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arxiv: 2606.27285 · v2 · pith:GPKUPKAJnew · submitted 2026-06-25 · 💻 cs.LG · cs.IT· math.CA· math.DS· math.IT

Recovering Governing Equations from Solution Data: Identifiability Bounds for Linear and Nonlinear ODEs

Yang Pan , Helmut B\"olcskei This is my paper

Pith reviewed 2026-06-29 04:38 UTC · model grok-4.3

classification 💻 cs.LG cs.ITmath.CAmath.DSmath.IT
keywords governing equationsODE identificationidentifiability boundsHausdorff distancesample complexityscientific machine learningmetric entropy
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The pith

The Hausdorff distance on solution sets determines when governing ODEs can be uniquely identified from data.

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

The paper introduces the Hausdorff distance on solution sets to compare differential equations by their worst-case separation over initial conditions. It establishes identifiability bounds for linear ODEs and nonlinear ones with Lipschitz or Hölder continuous vector fields, showing precisely when two equations can be distinguished from solution observations. The work then derives metric entropy estimates for these classes and provides sample complexity bounds for reliable recovery of the governing equation.

Core claim

We introduce the Hausdorff distance on solution sets as the natural metric for comparing differential equations, since it captures the worst-case separation between two equations over all admissible initial conditions. We establish identifiability bounds for governing ODEs ranging from linear to nonlinear classes with Lipschitz (Hölder)-continuous vector fields, characterizing when two distinct equations can be distinguished from solution data, and derive metric entropy estimates and sample complexity bounds.

What carries the argument

Hausdorff distance on solution sets, which measures the maximum separation between trajectories of two ODEs over all initial conditions.

If this is right

  • Positive Hausdorff distance between solution sets implies the equations are distinguishable given enough solution observations.
  • Metric entropy estimates bound the number of distinguishable ODEs in a class, directly controlling the sample size needed for identification.
  • Sample complexity bounds quantify the number of observed trajectories required to recover the governing equation with high probability.
  • The bounds hold uniformly for linear ODEs and for nonlinear ODEs whose vector fields satisfy Lipschitz or Hölder continuity.

Where Pith is reading between the lines

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

  • The same distance could be used to derive identifiability results for systems with time-varying or control inputs.
  • Practical recovery algorithms might use the derived entropy numbers to set stopping criteria or regularization strengths.
  • The minimax character of the Hausdorff metric suggests that worst-case initial-condition sampling is necessary for robust identification guarantees.

Load-bearing premise

The Hausdorff distance on solution sets is the natural metric for comparing differential equations because it captures the worst-case separation between two equations over all admissible initial conditions.

What would settle it

Observing two distinct ODEs whose solution sets have small Hausdorff distance yet yield data that allows reliable distinction under some sampling of initial conditions would contradict the metric's claimed suitability for the identification problem.

Figures

Figures reproduced from arXiv: 2606.27285 by Helmut B\"olcskei, Yang Pan.

Figure 1
Figure 1. Figure 1: Lipschitz functions f and ˜f We have [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Red points represent the randomly generated samples. Yellow and blue [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Red points represent the randomly generated samples. Yellow and blue [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
read the original abstract

Learning governing equations from observed solution data is a fundamental challenge in scientific machine learning, yet the theoretical conditions under which a ground-truth ODE can be uniquely and stably identified from multiple solution observations remain largely undeveloped, and no quantitative analysis of the sample complexity of such learning tasks exists in the literature. To address this gap, we introduce the Hausdorff distance on solution sets as the natural metric for comparing differential equations, since it captures the worst-case separation between two equations over all admissible initial conditions and thus encodes the minimax structure of the identification problem. We establish identifiability bounds for governing ODEs across a wide class of structure equations--ranging from linear ODEs to nonlinear classes with Lipschitz (H\"older)-continuous vector fields--characterizing precisely when two distinct equations can be distinguished from solution data. Using this metric, we derive metric entropy estimates for the relevant ODE classes and analyze sample complexity bounds, quantifying how many solution observations are needed to reliably recover the governing equation.

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

2 major / 1 minor

Summary. The paper introduces the Hausdorff distance on solution sets as the natural metric for comparing differential equations and uses it to establish identifiability bounds for governing ODEs ranging from linear systems to nonlinear classes with Lipschitz or Hölder-continuous vector fields. It characterizes when two distinct equations can be distinguished from solution data, derives metric entropy estimates for the relevant ODE classes, and analyzes sample complexity bounds quantifying the number of solution observations needed to recover the governing equation.

