optimal transport Monge-Amp\`ere problem with transport boundary conditions.