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What Uncertainties Do We Need for Dynamical Systems?

This paper presents a machine learning perspective on uncertainty modeling for dynamical systems by distinguishing between aleatoric and epistemic uncertainties, clarifying their sources, and examining how uncertainty quantification objectives vary across different tasks.

Original authors: Yusuf Sale, Christopher Bülte, Felix Czaja, Joshua Stiller, Eyke Hüllermeier

Published 2026-06-11

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yusuf Sale, Christopher Bülte, Felix Czaja, Joshua Stiller, Eyke Hüllermeier

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Predicting the Future is Hard

Imagine you are trying to predict where a ball will roll next. In a perfect world, if you know exactly where the ball starts, how hard you pushed it, and the shape of the hill, you could calculate its exact path forever. This is a deterministic system: no surprises, just math.

But in the real world, things are messy. We don't know everything perfectly. This paper asks a simple but deep question: When we try to predict how things change over time (like a swinging pendulum, the weather, or a robot's movement), what kind of "not knowing" are we actually dealing with?

The authors break down "not knowing" into two main types:

Aleatoric Uncertainty (The "Noise"): This is randomness that is built into the system itself. It's like a gust of wind hitting the ball. No matter how much you study or how many sensors you add, you can't predict the wind. It's irreducible.
Epistemic Uncertainty (The "Ignorance"): This is a lack of knowledge. It's like not knowing exactly how hard you pushed the ball. If you had better tools or more data, you could figure this out. It is reducible.

The paper argues that in dynamic systems (things that change over time), these two types of uncertainty mix together in complex ways, and we need to treat them differently depending on what we are trying to do.

The Five Sources of "Not Knowing"

The authors identify five specific places where uncertainty sneaks into our predictions. Let's use the analogy of a Swinging Pendulum to explain them.

1. The Starting Point (Initial Condition)

The Scenario: You want to predict the pendulum's swing, but you aren't 100% sure where it was when you started the timer. Maybe your ruler was slightly off.
The Uncertainty: This is Epistemic. If you had a better camera or a more precise ruler, you could know the starting point exactly.
How it spreads: Once you start, that small error gets carried along. If the pendulum is stable, the error might stay small. If it's chaotic (like a double pendulum), that tiny error can grow huge very quickly.

2. The Random Jiggles (Process Noise)

The Scenario: Even if you know the starting point perfectly, the pendulum is being hit by tiny, random air currents or vibrations from the floor.
The Uncertainty: This is Aleatoric. It's the "noise" of the universe. You can't measure every air molecule, so this randomness is permanent.
How it spreads: Unlike the starting point error (which happens once), this noise happens every single second. It's like trying to walk a straight line while someone keeps gently pushing you from the side. The path becomes a fuzzy cloud rather than a single line.

3. The Blurry Glasses (Partial Observations)

The Scenario: You can't see the whole pendulum. You only have a sensor that tells you the angle, but not the speed. Or maybe your sensor is noisy and gives you a slightly wrong reading.
The Uncertainty: This is Epistemic. You are "blind" to parts of the system.
The Fix: This is a puzzle called "filtering." You use what you do see to guess what you don't see. If the system is stable, your guesses get better over time. If it's chaotic, you might never be sure.

4. The Wrong Manual (Parameter Uncertainty)

The Scenario: You know the formula for how the pendulum moves, but you don't know the exact numbers. Is the gravity 9.8 or 9.81? Is the friction 0.5 or 0.6?
The Uncertainty: This is Epistemic. The "true" numbers exist; you just haven't found them yet.
The Danger: This is tricky because a small mistake in the numbers creates a systematic bias. It's not just a random wobble; it's like driving a car with the steering wheel slightly crooked. Over a short trip, you're fine. Over a long trip, you end up in a completely different country.

5. The Wrong Map (Structural Uncertainty)

The Scenario: You are using the wrong formula entirely. Maybe you assumed the pendulum moves in a straight line, but it actually moves in a circle. Or maybe you forgot that there is a hidden weight inside the pendulum you can't see.
The Uncertainty: This is the hardest kind. It's Epistemic in the sense that you don't know the truth, but it's qualitatively different. You can't fix this just by collecting more data if your entire model is wrong. It's like trying to navigate with a map of the ocean when you are actually in a forest.

