Related papers: Robin Pre-Training for the Deep Ritz Method
We establish error estimates for the approximation of parametric $p$-Dirichlet problems deploying the Deep Ritz Method. Parametric dependencies include, e.g., varying geometries and exponents $p\in (1,\infty)$. Combining the derived error…
Using a capacity approach, and the theory of measure's perturbation of Dirichlet forms, we give the probabilistic representation of the General Robin boundary value problems on an arbitrary domain $\Omega$, involving smooth measures, which…
We present a new view onto the successive approximations' approach in study of the two-point nonlinear fractional boundary value problems. In order to reduce the original problem and further construct its approximate solution we use the…
Reinforcement learning (RL) is attracting attention as an effective way to solve sequential optimization problems that involve high dimensional state/action space and stochastic uncertainties. Many such problems involve constraints…
We consider the torsional rigidity and the principal eigenvalue related to the Laplace operator with Dirichlet and Robin boundary conditions. The goal is to find upper and lower bounds to products of suitable powers of the quantities above…
While much attention of neural network methods is devoted to high-dimensional PDE problems, in this work we consider methods designed to work for elliptic problems on domains $\Omega \subset \mathbb{R} ^d, $ $d=1,2,3$ in association with…
Learned representations in deep reinforcement learning (DRL) have to extract task-relevant information from complex observations, balancing between robustness to distraction and informativeness to the policy. Such stable and rich…
We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show…
Linear second order elliptic boundary value problems (BVP) on bounded Lipschitz domains are studied in the case of Gaussian white noise loads. Especially, Neumann and Robin BVPs are considered. The main obstacle for applying the usual…
In this paper, we study the Dirichlet boundary value problem of steady-state relativistic Boltzmann equation in half-line with hard potential model, given the data for the outgoing particles at the boundary and a relativistic global…
This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization…
Typical reinforcement learning (RL) methods show limited applicability for real-world industrial control problems because industrial systems involve various constraints and simultaneously require continuous and discrete control. To overcome…
Many state-of-the-art adversarial training methods for deep learning leverage upper bounds of the adversarial loss to provide security guarantees against adversarial attacks. Yet, these methods rely on convex relaxations to propagate lower…
In this paper, we present a new iterative approximate method of solving boundary value problems. The idea is to compute approximate polynomial solutions in the Bernstein form using least squares approximation combined with some properties…
The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…
A Neural Network (NN) based numerical method is formulated and implemented for solving Boundary Value Problems (BVPs) and numerical results are presented to validate this method by solving Laplace equation with Dirichlet boundary condition…
This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones…
In this paper, we prove that there exists a unique, bounded continuous weak solution to the Dirichlet boundary value problem for a general class of second-order elliptic operators with singular coefficients, which does not necessarily have…
In this article we establish an approximation result involving the Laplacian with Robin boundary conditions. It informs about the weak solutions dependence from the input function on the boundary.
Strong worst-case performance bounds for episodic reinforcement learning exist but fortunately in practice RL algorithms perform much better than such bounds would predict. Algorithms and theory that provide strong problem-dependent bounds…