Low-fidelity models speed neural solving of complex eigenvalue problems
Low-Fidelity FDM Spectral Guidance for Neural Eigenvalue Solvers
Machine Learning
Summary
Finding special numbers called eigenvalues in complex systems is crucial in science but can be very slow and use lots of computer memory. The authors show that combining a rough, cheap method to estimate those numbers with a neural network that learns to find them more precisely can make the process much faster and more accurate. They also introduce a new way to train the neural network that works better for problems with many dimensions. Their combined approach solves test problems more efficiently than previous neural methods.
What this means in practice
- •For computational physicists: Accelerate solving high-dimensional eigenvalue problems in simulations by combining coarse numerical models with neural networks.
- •For engineering simulation teams: Use stabilized neural network training aided by low-fidelity guidance to improve accuracy when modeling complex systems involving eigenvalue calculations.
Authors
Aryan Chaudhary, Manikandan Padmanaban, Jagabondhu Hazra
Abstract
Operator eigenvalue problems appear throughout science. Classical methods usually discretize the operator into a matrix and then solve the resulting matrix eigenvalue problem. This works well in low dimensions, but fine grids quickly become expensive in both memory and computation as the dimension grows. Neural network based solvers avoid storing these large grids, but recent state of the art neural methods can require hundreds of thousands of training steps and may struggle to find the desired eigenvalues. We show that the two approaches can help each other. A coarse finite difference method (FDM) calculation acts as a cheap numerical model of the operator spectrum. We use the approximate eigenvalues as fixed shifts during the training of the neural solver, as they only need to locate the relevant part of the spectrum. We also introduce Stabilized Inverse Power Method Neural Network (SIPMNN), a more stable training procedure for higher-dimensional problems. Across five test problems at $d=10$, the combined approach is more accurate overall than the tested fully neural alternatives while using eight to ten times fewer iterations.