CNNs achieve dimension-dependent Sobolev approximation rates on manifolds, and a spectral boundary loss using Laplace-Beltrami eigenmodes enables stable PINN solutions for elliptic problems with improved accuracy over standard approaches.
On the rates of convergence for learning with convolutional neural networks.SIAM Journal on Mathematics of Data Science, 7 (4):1755–1772
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Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems
CNNs achieve dimension-dependent Sobolev approximation rates on manifolds, and a spectral boundary loss using Laplace-Beltrami eigenmodes enables stable PINN solutions for elliptic problems with improved accuracy over standard approaches.