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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. 

This paper aims to efficiently compute transport maps between probability distributions arising from particle-based representations of bio-physical problems. We develop a Bidirectional DeepParticle (BDP) method to learn and generate solutions under varying physical parameters, where solutions are approximated as empirical measures of particles that adaptively concentrate in high-gradient regions. The core idea of the BDP method is to learn both forward and reverse maps (between a uniform reference distribution and a non-trivial target distribution) by minimizing the discrete 2-Wasserstein (W2) distance and optimizing the transition map using a mini-batch optimization technique. We present numerical results to demonstrate the effectiveness of the BDP method for learning and generating solutions to the Keller–Segel chemotaxis systems in the presence of laminar flows and Kolmogorov flows with chaotic streamlines in three-dimensional (3D) space. Compared to recent representative single-step flow matching and generative models (rectified flow and shortcut diffusion models), the BDP method achieves superior accuracy with compact neural networks. We also find that for high-dimensional target distributions (4D and above, e.g., Gaussian mixtures), single-step diffusion models exhibit better scalability than the BDP method in terms of W2 accuracy. 

Tumor angiogenesis involves a collection of tumor cells moving towards blood vessels for nutrients to grow. Angiogenesis, and in general chemotaxis systems have been modeled using partial differential equations (PDEs) and as such require numerical methods to approximate their solutions in 3 space dimensions (3D). This is an expensive computation when solutions develop large gradients at unknown locations, and so efficient algorithms to capture the main dynamical behavior are valuable. Here as a case study, we consider a parabolic-hyperbolic Keller-Segel (PHKS) system in the angiogenesis literature, and develop a mesh-free particle-based neural network algorithm that scales better to 3D than traditional mesh based solvers. From a regularized approximation of PHKS, we derive a neural stochastic interacting particle-field (NSIPF) algorithm where the bacterial density is represented as empirical measures of particles and the field variable (concentration of chemo-attractant) by a convolutional neural network trained on low cost synthetic data. As a new model, NSIPF preserves total mass and non-negativity of the density, and captures the dynamics of 3D multi-bump solutions at much faster speeds compared with classical finite difference and spline based SIPF methods. 

Transformer-based models are at the forefront in long timeseries forecasting (LTSF). Despite their powerful modeling capacity, they are hindered in this domain by a bias toward high-energy, low-frequency features in the data. Recent work has established that learnable frequency filters can strengthen deep forecasting models by enhancing their spectral utilization. These works choose to use multilayer perceptrons to process their filtered signals and thus do not address the issues found with transformer-based models. In this paper, we demonstrate that applying learnable frequency filters to embedded signals enhance the performance of transformer-based forecasters across a number of architectures with negligible increases in memory and compute. We also conduct synthetic experiments that demonstrate learnable filters enhance transformer-based models by amplifying primarily middle and high-frequency features in the data. 

We develop an interacting particle method (IPM) for computing the large deviation rate function of entropy production for diffusion processes, with emphasis on the vanishing-noise limit and high dimensions. The crucial ingredient for obtaining the rate function is the computation of the principal eigenvalue \lambda of elliptic, non-self-adjoint operators. We show that this principal eigenvalue can be approximated in terms of the spectral radius of a discretized evolution operator, which is obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size. We also show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, which is suitable for computation using the IPM. The IPM applies naturally to problems in unbounded domains and scales easily to high dimensions. We show numerical examples of dimensions up to 16, and the results show that our numerical approximation of \lambda converges to the analytical vanishing-noise limit within visual tolerance with a fixed number of particles and a fixed time step size. It is numerically shown that the IPM can adapt to singular behaviors in the vanishing-noise limit. We also apply the IPM to explore situations with no explicit formulas of the vanishing-noise limit. Our paper appears to be the first to obtain numerical results of principal eigenvalue problems for non-self-adjoint operators in such high dimensions.