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arxiv: 2606.10045 · v1 · pith:IAPVGYZTnew · submitted 2026-06-08 · 📡 eess.SY · cs.SY· math.OC

A constrained symbolic regression approach for Lyapunov function discovery

Ilias Mitrai , Wentao Tang This is my paper

Pith reviewed 2026-06-27 15:13 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords Lyapunov functionssymbolic regressiondynamical systemsstability analysisexpression treesbranch-and-bounddata-driven discoveryautonomous systems
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The pith

Constrained symbolic regression discovers valid Lyapunov functions for autonomous dynamical systems without assuming their form.

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

The paper sets up Lyapunov function discovery as a constrained self-supervised symbolic regression task where candidate functions are represented as fixed-depth expression trees. Constraints inside the objective enforce that the function is positive definite and that its derivative along system trajectories is negative, directly encoding the stability certificate requirements. This removes the need for any pre-chosen functional form and yields an explicit symbolic expression that can be inspected. A branch-and-bound-and-check algorithm solves the resulting combinatorial problem, and the method is shown to succeed on multiple case studies of continuous autonomous systems.

Core claim

The central claim is that formulating Lyapunov function discovery as a constrained self-supervised symbolic regression problem with expression trees of fixed depth and solving it with a branch-and-bound-and-check procedure discovers valid Lyapunov functions for autonomous dynamical systems without a priori assumptions on functional form, as demonstrated on several case studies.

What carries the argument

Constrained symbolic regression on fixed-depth expression trees whose objective directly encodes both function values and the two Lyapunov stability conditions, solved by branch-and-bound-and-check.

If this is right

  • The approach applies in principle to any continuous dynamical system.
  • Discovered functions appear in explicit symbolic form and are therefore directly interpretable.
  • No assumption about the shape of the Lyapunov function is required before the search begins.
  • The branch-and-bound-and-check procedure makes the combinatorial search tractable for the tested examples.

Where Pith is reading between the lines

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

  • The same constraint-encoding idea could be tried on systems that include control inputs or external disturbances.
  • Hybrid schemes that first use the symbolic method on low-dimensional subsystems and then refine with other techniques become conceivable.
  • Automated generation of stability certificates might shorten the design loop in control applications where manual Lyapunov construction is currently required.

Load-bearing premise

The two Lyapunov stability conditions can be written as hard constraints inside the symbolic regression objective without ruling out all workable functions or requiring separate verification that the returned expression works everywhere.

What would settle it

A concrete continuous autonomous system that is known to possess a Lyapunov function but for which the solver returns either no feasible expression or an expression whose derivative along trajectories is positive at some tested point.

Figures

Figures reproduced from arXiv: 2606.10045 by Ilias Mitrai, Wentao Tang.

Figure 1
Figure 1. Figure 1: Symbolic expression tree representation of two Lyapunov functions. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Symbolic expression tree representation of the Lyapunov function [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Learning trajectory for case 1 strategy of the solver. In such cases, one can add integer cuts to exclude certain functional forms for the Lyapunov function and resolve the problem. 4. Computational results In this section, we use the proposed approach to discover Lyapunov functions for various dynamical systems and explore its computational efficiency across different numbers of data points and objectives… view at source ↗
Figure 5
Figure 5. Figure 5: Sign of dV/dt for the identified Lyapunov function V(x1, x2) = 2x 2 1 + 8x 2 2 for the dynamical system in case 1. First, we set the depth of the tree equal to 1 and the prob￾lem has 419 variables (12 binary) and 1828 constraints. Gurobi declares this problem as infeasible after presolve. Next, we increase the depth of the tree to two (seven nodes in the ex￾pression tree) and the problem is declared infeas… view at source ↗
Figure 6
Figure 6. Figure 6: Learning trajectory for case 1 using snapshots from 10 initial con [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: State trajectories for case 2. identified in Section 4.1. For NC = 13 the identified function is x 2 1 + 4x 2 2 whereas for NC = 11 the function is 1.25x 2 1 + 5x 2 2 . Comparing the solution time, we observe that as the maximum number of operators decreases, the solution time also decreases since the search space is more constrained. We note that if the tree depth is 2, which corresponds to a maximum of 7… view at source ↗
read the original abstract

