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1. Operator-Based Uncertainty Quantification of Stochastic Fractional Partial Differential Equations

   https://doi.org/10.1115/1.4046093  

   Kharazmi, Ehsan; Zayernouri, Mohsen (December 2019, Journal of Verification, Validation and Uncertainty Quantification)

   null (Ed.)

   Abstract Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional orders as uncertain variables, we develop an operator-based uncertainty quantification framework in the context of stochastic fractional partial differential equations (SFPDEs), subject to additive random noise. We characterize different sources of uncertainty and then, propagate their associated randomness to the system response by employing a probabilistic collocation method (PCM). We develop a fast, stable, and convergent Petrov–Galerkin spectral method in the physical domain in order to formulate the forward solver in simulating each realization of random variables in the sampling procedure. 

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2. Lagrangian Averaged Stochastic Advection by Lie Transport for Fluids

   https://doi.org/10.1007/s10955-020-02493-4  

   Drivas, Theodore D.; Holm, Darryl D.; Leahy, James-Michael (June 2020, Journal of Statistical Physics)

   null (Ed.)

   Abstract We formulate a class of stochastic partial differential equations based on Kelvin’s circulation theorem for ideal fluids. In these models, the velocity field is randomly transported by white-noise vector fields, as well as by its own average over realizations of this noise. We call these systems the Lagrangian averaged stochastic advection by Lie transport (LA SALT) equations. These equations are nonlinear and non-local, in both physical and probability space. Before taking this average, the equations recover the Stochastic Advection by Lie Transport (SALT) fluid equations introduced by Holm (Proc R Soc A 471(2176):20140963, 2015). Remarkably, the introduction of the non-locality in probability space in the form of momentum transported by its own mean velocity gives rise to a closed equation for the expectation field which comprises Navier–Stokes equations with Lie–Laplacian ‘dissipation’. As such, this form of non-locality provides a regularization mechanism. The formalism we develop is closely connected to the stochastic Weber velocity framework of Constantin and Iyer (Commun Pure Appl Math 61(3):330–345, 2008) in the case when the noise correlates are taken to be the constant basis vectors in $$\mathbb {R}^3$$ R 3 and, thus, the Lie–Laplacian reduces to the usual Laplacian. We extend this class of equations to allow for advected quantities to be present and affect the flow through exchange of kinetic and potential energies. The statistics of the solutions for the LA SALT fluid equations are found to be changing dynamically due to an array of intricate correlations among the physical variables. The statistical properties of the LA SALT physical variables propagate as local evolutionary equations which when spatially integrated become dynamical equations for the variances of the fluctuations. Essentially, the LA SALT theory is a non-equilibrium stochastic linear response theory for fluctuations in SALT fluids with advected quantities. 

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3. Spike Variations for Stochastic Volterra Integral Equations

   https://doi.org/10.1137/22M1522097  

   Wang, Tianxiao; Yong, Jiongmin (December 2023, SIAM Journal on Control and Optimization)

   The spike variation technique plays a crucial role in deriving Pontryagin's type maximum principle of optimal controls for ordinary differential equations (ODEs), partial differential equations (PDEs), stochastic differential equations (SDEs), and (deterministic forward) Volterra integral equations (FVIEs), when the control domains are not assumed to be convex. It is natural to expect that such a technique could be extended to the case of (forward) stochastic Volterra integral equations (FSVIEs). However, by mimicking the case of SDEs, one encounters an essential difficulty of handling an involved quadratic term. To overcome this difficulty, we introduce an auxiliary process for which one can use It\^o's formula, and develop new technologies inspired by stochastic linear-quadratic optimal control problems. Then the suitable representation of the above-mentioned quadratic form is obtained, and the second-order adjoint equations are derived. Consequently, the maximum principle of Pontryagin type is established. Some relevant extensions are investigated as well. 

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4. Cyber-Physical Trust Anchors in Additive Manufacturing: Secure, Low-Cost, and Educational

   https://doi.org/10.1115/MSEC2023-105046  

   Maasberg, Michele; Birch, Brendan; Janes, Daniel; Stor, Kirsten; Ham, Kyungmin; Butler, Leslie G. (June 2023, American Society of Mechanical Engineers)

   Abstract We report progress towards development of a cyber-physical trust anchor for additive manufacturing systems. The additive manufacturing commercial sector needs cyber-physical trust anchors to establish a secure supply chain, to detect counterfeiting and to ensure part provenance. However, the underlying technology of cyber-physical trust anchors requires optimization and spans several sectors ranging from mathematics, additive manufacturing, materials science, nondestructive evaluation, to cyber science. The fast and effective deployment of cyber-physical trust anchors requires an educational component. This project present a novel method for authenticating additively manufactured parts. Features are extracted using advanced X-ray imaging, transformed into unique identifiers, and bound with security features for cloud-based blockchain authentication. A plan for the low-cost and safe incorporation of cyber-physical trust anchor research in education is included. The anticipated outcome is an optimized trust anchor prototype and educational product suitable for interdisciplinary research and coursework to develop the workforce needed for cyber-secured physical supply chainsd. 

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5. Partial differential equations in data science

   https://doi.org/10.1098/rsta.2024.0249  

   Bertozzi, Andrea L; Drenska, Nadejda; Latz, Jonas; Thorpe, Matthew (June 2025, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   The advent of artificial intelligence and machine learning has led to significant technological and scientific progress, but also to new challenges. Partial differential equations, usually used to model systems in the sciences, have shown to be useful tools in a variety of tasks in the data sciences, be it just as physical models to describe physical data, as more general models to replace or construct artificial neural networks, or as analytical tools to analyse stochastic processes appearing in the training of machine-learning models. This article acts as an introduction of a theme issue covering synergies and intersections of partial differential equations and data science. We briefly review some aspects of these synergies and intersections in this article and end with an editorial foreword to the issue. This article is part of the theme issue ‘Partial differential equations in data science’. 

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