Machine learning (ML) has been used to solve multiphysics problems like thermoelasticity through multi-layer perceptron (MLP) networks. However, MLPs have high computational costs and need to be ...trained for each prediction instance. To overcome these limitations, we introduced an integrated finite element neural network (I-FENN) framework to solve transient thermoelasticity problems in Abueidda and Mobasher (2024). This approach used a physics-informed temporal convolutional network (PI-TCN) within a finite element scheme for solving transient thermoelasticity problems. In this paper, we introduce an I-FENN framework using a new variational TCN model trained to minimize the thermoelastic variational form rather than the strong form of the energy balance. We mathematically prove that the I-FENN setup based on minimizing the variational form of transient thermoelasticity still leads to the same solution as the strong form. Introducing the variational form to the ML model brings the advantages of lower requirement for the differentiability of the basis function and, thus, lower memory requirement and higher computational efficiency. Also, it automatically satisfies zero Neumann boundary conditions, thus reducing the complexity of the loss function. The formulation based on the variational form complies with thermodynamic requirements. The proposed loss function reduces the difference between predicted and target data while minimizing the variational form of thermoelasticity equations, combining the benefits of both data-driven and variational methods. In addition, this study uses finite element shape functions for spatial gradient calculations and compares their performance against automatic differentiation. Our results reveal that models leveraging shape functions exhibit higher accuracy in capturing the behavior of the thermoelasticity problem and faster convergence. Adding the variational term and using shape functions for gradient calculations ensure better adherence to the underlying physics. We demonstrate the capabilities of this I-FENN framework through multiple numerical examples. Additionally, we discuss the convergence of the proposed variational TCN model and the impact of hyperparameters on its performance. The proposed approach offers a well-founded and flexible platform for solving fully coupled thermoelasticity problems while retaining computational efficiency, where the efficiency scales proportional to the model size.
•A novel I-FENN framework is developed to speed up solving thermoelasticity problems.•A TCN model for temperature predictions is trained using the variational form.•Training the TCN model using the variational form reduces error and computation time.•Using FE shape functions for computing gradients yields better results than AD.•A convergence study for the variational TCN model has been conducted.
Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two ...decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift towards modern nonlinear regularization methods, including their analysis, applications and issues for future research. In particular we will discuss variational methods and techniques derived from them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.
We consider controlling a graph-based Markov decision process (GMDP) with a control capacity constraint given only uncertain measurements of the underlying state. We also consider two special ...structural properties of GMDPs, called Anonymous Influence and Symmetry. Large-scale spatial processes such as forest wildfires, disease epidemics, opinion dynamics, and robot swarms are well-modeled by GMDPs with these properties. We adopt a certainty-equivalence approach and derive efficient and scalable algorithms for estimating the GMDP state given uncertain measurements, and for computing approximately optimal control policies given a maximum-likelihood state estimate. We also derive sub-optimality bounds for our estimation and control algorithms. Unlike prior work, our methods scale to GMDPs with large state-spaces and explicitly enforce a control constraint. We demonstrate the effectiveness of our estimation and control approach in simulations of controlling a forest wildfire using a model with <inline-formula><tex-math notation="LaTeX">10^{1192}</tex-math></inline-formula> total states.
We study the following double-phase problem
where
and 1<p<q<N. The aim is to determine the precise positive interval of λ for which the problem admits at least two nontrivial solutions via ...variational methods for the above equation. The primitive of the nonlinearity f is of super-q growth near infinity in u and allowed to be sign-changing. Furthermore, our assumptions are suitable and different from those studied previously.
In this paper we use variational methods to establish existence of solitary waves for a class of generalized Kadomtsev-Petviashvili (GKP) equation in RN. The positive and zero mass cases are ...considered. The main argument is to find a Palais Smale sequence satisfying a property related to Pohozaev identity, as in 21, which was used for the first time by 23.
The partitioning of the energy in ab initio quantum mechanical calculations into its chemical origins (e.g., electrostatics, exchange-repulsion, polarization, and charge transfer) is a relatively ...recent development; such concepts of isolating chemically meaningful energy components from the interaction energy have been demonstrated by variational and perturbation based energy decomposition analysis approaches. The variational methods are typically derived from the early energy decomposition analysis of Morokuma Morokuma, J. Chem. Phys., 1971, 55, 1236, and the perturbation approaches from the popular symmetry-adapted perturbation theory scheme Jeziorski et al., Methods and Techniques in Computational Chemistry: METECC-94, 1993, ch. 13, p. 79. Since these early works, many developments have taken place aiming to overcome limitations of the original schemes and provide more chemical significance to the energy components, which are not uniquely defined. In this review, after a brief overview of the origins of these methods we examine the theory behind the currently popular variational and perturbation based methods from the point of view of biochemical applications. We also compare and discuss the chemical relevance of energy components produced by these methods on six test sets that comprise model systems that display interactions typical of biomolecules (such as hydrogen bonding and π-π stacking interactions) including various treatments of the dispersion energy.
In this paper, we consider the existence of nontrivial solutions for a class of fractional advection–dispersion equations. A new existence result is established by introducing a suitable fractional ...derivative Sobolev space and using the critical point theorem.
A Variational Approach for Pan-Sharpening Fang, Faming; Li, Fang; Shen, Chaomin ...
IEEE transactions on image processing,
07/2013, Letnik:
22, Številka:
7
Journal Article
Recenzirano
Pan-sharpening is a process of acquiring a high resolution multispectral (MS) image by combining a low resolution MS image with a corresponding high resolution panchromatic (PAN) image. In this ...paper, we propose a new variational pan-sharpening method based on three basic assumptions: 1) the gradient of PAN image could be a linear combination of those of the pan-sharpened image bands; 2) the upsampled low resolution MS image could be a degraded form of the pan-sharpened image; and 3) the gradient in the spectrum direction of pan-sharpened image should be approximated to those of the upsampled low resolution MS image. An energy functional, whose minimizer is related to the best pan-sharpened result, is built based on these assumptions. We discuss the existence of minimizer of our energy and describe the numerical procedure based on the split Bregman algorithm. To verify the effectiveness of our method, we qualitatively and quantitatively compare it with some state-of-the-art schemes using QuickBird and IKONOS data. Particularly, we classify the existing quantitative measures into four categories and choose two representatives in each category for more reasonable quantitative evaluation. The results demonstrate the effectiveness and stability of our method in terms of the related evaluation benchmarks. Besides, the computation efficiency comparison with other variational methods also shows that our method is remarkable.
In this paper, we study the existence of normalized solutions to the following nonlinear Schrödinger systems with critical exponential ...growth:{−Δu+λ1u=f1(u)+βr1|u|r1−2u|v|r2,inR2,−Δv+λ2v=f2(v)+βr2|u|r1|v|r2−2v,inR2,∫R2u2dx=a2and∫R2v2dx=b2,u,v∈H1(R2), where 0<a,b<1,β>0,r1,r2>1 and r1+r2∈(4,+∞), f1,f2∈C(R,R) have critical exponential growth in the sense of Trudinger-Moser inequality. λ1,λ2∈R will arise as Lagrange multipliers. Under some suitable assumptions on f1,f2, we prove the existence of positive normalized ground state solutions for the problem when β>0 is sufficiently large via variational method. Our results improve and extend the previous results.
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