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The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric…
We adapt a symmetric interior penalty discontinuous Galerkin method using a patch reconstructed approximation space to solve elliptic eigenvalue problems, including both second and fourth order problems in 2D and 3D. It is a direct…
This paper explores the integration of Diophantine equations into neural network (NN) architectures to improve model interpretability, stability, and efficiency. By encoding and decoding neural network parameters as integer solutions to…
We introduce Neural Parameter Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs). Tailored for operator learning, this approach surpasses traditional DeepONets…
We introduce an adaptive finite element scheme for the efficient approximation of a (large) collection of eigenpairs of selfadjoint elliptic operators in which the adaptive refinement is driven by the solution of a single source problem --…
Recent years have witnessed rapid advances in the use of neural networks to solve combinatorial optimization problems. Nevertheless, designing the "right" neural model that can effectively handle a given optimization problem can be…
Neural networks (NNs) have gained significant attention across various engineering disciplines, particularly in design optimization, where they are used to build surrogate models for high-dimensional regression problems. Despite their power…
This paper introduces the Non-linear Partition of Unity Method, a novel technique integrating Radial Basis Function interpolation and Weighted Essentially Non-Oscillatory algorithms. It addresses challenges in high-accuracy approximations,…
Localized features such as singularities, sharp gradients, discontinuities, and moving sources require adaptive finite element discretizations. Conventional refinement strategies introduce significant computational overhead through…
This paper proposes a meshless deep learning algorithm, enriched physics-informed neural networks (EPINNs), to solve dynamic Poisson-Nernst-Planck (PNP) equations with strong coupling and nonlinear characteristics. The EPINNs takes the…
We discuss realization, properties and performance of the adaptive finite element approach to the design of nano-photonic components. Central issues are the construction of vectorial finite elements and the embedding of bounded components…
We develop the Randomized Neural Networks with Petrov-Galerkin Methods (RNN-PG methods) to solve linear elasticity problems. RNN-PG methods use Petrov-Galerkin variational framework, where the solution is approximated by randomized neural…
The proximal Galerkin finite element method is a high-order, low-iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of point-wise bound constraints in infinite-dimensional function spaces.…
In this paper, we explore the graph partitioning problem, a pivotal combina-torial optimization challenge with extensive applications in various fields such as science, technology, and business. Recognized as an NP-hard prob-lem, graph…
Finding a feasible and prompt solution to the Vehicle Routing Problem (VRP) is a prerequisite for efficient freight transportation, seamless logistics, and sustainable mobility. Traditional optimization methods reach their limits when…
An artificial intelligence-augmented Streamline Upwind/Petrov-Galerkin finite element scheme (AiStab-FEM) is proposed for solving singularly perturbed partial differential equations. In particular, an artificial neural network framework is…
Minimal-residual methods for PDEs with a residual in a dual space are non-trivial to guarantee stability. We present a minimal-residual finite element method in which the solution space is a standard finite element space, but neural…
This article explores the integration of deep learning models into combinatorial optimization pipelines, specifically targeting NP-hard problems. Traditional exact algorithms for such problems often rely on heuristic criteria to guide the…
Recently, there has been much work on the design of general heuristics for graph-based, combinatorial optimization problems via the incorporation of Graph Neural Networks (GNNs) to learn distribution-specific solution structures.However,…
This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter $\epsilon$ multiplying the highest derivative. We…