Significance. If the central constructions hold with the required restrictions, the work would provide the first quantitative identifiability and sample-complexity theory for ODE recovery from solution data, directly addressing a gap in scientific machine learning. The minimax framing via Hausdorff distance on solution sets is a coherent choice for the identification problem.

major comments (2)
  1. [Section 2 (definition of the metric)] The definition of the Hausdorff distance between solution sets (introduced to encode the minimax identification problem) does not specify a fixed finite time horizon T or a compact domain for admissible initial conditions. Without these, trajectories of distinct Lipschitz or Hölder fields can diverge arbitrarily far, rendering the Hausdorff distance infinite and the subsequent metric entropy and sample-complexity estimates vacuous. This is load-bearing for all claimed bounds.
  2. [Section 3 (nonlinear Hölder case)] For the Hölder-continuous case with exponent <1, uniqueness of solutions may fail, making the solution set multi-valued; the Hausdorff comparison then requires additional selection rules or Filippov regularization that are not addressed, undermining the identifiability claims for that class.
minor comments (1)
  1. Notation for the structure equation classes and the precise function spaces (e.g., C^{0,α} vs. Lip) should be stated explicitly at first use to avoid ambiguity in the entropy calculations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments identify important technical points that must be made fully explicit. We address each below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Section 2 (definition of the metric)] The definition of the Hausdorff distance between solution sets (introduced to encode the minimax identification problem) does not specify a fixed finite time horizon T or a compact domain for admissible initial conditions. Without these, trajectories of distinct Lipschitz or Hölder fields can diverge arbitrarily far, rendering the Hausdorff distance infinite and the subsequent metric entropy and sample-complexity estimates vacuous. This is load-bearing for all claimed bounds.

    Authors: We agree that a fixed finite horizon T and a compact set of admissible initial conditions are required for the Hausdorff distance to be finite and for the subsequent entropy and sample-complexity results to be meaningful. While these restrictions are implicit in the problem formulation (standard for global existence and boundedness arguments in ODE theory), they are not stated explicitly in Section 2. We will add a precise statement of the standing assumptions on T and the initial-condition domain at the beginning of Section 2, together with a short remark confirming that all subsequent bounds are understood to hold under these restrictions. This change will be made in the revised manuscript. revision: yes

  2. Referee: [Section 3 (nonlinear Hölder case)] For the Hölder-continuous case with exponent <1, uniqueness of solutions may fail, making the solution set multi-valued; the Hausdorff comparison then requires additional selection rules or Filippov regularization that are not addressed, undermining the identifiability claims for that class.

    Authors: The referee correctly notes that for Hölder exponents α < 1 the classical Picard–Lindelöf theorem does not guarantee uniqueness, so the solution set may be multi-valued. The manuscript treats the solution set as the collection of all absolutely continuous curves satisfying the integral equation; the Hausdorff distance is then taken between these (possibly multi-valued) reachable sets. This construction remains well-defined without additional selection rules. Nevertheless, to avoid any ambiguity we will insert a clarifying paragraph in Section 3 that (i) recalls the possible non-uniqueness for α < 1, (ii) states that the Hausdorff metric is applied directly to the full reachable sets, and (iii) notes that the identifiability and entropy bounds continue to hold for the set-valued case. This revision will be included. revision: yes

Circularity Check

0 steps flagged

No circularity; metric definition followed by independent mathematical derivation

full rationale

The paper defines the Hausdorff distance on solution sets as its comparison metric for ODEs and then derives identifiability bounds, metric entropy estimates, and sample-complexity results from that definition across linear and nonlinear classes. No self-citations appear in the provided text, no parameters are fitted to data and then relabeled as predictions, and no step reduces by construction to its own inputs. The derivation chain is therefore self-contained: the metric is introduced explicitly, and subsequent bounds follow from standard functional-analytic arguments on the chosen function classes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the assumption that the Hausdorff metric captures the minimax structure of the identification problem; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Hausdorff distance on solution sets is the natural metric for comparing differential equations as it captures the worst-case separation over all admissible initial conditions
    Explicitly stated in the abstract as the natural metric encoding the minimax structure.

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