This study reveals the inherent tolerance of contrastive learning (CL) towards sampling bias, wherein negative samples may encompass similar semantics (\eg labels). However, existing theories fall short in providing explanations for this phenomenon. We bridge this research gap by analyzing CL through the lens of distributionally robust optimization (DRO), yielding several key insights: (1) CL essentially conducts DRO over the negative sampling distribution, thus enabling robust performance across a variety of potential distributions and demonstrating robustness to sampling bias; (2) The design of the temperature $\tau$ is not merely heuristic but acts as a Lagrange Coefficient, regulating the size of the potential distribution set; (3) A theoretical connection is established between DRO and mutual information, thus presenting fresh evidence for ``InfoNCE as an estimate of MI'' and a new estimation approach for $\phi$-divergence-based generalized mutual information. We also identify CL's potential shortcomings, including over-conservatism and sensitivity to outliers, and introduce a novel Adjusted InfoNCE loss (ADNCE) to mitigate these issues.

This study reveals the inherent tolerance of contrastive learning (CL) towards sampling bias, wherein negative samples may encompass similar semantics (\eg labels). However, existing theories fall short in providing explanations for this phenomenon. We bridge this research gap by analyzing CL through the lens of distributionally robust optimization (DRO), yielding several key insights: (1) CL essentially conducts DRO over the negative sampling distribution, thus enabling robust performance across a variety of potential distributions and demonstrating robustness to sampling bias; (2) The design of the temperature \tau is not merely heuristic but acts as a Lagrange Coefficient, regulating the size of the potential distribution set; (3) A theoretical connection is established between DRO and mutual information, thus presenting fresh evidence for InfoNCE as an estimate of MI'' and a new estimation approach for \phi -divergence-based generalized mutual information. We also identify CL's potential shortcomings, including over-conservatism and sensitivity to outliers, and introduce a novel Adjusted InfoNCE loss (ADNCE) to mitigate these issues.

This paper introduces a hybrid learning framework that combines convolutional neural networks (CNNs) and physics-informed neural networks (PINNs) to address the challenging problem of full-inverse electrical impedance tomography (EIT). EIT is a noninvasive imaging technique that reconstructs the spatial distribution of internal conductivity based on boundary voltage measurements from injected currents. This method has applications across medical imaging, multiphase flow detection, and tactile sensing. However, solving EIT involves a nonlinear partial differential equation (PDE) derived from Maxwell's equations, posing significant computational challenges as an ill-posed inverse problem. Existing PINN approaches primarily address semi-inverse EIT, assuming full access to internal potential data, which limits practical applications in realistic, full-inverse scenarios. Our framework employs a forward CNN-based supervised network to map differential boundary voltage measurements to a discrete potential distribution under fixed Neumann boundary conditions, while an inverse PINN-based unsupervised network enforces PDE constraints for conductivity reconstruction. Instead of traditional automatic differentiation, we introduce discrete numerical differentiation to bridge the forward and inverse networks, effectively decoupling them, enhancing modularity, and reducing computational demands. We validate our framework under realistic conditions, using a 16-electrode setup and rigorous testing on complex conductivity distributions with sharp boundaries, without Gaussian smoothing. This approach demonstrates robust flexibility and improved applicability in full-inverse EIT, establishing a practical solution for real-world imaging challenges.

Physics-Informed Neural Networks (PINNs) are a machine learning technique for solving partial differential equations (PDEs) by incorporating PDEs as loss terms in neural networks and minimizing the loss function during training. Tomographic imaging, a method to reconstruct internal properties from external measurement data, is highly complex and ill-posed, making it an inverse problem. Recently, PINNs have shown significant potential in computational fluid dynamics (CFD) and have advantages in solving inverse problems. However, existing research has primarily focused on semi-inverse Electrical Impedance Tomography (EIT), where internal electric potentials are accessible. The practical full inverse EIT problem, where only boundary voltage measurements are available, remains challenging. To address this, we propose a two-stage hybrid learning framework combining Convolutional Neural Networks (CNNs) and PINNs to solve the full inverse EIT problem. This framework integrates data-driven and model-driven approaches, combines supervised and unsupervised learning, and decouples the forward and inverse problems within the PINN framework in EIT. Stage I: a U-Net constructs an end-to-end mapping from boundary voltage measurements to the internal potential distribution using supervised learning. Stage II: a Multilayer Perceptron (MLP)-based PINN takes the predicted internal potentials as input to solve for the conductivity distribution through unsupervised learning.

Simulating the dynamic characteristics of a PN junction at the microscopic level requires solving the Poisson's equation at every time step. Solving at every time step is a necessary but time-consuming process when using the traditional finite difference (FDM) approach. Deep learning is a powerful technique to fit complex functions. In this work, deep learning is utilized to accelerate solving Poisson's equation in a PN junction. The role of the boundary condition is emphasized in the loss function to ensure a better fitting. The resulting I-V curve for the PN junction, using the deep learning solver presented in this work, shows a perfect match to the I-V curve obtained using the finite difference method, with the advantage of being 10 times faster at every time step.