A probabilistic approach to solving unknown nonlinear stochastic differential equations with Bayesian uncertainty quantification

Résumé

This study develops a unified Bayesian framework for solving nonlinear stochastic differential equations (SDEs) with unknown dynamics, treating their solution as a problem of probabilistic inference rather than deterministic computation. Classical numerical solvers typically assume known system dynamics and ignore modeling or discretization errors, leading to overconfident predictions. To address these limitations, the proposed approach simultaneously reconstructs hidden state trajectories, estimates unknown drift and diffusion terms, and rigorously quantifies uncertainty. The method integrates the Fokker–Planck and Kushner–Stratonovich equations within a continuous-time Bayesian filtering framework, enabling full propagation of state distributions and recursive assimilation of measurements. Analytical derivations for the mean and covariance provide practical tools for uncertainty quantification, while the probabilistic formulation ensures consistent joint estimation of states and parameters. Results show that embedding uncertainty directly into the solution process enhances reliability, facilitates data-driven model discovery, and provides a principled foundation for sensitivity analysis and risk-aware decision-making in complex stochastic systems.