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Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
In this paper we introduce a procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations. The procedure maximizes accuracy by adaptively computing…
We introduce a Partial Integral Equation (PIE) representation of Partial Differential Equations (PDEs) in two spatial variables. PIEs are an algebraic state-space representation of infinite-dimensional systems and have been used to model 1D…
A linearizable version of multidimensional system of $n$-wave type nonlinear PDEs is proposed. This system is derived using the spectral representation of its solution via the procedure similar to the dressing method for the ISTM-integrable…
In this paper, non-variational systems of differential equations containing small terms are considered, and a consistent approach for deriving approximate conservation laws through the introduction of approximate Lagrange multipliers is…
We explore the derivation of distributed parameter system evolution laws (and in particular, partial differential operators and associated partial differential equations, PDEs) from spatiotemporal data. This is, of course, a classical…
In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational…
This note describes a method for generating an infinite-dimensional family of nonlinear control laws for underactuated systems. For a ball and beam system, the entire family is found explicitly.
The method of nonlinear self-adjointness is applied to the Kadomtsev-Petviashvili equation. The infinite set of conservation laws associated with the infinite algebra of Lie point symmetry of the KP equation is constructed.
I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such…
In this paper the numerical approximation of solutions of Liouville-Master Equations for time-dependent distribution functions of Piecewise Deterministic Processes with memory is considered. These equations are linear hyperbolic PDEs with…
The reaction diffusion equation arises in physical situations in problems from population growth, genetics and physical sciences. We consider the generalised Fisher equation in cylindrical coordinates from Lie theory stand point. An…
The working mechanisms of complex natural systems tend to abide by concise and profound partial differential equations (PDEs). Methods that directly mine equations from data are called PDE discovery, which reveals consistent physical laws…
We propose a new method to design adaptation algorithms that guarantee a certain prescribed level of performance and are applicable to systems with nonconvex parameterization. The main idea behind the method is, given the desired…
A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
In this paper, we present solvable, convex formulations of $H_2$-optimal state estimation and state-feedback control problems for a general class of linear Partial Differential Equations (PDEs) with one spatial dimension. These convex…
We consider parameter identification problems in parametrized partial differential equations (PDE). This leads to nonlinear ill-posed inverse problems. One way to solve them are iterative regularization methods, which typically require…
We consider optimization problems constrained by partial differential equations (PDEs) with additional constraints placed on the solution of the PDEs. We develop a general and versatile framework using infinite-valued penalization functions…