In this work, we develop a machine-learning-based predictive control design for nonlinear parabolic partial differential equation (PDE) systems using process state measurement time-series data. First, the Karhunen-Loève expansion is used to derive dominant spatial empirical eigenfunctions of PDE system from Then, these are as basis functions within Galerkin's model reduction framework temporal ...

in this paper, we investigate some properties of eigenvalues and eigenfunctions of boundary value problems with separated boundary conditions. also, we obtain formal series solutions for some partial differential equations associated with the second order differential equation, and study necessary and sufficient conditions for the negative and positive eigenvalues of the boundary value problem....

The de nition of an Integrated Mean-Squared Error (IMSE) criterion for the learning of a random eld model yields a particular Karhunen-Loève expansion of the underlying eld. The model can thus also be interpreted as a Bayesian (or regularised) linear model based on eigenfunctions of this Karhunen-Loève expansion, and can be approximated by a linear model involving orthogonal observation errors....

The Calogero–Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as t...

In this paper, we investigate some properties of eigenvalues and eigenfunctions of boundary value problems with separated boundary conditions. Also, we obtain formal series solutions for some partial differential equations associated with the second order differential equation, and study necessary and sufficient conditions for the negative and positive eigenvalues of the boundary value problem....

We analyze the Scarf potential, which exhibits both discrete energy bound states and energy bands, through the quantum Hamilton-Jacobi approach. The singularity structure and the boundary conditions in the above approach, naturally isolate the bound and periodic states, once the problem is mapped to the zero energy sector of another quasi-exactly solvable quantum problem. The energy eigenvalues...

Extended free energy Lagrangians are proposed for first principles molecular dynamics simulations at finite electronic temperatures for plane-wave pseudopotential and local orbital density matrix-based calculations. Thanks to the extended Lagrangian description, the electronic degrees of freedom can be integrated by stable geometric schemes that conserve the free energy. For the local orbital r...

This paper studies surface gravity waves in a channel with corrugated sidewalls in water of constant depth according to linear theory. Waves propagating along the channel encounter a nite patch of step-like corrugations in one of the channel walls. The method of solution employed is one of expansion in terms of eigenfunctions in each region and matching at the boundaries between regions. It is ...

We discuss a novel strategy for computing the eigenvalues and eigenfunctions of the relativistic Dirac operator with a radially symmetric potential. The virtues of this strategy lie on the fact that it avoids completely the phenomenon of spectral pollution and it always provides two-side estimates for the eigenvalues with explicit error bounds on both eigenvalues and eigenfunctions. We also dis...