Yuan Chen

Yuan Chen, Xu Zhang

In this paper, we introduce an immersed $C^0$ interior penalty method for solving two-dimensional biharmonic interface problems on unfitted meshes. To accommodate the biharmonic interface conditions, high-order immersed finite element (IFE) spaces are constructed in the least-squares sense. We establish key properties of these spaces including unisolvency and partition of unity are, and verify their optimal approximation capability. These spaces are further incorporated into a modified $C^0$ interior penalty scheme with additional penalty terms on interface segments. The well-posedness of the discrete solution is proved. Numerical experiments with various interface geometries confirm optimal convergence of the proposed method in $L^2$, $H^1$ and $H^2$ norms.

Zhongshu Xu, Yuan Chen, Qifan Chen et al.

We present a numerical method to learn an accurate predictive model for an unknown stochastic dynamical system from its trajectory data. The method seeks to approximate the unknown flow map of the underlying system. It employs the idea of autoencoder to identify the unobserved latent random variables. In our approach, we design an encoding function to discover the latent variables, which are modeled as unit Gaussian, and a decoding function to reconstruct the future states of the system. Both the encoder and decoder are expressed as deep neural networks (DNNs). Once the DNNs are trained by the trajectory data, the decoder serves as a predictive model for the unknown stochastic system. Through an extensive set of numerical examples, we demonstrate that the method is able to produce long-term system predictions by using short bursts of trajectory data. It is also applicable to systems driven by non-Gaussian noises.

Yuan Chen, Abdul Khaliq, Khaled M. Furati

Nonlinear time-dependent partial differential equations are essential in modeling complex phenomena across diverse fields, yet they pose significant challenges due to their computational complexity, especially in higher dimensions. This study explores Quantum Recurrent Neural Networks within an encoder-decoder framework, integrating Variational Quantum Circuits into Gated Recurrent Units and Long Short-Term Memory networks. Using this architecture, the model efficiently compresses high-dimensional spatiotemporal data into a compact latent space, facilitating more efficient temporal evolution. We evaluate the algorithms on the Hamilton-Jacobi-Bellman equation, Burgers' equation, the Gray-Scott reaction-diffusion system, and the three dimensional Michaelis-Menten reaction-diffusion equation. The results demonstrate the superior performance of the quantum-based algorithms in capturing nonlinear dynamics, handling high-dimensional spaces, and providing stable solutions, highlighting their potential as an innovative tool in solving challenging and complex systems.

Yuan Chen, Dongbin Xiu

We present a numerical framework for learning unknown stochastic dynamical systems using measurement data. Termed stochastic flow map learning (sFML), the new framework is an extension of flow map learning (FML) that was developed for learning deterministic dynamical systems. For learning stochastic systems, we define a stochastic flow map that is a superposition of two sub-flow maps: a deterministic sub-map and a stochastic sub-map. The stochastic training data are used to construct the deterministic sub-map first, followed by the stochastic sub-map. The deterministic sub-map takes the form of residual network (ResNet), similar to the work of FML for deterministic systems. For the stochastic sub-map, we employ a generative model, particularly generative adversarial networks (GANs) in this paper. The final constructed stochastic flow map then defines a stochastic evolution model that is a weak approximation, in term of distribution, of the unknown stochastic system. A comprehensive set of numerical examples are presented to demonstrate the flexibility and effectiveness of the proposed sFML method for various types of stochastic systems.

Yuan Chen, Jiaqi Li, Guorui Xu et al.

Since its debut, SGX has been used in many applications, e.g., secure data processing. However, previous systems usually assume a trusted enclave and ignore the security issues caused by an untrusted enclave. For instance, a vulnerable (or even malicious) third-party enclave can be exploited to attack the host application and the rest of the system. In this paper, we propose an efficient mechanism to confine an untrusted enclave's behaviors. The threats of an untrusted enclave come from the enclave-host asymmetries. They can be abused to access arbitrary memory regions of its host application, jump to any code location after leaving the enclave and forge the stack register to manipulate the saved context. Our solution breaks such asymmetries and establishes mutual distrust between the host application and the enclave. It leverages Intel MPK for efficient memory isolation and the x86 single-step debugging mechanism to capture the event when an enclave is existing. It then performs the integrity check for the jump target and the stack pointer. We have solved two practical challenges and implemented a prototype system. The evaluation with multiple micro-benchmarks and representative real-world applications demonstrated the efficiency of our system, with less than 4% performance overhead.