What Uncertainties Do We Need for Dynamical Systems?

The study of dynamical systems has evolved from classical deterministic models, such as Newtonian mechanics and linear control systems, to incorporate stochasticity and data-driven approaches. Early works focused on stability analysis and control design using Lyapunov functions and transfer functions. With the advent of statistical inference and machine learning, researchers began integrating probabilistic models, including Bayesian networks, Gaussian processes, and neural networks, to better capture uncertainties inherent in real systems. Notable developments include the introduction of neural ODEs by Chen et al. (2018), which enabled end-to-end learning of continuous-time dynamics, and Gaussian process state-space models by Frigola et al. (2013), which provided probabilistic predictions with uncertainty quantification. Despite these advances, challenges remain in modeling the propagation of multiple uncertainty sources, especially in nonlinear, high-dimensional, and partially observed systems. The need for a unified theoretical framework that can handle diverse uncertainties and their interactions in a dynamic context has driven recent research efforts.

The core challenge addressed in this paper is the comprehensive modeling and analysis of multiple, interacting sources of uncertainty within dynamical systems. Existing methods often treat uncertainties independently or focus on static distributions, neglecting their temporal evolution and interactions. This limits the ability to accurately predict long-term behaviors, assess system robustness, and design effective control strategies. Moreover, distinguishing between aleatoric (inherent randomness) and epistemic (knowledge-based) uncertainties is crucial for tasks like system verification and decision-making. The difficulty lies in formalizing these distinctions mathematically, capturing their propagation over time, and developing scalable algorithms capable of real-time inference. Addressing these issues is essential for advancing applications in climate science, robotics, finance, and beyond, where uncertainty quantification directly impacts safety and reliability.

The paper introduces a novel, unified framework that systematically classifies and models multiple sources of uncertainty in dynamical systems. It combines classical PDEs (Liouville and Fokker-Planck equations) with Bayesian inference techniques to describe how uncertainties propagate and interact over time. This approach explicitly distinguishes between aleatoric and epistemic uncertainties, providing formal mathematical descriptions and scalable algorithms. The framework supports both continuous and discrete systems, accommodating complex nonlinear dynamics and high-dimensional states. By integrating neural network parametrizations (e.g., neural ODEs) with probabilistic models, it achieves high flexibility and interpretability. The methodology enables task-specific uncertainty quantification, improving prediction accuracy, robustness, and decision-making in real-world applications.

• �� Identify sources of uncertainty: initial conditions, process noise, parameters, structural errors, and observation noise.
• �� Model system dynamics using differential equations, incorporating stochastic terms where appropriate.
• �� Derive Liouville equations for initial condition uncertainty propagation, describing how probability densities evolve in phase space.
• �� Use Fokker-Planck equations to model the evolution of probability densities under stochastic influences, capturing diffusion effects.
• �� Implement Bayesian filtering algorithms (particle filters, extended Kalman filter) for recursive state estimation, integrating observational data.
• �� Develop joint models for multiple uncertainty sources, leveraging Gaussian processes, neural ODEs, and set-based representations.
• �� Validate models through simulations of nonlinear pendulum dynamics, analyzing the impact of different uncertainty sources on prediction accuracy and stability.
• �� Explore task-specific modeling strategies for forecasting, control, and verification, emphasizing the role of uncertainty quantification in each context.

The experimental setup involves simulating a nonlinear pendulum with damping (γ=0.5, g/ℓ=1.0), introducing various uncertainties such as random initial angles, process noise, and parameter perturbations. The models employ Gaussian process priors for system dynamics, particle filters for state estimation, and neural ODEs for flexible modeling. The evaluation metrics include mean squared error (MSE), trajectory divergence, and uncertainty calibration scores. Baseline comparisons involve traditional deterministic models, static probabilistic models, and recent deep learning approaches. Ablation studies assess the contribution of each uncertainty source, and sensitivity analyses examine the impact of observation noise and data scarcity. Results demonstrate that explicit modeling of multiple uncertainties improves long-term forecast reliability and system robustness, with predictive errors reduced by over 20% compared to baseline methods.

The key findings indicate that short-term predictions are predominantly influenced by initial condition uncertainty, accounting for approximately 60% of total variance, while long-term behaviors are shaped mainly by parameter and structural uncertainties. Incorporating process noise explicitly via stochastic differential equations enhances predictive accuracy by 20%, especially in highly nonlinear regimes. Bayesian filtering methods, such as particle filters, reduce state estimation errors by 15% under partial observation scenarios. The joint modeling framework captures the complex interplay among multiple uncertainty sources, leading to more reliable long-term forecasts and improved robustness against model misspecification. These results validate the theoretical predictions and demonstrate practical applicability in real-world systems.

The proposed framework is directly applicable to climate modeling, where quantifying uncertainties in temperature and precipitation forecasts can inform policy decisions. In robotics, it enhances autonomous navigation by providing probabilistic safety margins under sensor noise and environmental variability. Financial risk management benefits from improved modeling of market volatility and systemic uncertainties, aiding in robust portfolio optimization. The methodology can also be integrated into control systems for industrial automation, enabling adaptive strategies that account for model inaccuracies and external disturbances. Long-term, these approaches could underpin the development of fully autonomous, uncertainty-aware decision-making systems across multiple domains.

The reliance on continuous-time differential equations limits applicability to systems with discrete events or hybrid dynamics. Parameter and structural uncertainty estimation requires extensive data, which may be unavailable or noisy in practice, reducing model reliability. Computational complexity increases with system size and the number of uncertainty sources, hindering real-time deployment in large-scale or high-dimensional systems. Additionally, the framework assumes model correctness in the form of differential equations, which may not hold in systems with unmodeled dynamics or abrupt regime shifts. Future work must address these limitations through algorithmic innovations, data-efficient methods, and broader model classes.