Semi-Supervised Deep Sobolev Regression: Estimation, Variable Selection and Beyond

We propose SDORE, a semi-supervised deep Sobolev regressor, for the nonparametric estimation of the underlying regression function and its gradient. SDORE employs deep neural networks to minimize empirical risk with gradient norm regularization, allowing computation of the gradient norm on unlabeled data. We conduct a comprehensive analysis of the convergence rates of SDORE and establish a minimax optimal rate for the regression function. Crucially, we also derive a convergence rate for the associated plug-in gradient estimator, even in the presence of significant domain shift. These theoretical findings offer valuable prior guidance for selecting regularization parameters and determining the size of the neural network, while showcasing the provable advantage of leveraging unlabeled data in semi-supervised learning. To the best of our knowledge, SDORE is the first provable neural network-based approach that simultaneously estimates the regression function and its gradient, with diverse applications including nonparametric variable selection and inverse problems. The effectiveness of SDORE is validated through an extensive range of numerical simulations and real data analysis.

Current density impedance imaging with PINNs

In this paper, we introduce CDII-PINNs, a computationally efficient method for solving CDII using PINNs in the framework of Tikhonov regularization. This method constructs a physics-informed loss function by merging the regularized least-squares output functional with an underlying differential equation, which describes the relationship between the conductivity and voltage. A pair of neural networks representing the conductivity and voltage, respectively, are coupled by this loss function. Then, minimizing the loss function provides a reconstruction. A rigorous theoretical guarantee is provided. We give an error analysis for CDII-PINNs and establish a convergence rate, based on prior selected neural network parameters in terms of the number of samples. The numerical simulations demonstrate that CDII-PINNs are efficient, accurate and robust to noise levels ranging from $1\%$ to $20\%$.