Randomized Neural Networks with Petrov–Galerkin Methods for Solving Linear Elasticity and Navier–Stokes Equations

We develop randomized neural networks (RNNs) with Petrov–Galerkin (RNN-PG) methods to solve linear elasticity and Navier–Stokes equations. RNN-PGs use the Petrov–Galerkin variational framework, where the solution is approximated by randomized neural networks and the test functions are piecewise polynomials. Unlike conventional neural networks, the parameters of the hidden layers of randomized neural networks are fixed randomly while the parameters of the output layer are determined by the least-squares method, which can effectively approximate the solution. We also develop mixed RNN-PG (M-RNN-PG) methods for linear elasticity problems to ensure symmetry of the stress tensor and avoid locking effects. For the Stokes problem, we present various M-RNN-PG methods that enforce the divergence-free constraint by different techniques. For the Navier–Stokes equations, we propose a space-time M-RNN-PG that uses Picard or Newton iteration to deal with the nonlinear term. Using several examples, we compare RNN-PG methods with the finite-element method, the mixed discontinuous Galerkin method, and the physics-informed neural network. The numerical results demonstrate that RNN-PG methods achieve higher accuracy and efficiency.