Neural network guided adjoint computations in dual weighted residual error estimation

Abstract

In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint problem is solved in strong form with a feedforward neural network using two or three hidden layers. The main objective of our approach is to explore alternatives for solving the adjoint problem with greater potential of a numerical cost reduction. The proposed algorithm is based on the general goal-oriented error estimation theorem including both linear and nonlinear stationary partial differential equations and goal functionals. Our developments are substantiated with some numerical experiments that include comparisons of neural network computed adjoints and classical finite element solutions of the adjoints. In the programming software, the open-source library deal.II is successfully coupled with LibTorch, the PyTorch C++ application programming interface. Article Highlights Adjoint approximation with feedforward neural network in dual-weighted residual error estimation. Side-by-side comparisons for accuracy and computational cost with classical finite element computations. Numerical experiments for linear and nonlinear problems yielding excellent effectivity indices.

Neural control of discrete weak formulations: Galerkin, least squares & minimal-residual methods with quasi-optimal weights

“…For those related to physics-informed neural networks (PINN) and least squares methods; see Sirignano & Spiliopoulos [37, Section 7], Mishra & Molinaro [21,50], Pousin [18, Section 4], Cai, Chen & Liu [22], and Berrone, Canuto & Pintore [51]. Recently, a posteriori error analysis has also been studied, in particular goal-oriented analysis using the dual-weighted residual (DWR) methodology; see, e.g., Roth, Schröder & Wick [52], and Minakowski & Richter [53]. We note that in our current work, while we have in mind the error analysis for neural-control approximations, the abstract analysis presented in Section 2 is essentially an extension of the above-mentioned a priori analysis to a certain class of problems involving a convex and differentiable cost functional.…”

E2N: Error Estimation Networks for Goal-Oriented Mesh Adaptation

“…These include the test function choice [7], forward solve [8], adjoint solve [9], derivative recovery procedure [10], error estimation [11], metric/monitor function/sizing field construction step [12,13,14], and the entire mesh adaptation loop [6,15,16].…”

“…A similar 'focused' approach is also used in [9], which emulates the adjoint solve procedure. This is done on the base mesh and the data-driven adjoint solution is projected into an enriched space, where error indicators are assembled and thereby used to drive mesh adaptation.…”