Related papers: A State-Space Representation of Coupled Linear Mul…
Ensuring that a PDE model is well-posed is a necessary precursor to any form of analysis, control, or numerical simulation. Although the Lumer-Phillips theorem provides necessary and sufficient conditions for well-posedness of dissipative…
This paper proposes a novel low-rank approximation to the multivariate State-Space Model. The Stochastic Partial Differential Equation (SPDE) approach is applied component-wise to the independent-in-time Mat\'ern Gaussian innovation term in…
We analysis some singular partial differential equations systems(PDAEs) with boundary conditions in high dimension bounded domain with sufficiently smooth boundary. With the eigenvalue theory of PDE the systems initially is formulated as an…
This paper investigates a class of controlled stochastic partial differential equations (SPDEs) arising in the modeling of composite materials with spatially varying properties. The state equation describes the evolution of a material…
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…
In many scientific fields, the generation and evolution of data are governed by partial differential equations (PDEs) which are typically informed by established physical laws at the macroscopic level to describe general and predictable…
Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant…
Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural…
We consider the stability analysis of a large class of linear 1-D PDEs with polynomial data. This class of PDEs contains, as examples, parabolic and hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary coupled…
In this paper, we address stability of parabolic linear Partial Differential Equations (PDEs). We consider PDEs with two spatial variables and spatially dependent polynomial coefficients. We parameterize a class of Lyapunov functionals and…
State-dependent parameter identification, where unknown model parameters depend on one or more state variables in partial differential equations (PDEs) or coupled PDE systems, is fundamental to a wide range of problems in physics,…
In this paper, we present a convex formulation of $H_{\infty}$-optimal control problem for coupled linear ODE-PDE systems with one spatial dimension. First, we reformulate the coupled ODE-PDE system as a Partial Integral Equation (PIE)…
Impulse-to-peak response (I2P) analysis for state-space ordinary differential equation (ODE) systems is a well-studied classical problem. However, the techniques employed for I2P optimal control of ODEs have not been extended to partial…
In many science and engineering settings, system dynamics are characterized by governing PDEs, and a major challenge is to solve inverse problems (IPs) where unknown PDE parameters are inferred based on observational data gathered under…
Foundation models, such as large language models, have demonstrated success in addressing various language and image processing tasks. In this work, we introduce a multi-modal foundation model for scientific problems, named PROSE-PDE. Our…
Over the past decade, Finite Element Method (FEM) has served as a foundational numerical framework for approximating the terms of Time Series Expansion (TSE) as solutions to transient Partial Differential Equation (PDE). However, the…
Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications the…
Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…
We study the Allen-Cahn equation with a cubic-quintic nonlinear term and a stochastic $Q$-trace-class stochastic forcing in two spatial dimensions. This stochastic partial differential equation (SPDE) is used as a test case to understand,…
Many safety-critical scientific and engineering systems evolve according to differential-algebraic equations (DAEs), where dynamical behavior is constrained by physical laws and admissibility conditions. In practice, these systems operate…