Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which remain fixed throughout the computation, and uses a linear least squares method for training the parameters of the output layer of the neural network. It is known to be much faster than Physics informed neural networks. However, classical ELM is still computationally expensive when a high level of representation is desired in the solution as this requires solving a large least squares system. In this paper, we propose a nonoverlapping domain decomposition method (DDM) for ELMs that not only reduces the training time of ELMs, but is also suitable for parallel computation. In numerical analysis, DDMs have been widely studied to reduce the time to obtain finite element solutions for elliptic PDEs through parallel computation. Among these approaches, nonoverlapping DDMs are attracting the most attention. Motivated by these methods, we introduce local neural networks, which are valid only at corresponding subdomains, and an auxiliary variable at the interface. We construct a system on the variable and the parameters of local neural networks. A Schur complement system on the interface can be derived by eliminating the parameters of the output layer. The auxiliary variable is then directly obtained by solving the reduced system after which the parameters for each local neural network are solved in parallel. A method for initializing the hidden layer parameters suitable for high approximation quality in large systems is also proposed. Numerical results that verify the acceleration performance of the proposed method with respect to the number of subdomains are presented.

In this paper, a deep learning method for solving an improved one-dimensional Poisson-Nernst-Planck ion channel (PNPic) model, called the PNPic deep learning solver, is presented. In particular, it combines a novel local neural network scheme with an effective PNPic finite element solver. Since the input data of the neural network scheme only involves a small local patch of coarse grid solutions, which the finite element solver can quickly produce, the PNPic deep learning solver can be trained much faster than any corresponding conventional global neural network solvers. After properly trained, it can output a predicted PNPic solution in a much higher degree of accuracy than the low cost coarse grid solutions and can reflect different perturbation cases on the parameters, ion channel subregions, and interface and boundary values, etc. Consequently, the PNPic deep learning solver can generate a numerical solution with high accuracy for a family of PNPic models. As an initial study, two types of numerical tests were done by perturbing one and two parameters of the PNPic model, respectively, as well as the tests done by using a few perturbed interface positions of the model as training samples. These tests demonstrate that the PNPic deep learning solver can generate highly accurate PNPic numerical solutions.

The increasing demand for on-device deep learning services calls for a highly efficient manner to deploy deep neural networks (DNNs) on mobile devices with limited capacity. The cloud-based solution is a promising approach to enabling deep learning applications on mobile devices where the large portions of a DNN are offloaded to the cloud. However, revealing data to the cloud leads to potential privacy risk. To benefit from the cloud data center without the privacy risk, we design, evaluate, and implement a cloud-based framework ARDEN which partitions the DNN across mobile devices and cloud data centers. A simple data transformation is performed on the mobile device, while the resource-hungry training and the complex inference rely on the cloud data center. To protect the sensitive information, a lightweight privacy-preserving mechanism consisting of arbitrary data nullification and random noise addition is introduced, which provides strong privacy guarantee. A rigorous privacy budget analysis is given. Nonetheless, the private perturbation to the original data inevitably has a negative impact on the performance of further inference on the cloud side. To mitigate this influence, we propose a noisy training method to enhance the cloud-side network robustness to perturbed data. Through the sophisticated design, ARDEN can not only preserve privacy but also improve the inference performance. To validate the proposed ARDEN, a series of experiments based on three image datasets and a real mobile application are conducted. The experimental results demonstrate the effectiveness of ARDEN. Finally, we implement ARDEN on a demo system to verify its practicality.