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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…
System identification through learning approaches is emerging as a promising strategy for understanding and simulating dynamical systems, which nevertheless faces considerable difficulty when confronted with power systems modeled by…
We propose a method that meta-learns a knowledge on matrix factorization from various matrices, and uses the knowledge for factorizing unseen matrices. The proposed method uses a neural network that takes a matrix as input, and generates…
This paper considers one of the fundamental parallel-in-time methods for the solution of ordinary differential equations, Parareal, and extends it by adopting a neural network as a coarse propagator. We provide a theoretical analysis of the…
Equivariant neural networks are neural networks with symmetry. Motivated by the theory of group representations, we decompose the layers of an equivariant neural network into simple representations. The nonlinear activation functions lead…
A {\it Lie system} is a nonautonomous system of first-order differential equations admitting a {\it superposition rule}, i.e., a map expressing its general solution in terms of a generic family of particular solutions and some constants.…
Deep neural networks come in many sizes and architectures. The choice of architecture, in conjunction with the dataset and learning algorithm, is commonly understood to affect the learned neural representations. Yet, recent results have…
In recent years, deep learning techniques have shown great success in various tasks related to inverse problems, where a target quantity of interest can only be observed through indirect measurements by a forward operator. Common approaches…
Learning large scale nonlinear ordinary differential equation (ODE) systems from data is known to be computationally and statistically challenging. We present a framework together with the adaptive integral matching (AIM) algorithm for…
The time evolution of dynamical systems is frequently described by ordinary differential equations (ODEs), which must be solved for given initial conditions. Most standard approaches numerically integrate ODEs producing a single solution…
Biological networks often change under different environmental and genetic conditions. Understanding how these networks change becomes an important problem in biological studies. In this paper, we model the network change as the difference…
Deep learning has shown promising results in many machine learning applications. The hierarchical feature representation built by deep networks enable compact and precise encoding of the data. A kernel analysis of the trained deep networks…
The recent introduction of the Least-Squares Support Vector Regression (LS-SVR) algorithm for solving differential and integral equations has sparked interest. In this study, we expand the application of this algorithm to address systems of…
We present a lightweighted neural PDE representation to discover the hidden structure and predict the solution of different nonlinear PDEs. Our key idea is to leverage the prior of ``translational similarity'' of numerical PDE differential…
The coefficient algebra of a finite-dimensional Lie algebra on a finite-dimensional representation is defined as the subalgebra generated by all coefficients of the corresponding characteristic polynomial. We explore connections between…
A classification theorem for linear differential equations in two variables (one real and one Grassmann) having polynomial solutions(the generalized Bochner problem) is given. The main result is based on the consideration of the eigenvalue…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
The interaction of neural networks with physical equations offers a wide range of applications. We provide a method which enables a neural network to transform objects subject to given physical constraints. Therefore an U-Net architecture…