相关论文: Deep Hilbert--Galerkin Methods for Infinite-Dimens…

Machine learning methods have been lately used to solve partial differential equations (PDEs) and dynamical systems. These approaches have been developed into a novel research field known as scientific machine learning in which techniques…

We analyze a fully discrete scheme based on the discontinuous (in time) Galerkin approach, which is combined with conforming finite element subspaces in space, for the distributed optimal control problem of the three-dimensional…

This paper addresses the optimal control problem of finite-horizon discrete-time nonlinear systems under state and control constraints. A novel numerical algorithm based on optimal control theory is proposed to achieve superior…

This work investigates the optimal control problem for reflected McKean-Vlasov SDEs and the viscosity solutions to Hamilton-Jacobi-Bellman(HJB) equations on the Wasserstein space in terms of intrinsic derivative. It follows from the flow…

This paper addresses the model-free nonlinear optimal problem with generalized cost functional, and a data-based reinforcement learning technique is developed. It is known that the nonlinear optimal control problem relies on the solution of…

In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty$ norm. The discretization method…

We introduce an extended discontinuous Galerkin discretization of hyperbolic-parabolic problems on multidimensional semi-infinite domains. Building on previous work on the one-dimensional case, we split the strip-shaped computational domain…

Many engineering and scientific fields have recently become interested in modeling terms in partial differential equations (PDEs) with neural networks, which requires solving the inverse problem of learning neural network terms from…

We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural…

In recent years deep artificial neural networks (DNNs) have been successfully employed in numerical simulations for a multitude of computational problems including, for example, object and face recognition, natural language processing,…

This paper analyzes the numerical approximation of the Lindblad master equation on infinite-dimensional Hilbert spaces. We employ a classical Galerkin approach for spatial discretization and investigate the convergence of the discretized…

The dynamic programming approach is one of the most powerful ones in optimal control. However, when dealing with optimal control problems of stochastic Volterra integral equations (SVIEs) with completely monotone kernels, deep mathematical…

This manuscript presents an algorithm for obtaining an approximation of a nonlinear high order control affine dynamical system. Controlled trajectories of the system are leveraged as the central unit of information via embedding them in…

This paper proposes a fully data-driven approach for optimal control of nonlinear control-affine systems represented by a stochastic diffusion. The focus is on the scenario where both the nonlinear dynamics and stage cost functions are…

In numerous robotics and mechanical engineering applications, among others, data is often constrained on smooth manifolds due to the presence of rotational degrees of freedom. Common datadriven and learning-based methods such as neural…

We investigate discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints on general polygonal domains. We show that discontinuous Galerkin methods for general…

In this paper, we aim to solve the high dimensional stochastic optimal control problem from the view of the stochastic maximum principle via deep learning. By introducing the extended Hamiltonian system which is essentially an FBSDE with a…

In this paper we consider time-dependent PDEs discretized by a special class of Physics Informed Neural Networks whose design is based on the framework of Runge--Kutta and related time-Galerkin discretizations. The primary motivation for…

In this paper we establish best approximation property of fully discrete Galerkin solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty(I;W^{1,\infty}(\Om))$ norm. The discretization method…

Operator learning focuses on approximating mappings $\mathcal{G}^\dagger:\mathcal{U} \rightarrow\mathcal{V}$ between infinite-dimensional spaces of functions, such as $u: \Omega_u\rightarrow\mathbb{R}$ and $v:…