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A proof of convergence is given for bulk--surface finite element semi-discretisation of the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary conditions in a smooth domain. The semi-discretisation is studied in the weak…
We present a non-intrusive gradient and a non-intrusive BFGS algorithm for parameter estimation problems in non-stationary elasticity. To avoid multiple (and potentially expensive) solutions of the underlying partial differential equation…
We present an algorithm for the calibration of local volatility from market option prices through deep self-consistent learning, by approximating both market option prices and local volatility using deep neural networks. Our method uses the…
A new implicit-explicit local differential transform method (IELDTM) is derived here for time integration of the nonlinear advection-diffusion processes represented by (2+1)-dimensional Burgers equation. The IELDTM is adaptively constructed…
Burgers equation is one of the simplest nonlinear partial differential equations-it combines the basic processes of diffusion and nonlinear steepening. In some applications it is appropriate for the diffusion coefficient to be a…
Self-similarity of Burgers' equation with some stochastic advection is studied. In self-similar variables a stationary solution is constructed which establishes the existence of a stochastically self-similar solution for the stochastic…
Deep learning models have been the subject of study from various perspectives, for example, their training process, interpretation, generalization error, robustness to adversarial attacks, etc. A trained model is defined by its decision…
We investigate the stability of large-scale structures in Burgers' equation under the perturbation of high wave-number noise in the initial conditions. Analytical estimates are obtained for random initial data with spatial spectral density…
Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection…
This study presents the application of variable-order (VO) fractional calculus to the modeling of nonlocal solids. The reformulation of nonlocal fractional-order continuum mechanic framework, by means of VO kinematics, enables a unique…
After reviewing the source-type solution of the Burgers equation with standard dissipativity, we study the hypoviscous counterpart of the Burgers equation. 1) We determine an equation that governs the near-identity transformation underlying…
The Local Randomized Neural Networks with Discontinuous Galerkin (LRNN-DG) methods, introduced in [42], were originally designed for solving linear partial differential equations. In this paper, we extend the LRNN-DG methods to solve…
In this work, we present a deep collocation method for three dimensional potential problems in nonhomogeneous media. This approach utilizes a physics informed neural network with material transfer learning reducing the solution of the…
In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases…
Partial Differential Equations (PDEs) describe phenomena ranging from turbulence and epidemics to quantum mechanics and financial markets. Despite recent advances in computational science, solving such PDEs for real-world applications…
In this paper, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints.…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
We propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents…
We study a system of forced viscous shallow water equations with nontrivial bathymetry in two spatial dimensions. We develop a well-posedness theory for small but arbitrary forcing data, as well as for a fixed data profile but large…
The aim of this work is to analyze the finite element approximation of the two-dimensional stationary Navier-Stokes equations with non-smooth Dirichlet boundary data. The discrete approximation is obtained by considering the Navier-Stokes…