How Time Changes the Game

The paper makes a crucial point: Which uncertainty matters most depends on how far into the future you are looking.

Short Term: If you want to know where the pendulum is in 1 second, the Starting Point matters most. You haven't had time for the wind or the wrong formula to mess things up yet.
Medium Term: If you look 10 seconds ahead, the Random Jiggles and Blind Spots start to dominate. The noise accumulates.
Long Term: If you look 1 hour ahead, the Wrong Manual and Wrong Map take over. Even if you knew the starting point perfectly, if your formula is slightly wrong or your model is missing a piece, your long-term prediction will be completely off.

Why Does This Distinction Matter?

You might ask, "Why do we need to label them 'Aleatoric' vs. 'Epistemic'?"

The answer lies in Action.

If you have Epistemic uncertainty (ignorance), the best action is to learn. Get better sensors, run more experiments, or refine your model. You can reduce the uncertainty.
If you have Aleatoric uncertainty (noise), learning won't help. You can't "learn away" the wind. Instead, you must adapt. You need to build a system that is robust enough to handle the randomness (e.g., a robot that can balance even when pushed).

The Machine Learning Connection

The paper concludes by looking at how Machine Learning (ML) fits into this.

ML helps Dynamical Systems: ML is great at finding complex patterns (like using a neural network to guess the "Wrong Manual" or "Wrong Map"). It can learn the rules of the game from data.
Dynamical Systems help ML: The math used to track moving objects (like Kalman filters) is now being used to make AI smarter at learning sequences and making decisions over time.

The Takeaway

The paper doesn't just say "uncertainty is bad." It says: Uncertainty is a process. It changes shape as time passes. To predict the future of a moving system, we need to know where the uncertainty comes from (the start, the noise, the sensors, the parameters, or the model itself) and when it will become the biggest problem. Only then can we decide whether to gather more data or build a more robust system.

Technical Summary: What Uncertainties Do We Need for Dynamical Systems?

Problem Statement

The distinction between aleatoric uncertainty (irreducible randomness inherent in a process) and epistemic uncertainty (reducible uncertainty arising from a lack of knowledge) is well-established in supervised machine learning. However, this dichotomy has received significantly less attention in the context of dynamical systems, despite the growing intersection between machine learning (ML) and dynamical systems (e.g., neural differential equations, scientific ML, and data-driven control).

The central problem addressed by this paper is the lack of a clear framework for identifying what uncertainties are necessary for dynamical systems. Specifically, the authors seek to:

Revisit the sources of uncertainty in dynamical systems.
Clarify whether these sources are aleatoric or epistemic.
Determine how the objectives of representing and quantifying uncertainty vary across different tasks (e.g., forecasting, control, verification).

The paper argues that the classification of a source as aleatoric or epistemic is not intrinsic to the source itself but depends heavily on the task, modeling assumptions, and the object of interest.

Methodology and Framework

The paper employs a conceptual analysis rather than proposing a new algorithm or conducting empirical experiments. The methodology involves:

Formalizing the Baseline: The authors start with the standard deterministic continuous-time autonomous dynamical system defined by the initial value problem $\dot{x}(t) = f(x(t)), x(0) = x_0$ .
Categorizing Sources of Uncertainty: They systematically introduce five distinct sources of uncertainty into this baseline, analyzing how each affects the evolution of the state distribution:
- S1: Initial Condition Uncertainty: Replacing the point-valued $x_0$ with a probability distribution $p_0$ . The evolution is governed by the Liouville equation.
- S2: Process Noise: Replacing the ODE with a Stochastic Differential Equation (SDE) where noise is injected continuously. The evolution is governed by the Fokker-Planck equation.
- S3: Partial Observations: Introducing an observation equation $y_k = h(x(t_k)) + v_k$ . This leads to the filtering problem, where uncertainty is updated via predict-update cycles (e.g., Kalman filters).
- S4: Parameter Uncertainty: Treating the parameters $\theta$ of the evolution law $f_\theta$ as unknown. This uncertainty propagates systematically through time, compounding errors and potentially altering the qualitative behavior (e.g., bifurcations) of the system.
- S5: Structural Uncertainty: Addressing cases where the model form itself is incorrect (e.g., wrong functional form, missing state variables, or wrong dynamics type).
Analyzing Interactions: The authors discuss how these sources compound and how their relative dominance shifts over different prediction horizons (short-term dominated by initial conditions, long-term by parameters and structure).
Task-Dependent Classification: The paper evaluates the aleatoric/epistemic nature of these sources based on specific tasks (forecasting, system identification, control, verification).

