# Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descent

@article{Lu2022SobolevAA, title={Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descent}, author={Yiping Lu and Jos{\'e} H. Blanchet and Lexing Ying}, journal={ArXiv}, year={2022}, volume={abs/2205.07331} }

In this paper, we study the statistical limits in terms of Sobolev norms of gradient descent for solving inverse problem from randomly sampled noisy observations using a general class of objective functions. Our class of objective functions includes Sobolev training for kernel regression, Deep Ritz Methods (DRM), and Physics Informed Neural Networks (PINN) for solving elliptic partial diﬀerential equations (PDEs) as special cases. We consider a potentially inﬁnite-dimensional parameterization…

## One Citation

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This paper establishes the information-theoretic lower bound in terms of the Sobolev Hilbert-Schmidt norm and shows that a regularization that learns the spectral components below the bias contour and ignores the ones that above the variance contour can achieve optimal learning rate.

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