A Hybrid Kernel-Free Boundary Integral Method with Operator Learning for Solving Parametric Partial Differential Equations In Complex Domains

The Kernel-Free Boundary Integral (KFBI) method presents an iterative solution to boundary integral equations arising from elliptic partial differential equations (PDEs). This method effectively addresses elliptic PDEs on irregular domains, including the modified Helmholtz, Stokes, and elasticity equations. The rapid evolution of neural networks and deep learning has invigorated the exploration of numerical PDEs. An increasing interest is observed in deep learning approaches that seamlessly integrate mathematical principles for investigating numerical PDEs. We propose a hybrid KFBI method, integrating the foundational principles of the KFBI method with the capabilities of deep learning. This approach, within the framework of the boundary integral method, designs a network to approximate the solution operator for the corresponding integral equations by mapping the parameters, inhomogeneous terms and boundary information of PDEs to the boundary density functions, which can be regarded as the solution of the integral equations. The models are trained using data generated by the Cartesian grid-based KFBI algorithm, exhibiting robust generalization capabilities. It accurately predicts density functions across diverse boundary conditions and parameters within the same class of equations. Experimental results demonstrate that the trained model can directly infer the boundary density function with satisfactory precision, obviating the need for iterative steps in solving boundary integral equations. Furthermore, applying the inference results of the model as initial values for iterations is also reasonable; this approach can retain the inherent second-order accuracy of the KFBI method while accelerating the traditional KFBI approach by reducing about 50% iterations.

Auto-ICell: An Accessible and Cost-Effective Integrative Droplet Microfluidic System for Real-Time Single-Cell Morphological and Apoptotic Analysis

The Auto-ICell system, a novel, and cost-effective integrated droplet microfluidic system, is introduced for real-time analysis of single-cell morphology and apoptosis. This system integrates a 3D-printed microfluidic chip with image analysis algorithms, enabling the generation of uniform droplet reactors and immediate image analysis. The system employs a color-based image analysis algorithm in the bright field for droplet content analysis. Meanwhile, in the fluorescence field, cell apoptosis is quantitatively measured through a combination of deep-learning-enabled multiple fluorescent channel analysis and a live/dead cell stain kit. Breast cancer cells are encapsulated within uniform droplets, with diameters ranging from 70 {\mu}m to 240 {\mu}m, generated at a high throughput of 1,500 droplets per minute. Real-time image analysis results are displayed within 2 seconds on a custom graphical user interface (GUI). The system provides an automatic calculation of the distribution and ratio of encapsulated dyes in the bright field, and in the fluorescent field, cell blebbing and cell circularity are observed and quantified respectively. The Auto-ICell system is non-invasive and provides online detection, offering a robust, time-efficient, user-friendly, and cost-effective solution for single-cell analysis. It significantly enhances the detection throughput of droplet single-cell analysis by reducing setup costs and improving operational performance. This study highlights the potential of the Auto-ICell system in advancing biological research and personalized disease treatment, with promising applications in cell culture, biochemical microreactors, drug carriers, cell-based assays, synthetic biology, and point-of-care diagnostics.