Key Contributions

1. A Taxonomy of Dynamical System Uncertainties

The paper provides a structured breakdown of uncertainty sources specific to dynamical systems, distinguishing them from static ML settings. A key insight is the temporal nature of uncertainty propagation:

Initial condition uncertainty is injected once and transformed by the flow.
Process noise is injected continuously.
Parameter uncertainty acts as a systematic bias that compounds over time, influencing the entire trajectory history.

2. Context-Dependent Classification of Uncertainty Types

The authors challenge the notion that a source is inherently aleatoric or epistemic. They demonstrate that the classification depends on modeling choices:

Aleatoric vs. Epistemic Initial Conditions: If $x_0$ is a fixed but unknown quantity estimated from imperfect sensors, the uncertainty is epistemic. If $x_0$ is a random variable representing natural variability (e.g., a human releasing a pendulum differently each time), it is aleatoric.
Structural Uncertainty: The paper highlights that structural uncertainty (uncertainty about "the right model") is qualitatively distinct. Unlike parameter uncertainty, which can be reduced within a model class, structural uncertainty involves "unknown unknowns" that cannot be resolved by additional data within a fixed hypothesis space.

3. Task-Driven Necessity of Uncertainty Quantification

The paper argues that the utility of disentangling aleatoric and epistemic uncertainty is task-dependent:

Forecasting/Filtering: The primary object of interest is the state $x(t)$ .
Control: The distinction is operationally critical. Epistemic uncertainty has a "value of information" and can be reduced by acting on the system (exploration), whereas aleatoric uncertainty cannot.
Verification: Uncertainty is translated into reachable sets to guarantee safety.

4. Modeling Perspectives

The authors review existing and potential modeling approaches:

Probability Theory: Standard distributions (first-order) capture aleatoric uncertainty well. Second-order distributions (distributions over distributions) are proposed for epistemic uncertainty but may be computationally challenging and conceptually unnecessary if the ground truth is a precise but unknown quantity.
Set-Based Representations: The paper notes the utility of differential inclusions and credal sets (sets of probability distributions) for representing epistemic uncertainty, arguing they may be more suitable for capturing ignorance than probability distributions in certain contexts.

Results and Findings

As a conceptual paper, the "results" are theoretical insights and clarifications:

Horizon-Dependent Dominance: The paper establishes that different uncertainty sources dominate at different time scales. Initial conditions dominate short horizons; process noise accumulates over intermediate horizons; and parameter/structural uncertainty dominates long horizons, determining the system's attractor.
Qualitative Shifts: Parameter uncertainty can lead to qualitative changes in system behavior (e.g., distinguishing between stable equilibrium and chaos) rather than just quantitative deviations.
Limitations of Standard Probabilistic Models: The paper suggests that standard probabilistic models in ML may be insufficient for dynamical systems if they fail to distinguish between reducible and irreducible uncertainty, particularly in control settings.

Significance and Claims

The paper claims significance in providing a machine learning perspective on uncertainty modeling for dynamical systems, a field that has been studied less than supervised learning.

Clarification of Concepts: It clarifies that the aleatoric-epistemic dichotomy is not a fixed property of a data source but a result of the modeling setup and the specific question being asked.
Bridging Fields: It highlights the two-way relationship between ML and dynamical systems, noting how ML provides flexible function approximators (e.g., Neural ODEs) for dynamical systems, while dynamical systems provide frameworks (e.g., filtering, reachability analysis) for ML.
Future Directions: The paper identifies opportunities for future work, including:
- Adapting second-order methods from supervised learning to the trajectory level of dynamical systems.
- Developing new uncertainty quantification measures that focus on derived quantities (e.g., time to stabilization, probability of entering a region) rather than just state uncertainty.
- Addressing the difficulty of representing structural uncertainty ("unknown unknowns").

The authors maintain a modest stance, framing the work as a perspective and a call to revisit fundamental assumptions rather than a proposal of a specific new algorithm or a demonstration of superior performance on benchmarks.