A homotopy continuation method that starts training at a large PDE parameter and tracks the solution to small values improves neural network solvers for singularly perturbed problems.
A variational analysis for the moving finite element method for gradient flows
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abstract
By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.
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2025 1verdicts
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Learn Singularly Perturbed Solutions via Homotopy Dynamics
A homotopy continuation method that starts training at a large PDE parameter and tracks the solution to small values improves neural network solvers for singularly perturbed problems.