A neural network method using the Rayleigh quotient with Gram-Schmidt orthogonalization solves differential eigenvalue problems in order, including parametric, nonlinear, and high-dimensional cases.
Physics Informed Neural Networks for heat conduction with phase change
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abstract
We study numerical algorithms to solve a specific Partial Differential Equation (PDE), namely the Stefan problem, using Physics Informed Neural Networks (PINNs). This problem describes the heat propagation in a liquid-solid phase change system. It implies a heat equation and a discontinuity at the interface where the phase change occurs. In the context of PINNs, this model leads to difficulties in the learning process, especially near the interface of phase change. We present different strategies that can be used in this context. We illustrate our results and compare with classical solvers for PDEs (finite differences).
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Solving engineering eigenvalue problems with neural networks using the Rayleigh quotient
A neural network method using the Rayleigh quotient with Gram-Schmidt orthogonalization solves differential eigenvalue problems in order, including parametric, nonlinear, and high-dimensional cases.