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Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next. (arXiv:2201.05624v1 [cs.LG]) arxiv.org/abs/2201.05624

Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next

Physic-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, and integral-differential equations. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages, the review also attempts to incorporate publications on a larger variety of issues, including physics-constrained neural networks (PCNN), where the initial or boundary conditions are directly embedded in the NN structure rather than in the loss functions. The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.

arxiv.org

Dynamics of small particle inertial migration in curved square ducts. (arXiv:2201.05645v1 [physics.flu-dyn]) arxiv.org/abs/2201.05645

Dynamics of small particle inertial migration in curved square ducts

Microchannels are well-known in microfluidic applications for the control and separation of microdroplets and cells. Often the objects in the flow experience inertial effects, resulting in dynamics that is a departure from the underlying channel flow dynamics. This paper considers small neutrally buoyant spherical particles suspended in flow through a curved duct having a square cross-section. The particle experiences a combination of inertial lift force induced by the disturbance from the primary flow along the duct, and drag from the secondary vortices in the cross-section, which drive migration of the particle within the cross-section. We construct a simplified model that preserves the core topology of the force field yet depends on a single parameter $κ$, quantifying the relative strength of the two forces. We show that $κ$ is a bifurcation parameter for the dynamical system that describes motion of the particle in the cross section of the duct. At large values of $κ$ there exists an attracting limit cycle, in each of the upper and lower halves of the duct. At small $κ$ we find that particles migrate to one of four stable foci. Between these extremes, there is an intermediate-range of $κ$ for which all particles migrate to a single stable focus. Noting that the positions of the limit cycles and foci vary with the value of $κ$, this behavior indicates that, for a suitable particle mixture, duct bend radius might be chosen to segregate particles by size. We evaluate the time and axial distance required to focus particles near the unique stable node, which determines the duct length required for particle segregation.

arxiv.org

Stationary States of the One-Dimensional Discrete-Time Facilitated Symmetric Exclusion Process. (arXiv:2201.05175v1 [math.PR]) arxiv.org/abs/2201.05175

Stationary States of the One-Dimensional Discrete-Time Facilitated Symmetric Exclusion Process

We describe the extremal translation invariant stationary (ETIS) states of the facilitated exclusion process on $\mathbb{Z}$. In this model all particles on sites with one occupied and one empty neighbor jump at each integer time to the empty neighbor site, and if two particles attempt to jump into the same empty site we choose one randomly to succeed. The ETIS states are qualitatively different for densities $ρ<1/2$, $ρ=1/2$, and $1/2<ρ<1$, but in each density region we find states which may be grouped into families, each of which is in natural correspondence with the set of all ergodic measures on $\{0,1\}^{\mathbb{Z}}$. For $ρ<1/2$ there is one such family, containing all the ergodic states in which the probability of two adjacent occupied sites is zero. For $ρ=1/2$ there are two families, in which configurations translate to the left and right, respectively, with constant speed 2. For the high density case there is a continuum of families. We show that all ETIS states at densities $ρ\le1/2$ belong to these families, and conjecture that also at high density there are no other ETIS states. We also study the possible ETIS states which might occur if the conjecture fails.

arxiv.org
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