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Abstract We introduce an efficient stochastic interacting particle-field (SIPF) algorithm with no history dependence for computing aggregation patterns and near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis system in three-dimensional (3D) space. In our algorithm, the KS solutions are approximated as empirical measures of particles coupled with a smoother field (concentration of chemo-attractant) variable computed by a spectral method. Instead of using heat kernels that cause history dependence and high memory cost, we leverage the implicit Euler discretization to derive a one-step recursion in time for stochastic particle positions and the field variable based on the explicit Green’s function of an elliptic operator of the form Laplacian minus a positive constant. In numerical experiments, we observe that the resulting SIPF algorithm is convergent and self-adaptive to the high-gradient part of solutions. Despite the lack of analytical knowledge (such as a self-similar ansatz) of a blowup, the SIPF algorithm provides a low-cost approach to studying the emergence of finite-time blowup in 3D space using only dozens of Fourier modes and by varying the amount of initial mass and tracking the evolution of the field variable. Notably, the algorithm can handle multi-modal initial data and the subsequent complex evolution involving the merging of particle clusters and the formation of a finite time singularity with ease. 

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. 

In this paper, we study the convergence analysis for a robust stochastic structure-preserving Lagrangian numerical scheme in computing effective diffusivity of time-dependent chaotic flows, which are modeled by stochastic differential equations (SDEs). Our numerical scheme is based on a splitting method to solve the corresponding SDEs in which the deterministic subproblem is discretized using a structure-preserving scheme while the random subproblem is discretized using the Euler-Maruyama scheme. We obtain a sharp and uniform-in-time convergence analysis for the proposed numerical scheme that allows us to accurately compute long-time solutions of the SDEs. As such, we can compute the effective diffusivity for time-dependent chaotic flows. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing effective diffusivity for the time-dependent Arnold-Beltrami-Childress (ABC) flow and Kolmogorov flow in three-dimensional space.