SuperAdjoint: Super-Resolution Neural Networks in Adjoint-based Output Error Estimation

Abstract

Numerical simulations and optimisation methods, such as mesh adaptation, rely on the accurate and inexpensive use of error estimation methods. Adjoint-based error estimation is the most accurate method, and generally the most costly. A strong contributor to this cost is the need to compute a higher resolution adjoint solution, using time dependent information. Here, recontruction methods applied to the primal and adjoint solutions are proposed to alleviate both the storage footprint of the primal problem and the adjoint computational cost. The method is compared to reference error estimators on an unsteady Burgers’ equation using the method of manufactured solutions. Two reconstruction methods, a proper orthogonal decomposition and a static convolutional neural network were used to demonstrate both the computational cost reduction and the potential for the reduction of the storage footprint of the primal problem. When reconstruction methods are applied to the primal problem, one can use both proposed approaches to reduce the footprint of the solution and reconstruct the effectively compressed primal solution to be recalled for the adjoint solution. The second approach consists in solving an adjoint solution from a coarse primal solution and using reconstruction methods to obtain a higher resolution adjoint solution, necessary for output error estimation and mesh adaptation. The obtained results give great confidence in the use of reconstruction methods for the reduction of both computational cost and storage requirements of adjoint-based error estimation, and goal-oriented mesh adaptation

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