A bidirectional DeepParticle method for efficiently solving low-dimensional transport map problems

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 technique. We present numerical results to demonstrate the effectiveness of the BDP method for learning and generating solutions to Keller–Segel chemotaxis systems in the presence of laminar flows and Kolmogorov flows with chaotic streamlines in three-dimensional 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.