Preprint submitted to Elsevier March 10, 2025 algorithms to solve such inverse-type problems advance different fields including inferring neural circuit dynamics from spiking data [42] in neuroscience, modeling and predicting complex weather patterns from historical data [9] in climate science, uncovering disease transmission dynamics from infection case counts over time [46] in epidemiology, and deducing reaction rates from experimental concentration-time profiles in reaction kinetics in biochemistry [30]. However, such inverse-type problems pose substantial mathematical and computational challenges, particularly when data are limited and noisy, motivating ongoing research into novel algorithms and theoretical frameworks to improve models' reconstruction accuracy and efficiency. In this paper, we study the inverse problem of inferring the distribution of model parameters for several dynamical systems including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs) from time-series data or spatiotemporal data. Existing methods for such problems can be broadly categorized into traditional statistical approaches and modern data-driven techniques. Traditional statistical methods often involve parameter estimation frameworks. For example, linear and nonlinear regression methods play a role in simpler systems where the functional form of the model is partially known [13]. Furthermore, maximum likelihood estimation and Bayesian inference methods [16, 33] are often adopted. Maximum likelihood estimation optimizes the likelihood of model parameter values in a proposed model from observed data, while Bayesian methods incorporate prior information and compute posterior distributions. These approaches are widely used in applications such as reaction network reconstruction and epidemiological modeling.

We analyze the Wasserstein distance ($W$-distance) between two probability distributions associated with two multidimensional jump-diffusion processes. Specifically, we analyze a temporally decoupled squared $W_2$-distance, which provides both upper and lower bounds associated with the discrepancies in the drift, diffusion, and jump amplitude functions between the two jump-diffusion processes. Then, we propose a temporally decoupled squared $W_2$-distance method for efficiently reconstructing unknown jump-diffusion processes from data using parameterized neural networks. We further show its performance can be enhanced by utilizing prior information on the drift function of the jump-diffusion process. The effectiveness of our proposed reconstruction method is demonstrated across several examples and applications.