Hybrid Analytical Numerical and Machine Learning Frameworks for Solving Deterministic and Stochastic Differential Equations with Stability, Convergence, and Uncertainty Quantification
Jul 2026· Journal of Intelligent Decision Making and Information Science· 2 citations
TL;DR
This work introduces a unified hybrid analytical–numerical–ML framework for deterministic and stochastic differential equations that provides a principled pathway toward physically consistent, scalable, and uncertainty-aware solvers for complex dynamical systems.
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.
Hybrid computational models can improve prediction when governing equations are incomplete, but their advantages are often evaluated on isolated systems and without simultaneous assessment of accuracy, stability, interpretability, and uncertainty. This study develops a modular hybrid mathematical model that combines a partially specified ordinary differential equation, a regularized neural residual, joint parameter calibration, physical constraints, and ensemble-based uncertainty quantification. The framework was evaluated through in silico experiments on five benchmark systems representing periodic, chaotic, stiff, ecological, and engineering dynamics: Van der Pol, Lorenz-63, Robertson kinetics, Lotka–Volterra, and a continuous stirred-tank reactor. Six hundred trajectories were generated using space-filling sampling, partitioned at the trajectory level, and tested under interpolation, extrapolation, measurement noise, data scarcity, and partial observability. The proposed model was compared with an incomplete mechanistic model, a neural ordinary differential equation, and sequential residual correction. In illustrative synthetic results, the hybrid model achieved a mean normalized root-mean-square error of 0.0678, reducing error by 37.3% relative to the strongest baseline. It also increased the mean time to divergence to 83.6% of the forecast horizon, reduced median mechanistic parameter error to 4.8%, limited physical-constraint violations to 0.6%, and attained 0.947 coverage for nominal 95% prediction intervals. Friedman and Holm-adjusted Wilcoxon tests indicated significant paired improvements with large effect sizes. Ablation analyses showed that residual regularization, physical constraints, and joint calibration each contributed materially. These findings illustrate how restricted data-driven correction can enhance heterogeneous dynamical-system prediction while preserving mechanistic meaning, although empirical execution and external validation are required before scientific claims are made
George Em Karniadakis, E. Torfs, L. Marchetti· Global Synthesis in Educatio...· 0 citations
This work proposes Neural Kolmogorov Equations (NKEs), a deterministic, infinite-dimensional reformulation of Neural SDEs based on the Kolmogorov Forward equation, transforming the learning problem from modelling individual stochastic trajectories to modelling the evolution of probability densities.
Nonlinear partial differential equations (PDEs) play a vital role in modeling complex fluid dynamics phenomena such as turbulence, heat transfer, and flow instability. Traditional numerical approaches, including finite difference and finite element methods, often require extensive computational resources and may struggle with large-scale or real-time simulations. Recent advancements in deep learning have introduced efficient alternatives for solving nonlinear PDEs through data-driven and physics-informed approaches. This paper presents a comprehensive study of deep learning-based numerical solutions for nonlinear PDEs in fluid dynamics. The proposed framework integrates neural network architectures with physics informed constraints to accurately approximate fluid behavior while reducing computational complexity. Various deep learning models, including Physics-Informed Neural Networks (PINNs), Deep Operator Networks, and Fourier Neural Operators, are analyzed and compared with conventional numerical techniques. Experimental results demonstrate improved prediction accuracy, faster convergence, and enhanced scalability in solving complex fluid flow problems. The study further discusses practical applications, current limitations, and future research opportunities in AI-driven scientific computing for fluid dynamics.
S. M.· International Journal of Mat...· 0 citations
Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.
Rishav Jha, Kameshwar Sahani, S. K. Sahani et al.· African Multidisciplinary Jo...· 0 citations
Accurate numerical integration remains a fundamental challenge in predictive mathematical modelling, particularly for nonlinear, high-dimensional, and data-driven systems. Conventional numerical integration techniques, including the Trapezoidal Rule, Simpson's Rule, and Gaussian Quadrature, are widely used to approximate integrals that cannot be solved analytically. However, their performance often depends on integration parameters, data quality, and computational resources, particularly for nonlinear, high-dimensional, and uncertain systems. This study proposes a hybrid system that integrates machine learning with numerical integration to enhance prediction accuracy, computational efficiency, and numerical robustness. The designed system uses numerical integration to model continuous system dynamics while machine learning algorithms learn from historical and simulated datasets to select suitable integration methods, optimize step sizes, and estimate numerical errors. Performance was evaluated using MAE, RMSE, MAPE, convergence rate, computational time, stability analysis, robustness analysis, and statistical significance testing. It reduced approximation errors by over 80%, improved predictive accuracy to approximately 99%, and maintained numerical stability across benchmark problems. The results demonstrate that machine learning-assisted numerical integration significantly improves computational efficiency and predictive performance while preserving the convergence and stability properties of classical numerical methods. The study develops a novel adaptive machine learning-assisted numerical integration algorithm that automatically selects numerical integration strategies while preserving theoretical convergence and stability across science, engineering, finance, and environmental systems.
Okwuchukwu Christabel Opara· FUDMA Journal of Sciences· 0 citations
For numerical investigation of dynamical systems, the formulation of the corresponding ordinary differential equations (ODE) based on physical principles is usually the first and most crucial step. However, if the underlying physics is not fully understood or the required expert knowledge for modeling is missing, setting up these differential equations fail. Sometimes, running either real‐world experiments or black‐box simulations with commercial of‐the‐shelf software are the only ways of system exploration, which can be time‐consuming and/or expensive. In such cases, based on the gathered data, a surrogate model for the ODE may be set up and trained with the goal of later substituting the missing differential equation for cheaper numerical investigations.
In this paper, an approach is presented which proposes the embedding of a neural network based architecture as a substitute for the state function of an ODE into established Runge–Kutta based integration method. For training the surrogate, the numerical integration method solves an initial value problem to map initial conditions onto target system states for comparison with training data. The corresponding loss is then backpropagated through the model graph spanned by the numerical integration scheme to update the adjustable weights of the neural network for minimizing the loss of the mapping. The optimized surrogate state function may finally be treated as a substitute for the differential equation under investigation.
Timo Bielitz, Dieter Bestle· Proceedings in Applied Mathe...· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
