Related papers: On bounds for the effective differential Nullstell…
The problem of designing optimal quantization rules for sequential detectors is investigated. First, it is shown that this task can be solved within the general framework of active sequential detection. Using this approach, the optimal…
A general setup for deterministic system identification problems on graphs with Dirichlet and Neumann boundary conditions is introduced. When control nodes are available along the boundary, we apply a discretize-then-optimize method to…
We prove constructively a Nullstellensatz giving an equivalence between the existence of a certain kind of algebraic identity on one hand, and the impossibility of finding an increasing sequence of irreducible varieties obeying certain…
The primary goal of this research is to propose a novel architecture for a deep neural network that can solve fractional differential equations accurately. A Gaussian integration rule and a $L_1$ discretization technique are used in the…
We provide bounds on the size of polynomial differential equations obtained by executing closure properties for D-algebraic functions. While it is easy to obtain bounds on the order of these equations, it requires some more work to derive…
We generalize the technique of [Solving Dirichlet boundary-value problems on curved domains by extensions from subdomains, SIAM J. Sci. Comput. 34, pp. A497--A519 (2012)] to elliptic problems with mixed boundary conditions and elliptic…
We introduce a novel kernel-based framework for learning differential equations and their solution maps that is efficient in data requirements, in terms of solution examples and amount of measurements from each example, and computational…
Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of…
Deriving sharp and computable upper bounds of the Lipschitz constant of deep neural networks is crucial to formally guarantee the robustness of neural-network based models. We analyse three existing upper bounds written for the $l^2$ norm.…
One of the fundamental challenges in the deep learning community is to theoretically understand how well a deep neural network generalizes to unseen data. However, current approaches often yield generalization bounds that are either too…
This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element (FE) and domain decomposition (DD) methods. In addition to a fully parallel computation, the proposed lower bounds separate…
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,…
In this article we present an algorithm to compute bounds on the marginals of a graphical model. For several small clusters of nodes upper and lower bounds on the marginal values are computed independently of the rest of the network. The…
We prove an existence and uniqueness result for Neumann boundary problem of a parabolic partial differential equation (PDE for short) with a singular nonlinear divergence term which can only be understood in a weak sense. A probabilistic…
In this article, we give a unified theory for constructing boundary layer expansions for dis-cretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
The incorporation of analytical kernel information is exploited in the construction of Nystr\"om discretization schemes for integral equations modeling planar Helmholtz boundary value problems. Splittings of kernels and matrices, coarse and…
The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes…
We introduce a novel monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. This problem is prevalent in…
Deep neural networks (DNNs) have achieved remarkable empirical success, yet the absence of a principled theoretical foundation continues to hinder their systematic development. In this survey, we present differential equations as a…