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1. Optimal Control, Numerics, and Applications of Fractional PDEs

   https://doi.org/10.1016/bs.hna.2021.12.003  

   Antil, H.; Brown T.; Khatri, R.; Onwunta, A.; Verma, D.; Warma, M. (February 2022, Handbook of numerical analysis)

   Trélat, E.; Zuazua, E. (Ed.)

   This chapter provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps, and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems. 

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2. Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

   https://doi.org/10.3934/cpaa.2021174  

   Scott, James M.; Mengesha, Tadele (January 2021, Communications on Pure & Applied Analysis)

   We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" Analysis & PDE, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript. 

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3. Uniqueness in an inverse problem of fractional elasticity

   https://doi.org/10.1098/rspa.2023.0474  

   Covi, Giovanni; de_Hoop, Maarten; Salo, Mikko (October 2023, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We study a nonlinear inverse problem for fractional elasticity. In analogy to the classical problem of linear elasticity, we consider the unique recovery of the Lamé parameters associated with a linear, isotropic fractional elasticity operator from fractional Dirichlet-to-Neumann data. In our analysis, we make use of a fractional matrix Schrödinger equation via a generalization of the so-called Liouville reduction to the case of fractional elasticity. We conclude that unique recovery is possible if the Lamé parameters agree and are constant in the exterior, and their Poisson ratios agree everywhere. Our study is motivated by the significant recent activity in the field of nonlocal elasticity. 

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4. Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discovery

   Yu, Yue; Liu, Ning; Lu, Fei; Gao, Tian; Jafarzadeh, Siavash; Silling, Stewart (December 2024, 38th Conference on Neural Information Processing Systems (NeurIPS 2024))

   Despite the recent popularity of attention-based neural architectures in core AI fields like natural language processing (NLP) and computer vision (CV), their potential in modeling complex physical systems remains underexplored. Learning problems in physical systems are often characterized as discovering operators that map between function spaces based on a few instances of function pairs. This task frequently presents a severely ill-posed PDE inverse problem. In this work, we propose a novel neural operator architecture based on the attention mechanism, which we refer to as the Nonlocal Attention Operator (NAO), and explore its capability in developing a foundation physical model. In particular, we show that the attention mechanism is equivalent to a double integral operator that enables nonlocal interactions among spatial tokens, with a data-dependent kernel characterizing the inverse mapping from data to the hidden parameter field of the underlying operator. As such, the attention mechanism extracts global prior information from training data generated by multiple systems, and suggests the exploratory space in the form of a nonlinear kernel map. Consequently, NAO can address ill-posedness and rank deficiency in inverse PDE problems by encoding regularization and achieving generalizability. We empirically demonstrate the advantages of NAO over baseline neural models in terms of generalizability to unseen data resolutions and system states. Our work not only suggests a novel neural operator architecture for learning interpretable foundation models of physical systems, but also offers a new perspective towards understanding the attention mechanism. Our code and data accompanying this paper are available at https://github.com/fishmoon1234/NAO. 

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5. Inverse Iteration for the Laplace Eigenvalue Problem With Robin and Mixed Boundary Conditions

   https://doi.org/10.46787/pump.v8i.4261  

   Lyons, Benjamin; Ruttenberg, Ephraim; Zitzelberger, Nicholas (May 2025, PUMP journal of undergraduate research)

   We apply the method of inverse iteration to the Laplace eigenvalue problem with Robin and mixed Dirichlet-Neumann boundary conditions, respectively. For each problem, we prove convergence of the iterates to a non-trivial principal eigenfunction and show that the corresponding Rayleigh quotients converge to the principal eigenvalue. We also propose a related iterative method for an eigenvalue problem arising from a model for optimal insulation and provide some partial results. 

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