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The paper is concerned with the adaptive finite element solution of linear elliptic differential equations using equidistributing meshes. A strategy is developed for defining this type of mesh based on residual-based a posteriori error…
Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace.…
We consider first-order linear systems of ordinary differential equations with periodic coefficients. Supposing that right-hand sides of equations are not known and subjected to some quadratic restrictions, we obtain optimal, in certain…
An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is…
Recent advances in solving ordinary differential equations (ODEs) with neural networks have been remarkable. Neural networks excel at serving as trial functions and approximating solutions within functional spaces, aided by gradient…
The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite…
We consider ordinary differential equations (ODEs) which involve expectations of a random variable. These ODEs are special cases of McKean-Vlasov stochastic differential equations (SDEs). A plain vanilla Monte Carlo approximation method for…
This study introduces new time-stepping strategies with built-in global error estimators. The new methods propagate the defect along with the numerical solution much like solving for the correction or Zadunaisky's procedure; however, the…
A posteriori error estimates are an important tool to bound discretization errors in terms of computable quantities avoiding regularity conditions that are often difficult to establish. For non-linear and non-differentiable problems,…
Continuous-depth neural networks, such as Neural ODEs, have refashioned the understanding of residual neural networks in terms of non-linear vector-valued optimal control problems. The common solution is to use the adjoint sensitivity…
Post-processing techniques are essential tools for enhancing the accuracy of finite element approximations and achieving superconvergence. Among these, recovery techniques stand out as vital methods, playing significant roles in both…
This paper is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all PDE types for which a stable variational formulation (SVF)…
Second-order two-scale expansions, a unified proof for the regularity of the correctors based on the translation invariant and a lemma for extracting $O(\epsilon)$ from the remainder term are presented for the second order nonlinear…
This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this…
This paper investigates the use of artificial neural networks (ANNs) to solve differential equations (DEs) and the construction of the loss function which meets both differential equation and its initial/boundary condition of a certain DE.…
The application of error-free transformation (EFT) is recently being developed to solve ill-conditioned problems. It can reduce the number of arithmetic operations required, compared with multiple precision arithmetic, and also be applied…
Stochastic differential equations (sdes) play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability. A general strategy for developing accurate and efficient schemes…
An ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution. Nonlinear ODE's with superposition formulas include matrix Riccati…