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G-equations are popular level set Hamilton–Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. Thelack of coercivityof Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomeshighly nonconvex and nonlinear. In the absence of coercivity and convexity, the PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging noncoercive, nonconvex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection. 

We study a regularized interacting particle method for computing aggregation patterns and near singular solutions of a Keller–Segel (KS) chemotaxis system in two and three space dimensions, then further develop the DeepParticle method to learn and generate solutions under variations of physical parameters. The KS solutions are approximated as empirical measures of particles that self-adapt to the high gradient part of solutions. We utilize the expressiveness of deep neural networks (DNNs) to represent the transform of samples from a given initial (source) distribution to a target distribution at a finite time 𝑇 prior to blowup without assuming the invertibility of the transforms. In the training stage, we update the network weights by minimizing a discrete 2-Wasserstein distance between the input and target empirical measures. To reduce the computational cost, we develop an iterative divide-and-conquer algorithm to find the optimal transition matrix in the Wasserstein distance. We present numerical results of the DeepParticle framework for successful learning and generation of KS dynamics in the presence of laminar and chaotic flows. The physical parameter in this work is either the evolution time or the flow amplitude in the advection-dominated regime.