Hybrid Analytical Numerical and Machine Learning Frameworks for Solving Deterministic and Stochastic Differential Equations with Stability, Convergence, and Uncertainty Quantification

Abstract

Differential equations underpin the modeling of dynamical systems across physics, engineering, biology, and finance. While deterministic ordinary and partial differential equations (ODEs/PDEs) describe systems governed by known physical laws, stochastic differential equations (SDEs) incorporate randomness to represent uncertainty, noise, and unresolved scales. Classical analytical methods are limited to restricted problem classes, and although numerical discretization provide general applicability, they can become computationally demanding for nonlinear, high-dimensional, stiff, or multiscale systems. In stochastic settings, accurate estimation further requires large ensembles of sample paths, amplifying computational cost. Conversely, purely data-driven machine learning (ML) approaches offer expressive approximation capabilities but often lack physical consistency, stability guarantees, and reliable generalization. This work introduces a unified hybrid analytical–numerical–ML framework for deterministic and stochastic differential equations. The approach integrates (i) analytical structure and prior knowledge (e.g., conservation laws and invariants), (ii) stable numerical discretization’s serving as computational backbones and multi-fidelity supervision sources, and (iii) physics-guided learning components acting as correction operators or drift–diffusion estimators. Governing equations and boundary/initial conditions are embedded directly into the learning objective, while stability constraints are enforced to preserve numerical robustness. An explicit error decomposition separates discretization, sampling, optimization, and generalization contributions, and sufficient conditions for stable and convergent hybrid approximations are derived. Numerical experiments on representative PDE and SDE benchmarks demonstrate improved accuracy and stability over backbone-only and ML-only baselines. The proposed framework provides a principled pathway toward physically consistent, scalable, and uncertainty-aware solvers for complex dynamical systems.