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Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory
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The problem of solving partial differential equations (PDEs) can be formulated into a least-squares minimization problem, where neural networks are used to parametrize PDE solutions. A global minimizer corresponds to a neural network that solves the given PDE. In this paper, we show that the gradient descent method can identify a global minimizer of the least-squares optimization for solving second-order linear PDEs with two-layer neural networks under the assumption of over-parametrization. We also analyze the generalization error of the least-squares optimization for second-order linear PDEs and two-layer neural networks, when the right-hand-side function of the PDE is in a Barron-type space and the least-squares optimization is regularized with a Barron-type norm, without the over-parametrization assumption.
Forward citations
Cited by 4 Pith papers
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Optimization and generalization analysis for two-layer physics-informed neural networks without over-parametrization
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LySep separates the layers and derivatives of a PINN into auxiliary variables, yielding a shallow, easier-to-optimize loss that remains provably consistent with the original PINN loss.
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Approximation Rates in Fr\'echet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliers
Two theorems give sufficient shallow-network width to reach a prescribed error in a Fréchet metric of semi-norms, applied to exponential spectral Barron, Gelfand-Shilov, and bandlimited (Paley-Wiener type) symbol classes.
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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.
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