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An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
Diffusive representations of fractional derivatives have proven to be useful tools in the construction of fast and memory efficient numerical methods for solving fractional differential equations. A common challenge in many of the known…
We introduce a general, analytical framework to express and to approximate partial differential equations (PDEs) numerically on graphs and networks of surfaces---generalized by the term hypergraphs. To this end, we consider PDEs on…
A system of partial differential equations (PDEs) is derived to compute the full-field stress from an observed kinematic field when the flow rule governing the plastic deformation is unknown. These equations generalize previously proposed…
Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and…
The dynamic mode decomposition (DMD) has become a leading tool for data-driven modeling of dynamical systems, providing a regression framework for fitting linear dynamical models to time-series measurement data. We present a simple…
Many physical processes can be expressed through partial differential equations (PDEs). Real-world measurements of such processes are often collected at irregularly distributed points in space, which can be effectively represented as…
A numerical framework is proposed for identifying partial differential equations (PDEs) governing dynamical systems directly from their observation data using Chebyshev polynomial approximation. In contrast to data-driven approaches such as…
The ability to differentiate through optimization problems has unlocked numerous applications, from optimization-based layers in machine learning models to complex design problems formulated as bilevel programs. It has been shown that…
Numerical methods of approximate solution of the Cauchy problem for coupled systems of evolution equations are considered. Separating simpler subproblems for individual components of the solution achieves simplification of the problem at a…
We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and…
This work establishes a robust mathematical foundation for compositional System Dynamics modeling, leveraging category theory to formalize and enhance the representation, analysis, and composition of system models. Here, System Dynamics…
A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary…
A graph theoretic perspective is taken for a range of phenomena in continuum physics in order to develop representations for analysis of large scale, high-fidelity solutions to these problems. Of interest are phenomena described by partial…
Quadratization for partial differential equations (PDEs) is a process that transforms a nonquadratic PDE into a quadratic form by introducing auxiliary variables. This symbolic transformation has been used in diverse fields to simplify the…
We propose a new numerical domain decomposition method for solving elliptic equations on compact Riemannian manifolds. One advantage of this method is its ability to bypass the need for global triangulations or grids on the manifolds.…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
Our study is dedicated to the probabilistic representation and numerical approximation of solutions to coupled systems of variational inequalities. The dynamics of each component of the solution is driven by a different linear parabolic…
Modeling real-world problems with partial differential equations (PDEs) is a prominent topic in scientific machine learning. Classic solvers for this task continue to play a central role, e.g. to generate training data for deep learning…
Bifurcation diagram is a powerful tool that visually gives information about the behavior of the equilibrium points of a dynamical system respect to the varying parameter. This paper proposes an educational algorithm by which the local…