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arxiv: 2606.11988 · v1 · pith:2PSU7UV5new · submitted 2026-06-10 · 💻 cs.LG · stat.ML

What Uncertainties Do We Need for Dynamical Systems?

Yusuf Sale , Christopher B\"ulte , Felix Czaja , Joshua Stiller , Eyke H\"ullermeier This is my paper

Pith reviewed 2026-06-27 10:18 UTC · model grok-4.3

classification 💻 cs.LG stat.ML
keywords dynamical systemsuncertainty quantificationaleatoric uncertaintyepistemic uncertaintymachine learningtime seriescontrol
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The pith

Dynamical systems require distinguishing aleatoric from epistemic uncertainty with task-specific objectives.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines uncertainty sources in dynamical systems through a machine learning lens. It classifies these sources as either aleatoric, arising from inherent randomness in the system, or epistemic, arising from incomplete knowledge about the system. It further shows that the goals for representing and quantifying uncertainty shift depending on the task, such as state prediction, parameter estimation, or controller design. A reader would care because dynamical systems underpin applications from robotics to finance, where mismatched uncertainty handling can produce overconfident or unsafe outputs. The discussion extends the aleatoric-epistemic framework beyond supervised learning to sequential, time-dependent settings.

Core claim

The paper claims that uncertainty modeling for dynamical systems must address multiple sources that fall into aleatoric or epistemic categories, and that the objectives of uncertainty representation and quantification are not fixed but depend on the concrete task, including forecasting, system identification, and control.

What carries the argument

The aleatoric-epistemic uncertainty distinction, separating irreducible randomness from reducible ignorance, applied to sources arising in dynamical systems.

If this is right

  • Prediction tasks require joint quantification of both uncertainty types to produce calibrated trajectory forecasts.
  • Control tasks can leverage epistemic uncertainty for exploration or robustness while treating aleatoric uncertainty as a safety constraint.
  • System identification primarily targets reduction of epistemic uncertainty through targeted data collection.
  • Generative modeling of system trajectories must capture aleatoric variability to reproduce observed stochasticity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same distinction may prove useful in related sequential domains such as reinforcement learning or video prediction.
  • Empirical benchmarks could measure performance drops when uncertainty types are misclassified in dynamical tasks.
  • Hybrid tasks combining prediction and control might benefit from uncertainty representations that adapt their emphasis dynamically.

Load-bearing premise

The aleatoric-epistemic distinction developed for supervised learning transfers to dynamical systems without needing substantial new formal definitions or task-specific redefinitions.

What would settle it

A concrete dynamical system example in which uncertainty sources cannot be classified as aleatoric or epistemic, or in which all tasks share identical uncertainty representation objectives.

Figures

Figures reproduced from arXiv: 2606.11988 by Christopher B\"ulte, Eyke H\"ullermeier, Felix Czaja, Joshua Stiller, Yusuf Sale.

Figure 1
Figure 1. Figure 1: (a) Physical setup. A pendulum of length ℓ and angle θ swings under gravity g with damping γ. It has a stable equilibrium (•) at the bottom and an unstable equilibrium (•) at the top (i.e., the inverted position). (b) Phase portrait of the system in the (θ, ω)-plane. The dynamics exhibit a stable focus (•) at the origin, two saddle points (×) at (±π, 0), and separatrices (—) that divide the phase plane int… view at source ↗
Figure 2
Figure 2. Figure 2: An ensemble of 1000 initial states is drawn from a Gaussian at t = 0 and evolved under the deterministic dynamics. (a) A cloud that stays well inside a separatrix (—) contracts as trajectories spiral toward the stable focus (•), and uncertainty about x(t) shrinks with time, compare t = 0 (•) with t = 14 (•). (b) A cloud straddling a separatrix is torn apart: part of the ensemble enters the oscillation (•) … view at source ↗
Figure 3
Figure 3. Figure 3: (a) Two trajectories from the same initial condition x0 (•) under the same dynamics: the deterministic (—) flow spirals smoothly toward the stable focus (•), while the stochastic (—) trajectory with σ = 0.6 jitters continuously and never settles to a point. (b) The stationary distribution p∞(θ, ω) ( ) obtained from a long stochastic simulation. S2 Process noise. Consider the pendulum (2) subject to additiv… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Time series of the angle: true θ(t) (—), discrete noisy observations yk (•), and the filtered estimate ˆθ(t) (—) with its ±2σ band. The band widens between observations (prediction step) and narrows when a measurement arrives (update step). (b) Time series of the angular velocity: no observations are available, but the filtered estimate ωˆ(t) (—) tracks the true ω(t) (—) by exploiting the relationship … view at source ↗
Figure 5
Figure 5. Figure 5: (a) Three trajectories from the same initial condition x0 (•) under different values of the damping coefficient: γ = 0.15 (—), γ = 0.50 (—), and γ = 1.50 (—). The same system produces qualitatively different paths to the stable focus (•), from many oscillations under light damping to a near-monotonic approach under heavy damping. (b) An ensemble of 60 trajectories from the same x0 with γ ∼ N (0.5, 0.182 ) … view at source ↗
Figure 6
Figure 6. Figure 6: (a) Same initial condition x0 (•) evolved under two different models: the nonlinear pendulum (—) and its linearized version sin θ ≈ θ (– –). Near the origin the two models agree, but at large amplitudes, where the trajectory approaches the separatrix (—), they diverge: the linearization has no separatrix and no notion of full rotation, so it takes a structurally different path through phase space. (b) Angu… view at source ↗
read the original abstract

The distinction between aleatoric and epistemic uncertainty has received considerable attention in machine learning research, mainly in the context of supervised learning but also in other settings such as generative modeling. In this paper, we offer a machine learning perspective on uncertainty modeling for dynamical systems, which has been studied much less so far. In particular, we ask: what uncertainties do we need for dynamical systems? We discuss sources of uncertainty, clarify their nature (aleatoric or epistemic), and consider how the objectives of representing and quantifying uncertainty vary across different tasks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The paper offers a machine learning perspective on uncertainty modeling for dynamical systems. It discusses sources of uncertainty, classifies them as aleatoric or epistemic, and examines how the objectives of representing and quantifying uncertainty vary across different tasks.

Significance. As a discussion piece, the work provides an organizing lens for an area that has received less attention than supervised learning. Its value lies in clarifying distinctions and task-specific objectives rather than new theorems, experiments, or quantitative claims; this framing could help structure future research on uncertainty in dynamical systems if the classification proves useful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the paper's value as a discussion piece, and recommendation to accept. We appreciate the framing of the work as providing an organizing lens for uncertainty in dynamical systems.

Circularity Check

0 steps flagged

No significant circularity

full rationale

This is a purely discursive discussion paper with no derivations, equations, fitted parameters, predictions, or quantitative claims. The central content is a qualitative classification of uncertainty sources in dynamical systems using the aleatoric/epistemic distinction as an organizing lens. No load-bearing step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain. The work is self-contained as conceptual analysis and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a high-level conceptual discussion and introduces no free parameters, mathematical axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5622 in / 948 out tokens · 22411 ms · 2026-06-27T10:18:19.141401+00:00 · methodology

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