In this paper, we consider the data-driven discovery of Lyapunov functions for autonomous dynamical systems. We represent the Lyapunov function as an expression tree of fixed depth and formulate the Lyapunov discovery task as a constrained self-supervised symbolic regression problem. The constraints model the output of the Lyapunov function for a given input as well as the Lyapunov stability conditions. This modeling approach makes no a priori assumptions about the functional form of the Lyapunov function, is inherently interpretable since the function is obtained in a symbolic form, and, in principle, can be applied to any continuous dynamical system. We also develop a tailored branch-and-bound-and-check solution approach to efficiently solve the resulting learning task. Applications to several case studies show the ability of the proposed approach to discover Lyapunov functions.

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 manuscript proposes a constrained symbolic regression approach for discovering Lyapunov functions in autonomous dynamical systems. Lyapunov functions are represented as expression trees of fixed depth, and the discovery is cast as a constrained self-supervised symbolic regression problem where constraints encode the function values and the Lyapunov stability conditions (positive definiteness and negative derivative along trajectories). A branch-and-bound-and-check solver is developed to solve the resulting optimization problem, and the method is demonstrated on several case studies.

Significance. Should the approach reliably produce functions that satisfy the Lyapunov conditions globally, it would represent a notable advance in data-driven methods for stability analysis, eliminating the need for a priori functional form assumptions and providing interpretable symbolic expressions. The development of a tailored solver for the constrained problem is a technical strength.

major comments (2)
  1. [Abstract] Abstract: The central claim that the method discovers valid Lyapunov functions (i.e., satisfying V(x)>0 and ˙V(x)<0 everywhere in a region) rests on the encoding of stability conditions as constraints. However, the abstract provides no information on whether these constraints are enforced only at discrete sample points or guaranteed symbolically over the continuum; satisfaction at samples does not imply global validity for continuous systems, which is load-bearing for the no-post-hoc-verification assertion.
  2. [Abstract] Abstract: No quantitative metrics (e.g., verification success rates over dense grids, comparison against baselines such as sum-of-squares or neural Lyapunov methods) are reported for the case studies. This absence makes it impossible to assess whether the discovered expressions actually meet the Lyapunov conditions beyond the training samples, directly undermining the claim of effective discovery.
minor comments (1)
  1. The abstract could briefly specify the dynamical systems used in the case studies (e.g., dimension, nonlinearity type) to contextualize the demonstrations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the abstract. We address each point below and will revise the manuscript accordingly to improve clarity and strengthen the presentation of results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the method discovers valid Lyapunov functions (i.e., satisfying V(x)>0 and ˙V(x)<0 everywhere in a region) rests on the encoding of stability conditions as constraints. However, the abstract provides no information on whether these constraints are enforced only at discrete sample points or guaranteed symbolically over the continuum; satisfaction at samples does not imply global validity for continuous systems, which is load-bearing for the no-post-hoc-verification assertion.

    Authors: We agree that the abstract should clarify the nature of the constraints. In the proposed approach, the Lyapunov conditions are encoded as constraints evaluated at a finite collection of sampled points in the state space. The optimization therefore enforces the conditions only at these discrete locations rather than providing a symbolic guarantee over the continuum. We will revise the abstract to explicitly note this sample-based enforcement and indicate that global validity requires separate verification (e.g., on a dense grid). This revision will also remove any implication of automatic global validity without post-discovery checks. revision: yes

  2. Referee: [Abstract] Abstract: No quantitative metrics (e.g., verification success rates over dense grids, comparison against baselines such as sum-of-squares or neural Lyapunov methods) are reported for the case studies. This absence makes it impossible to assess whether the discovered expressions actually meet the Lyapunov conditions beyond the training samples, directly undermining the claim of effective discovery.

    Authors: We concur that quantitative evaluation metrics are needed to substantiate the claims. In the revised version we will augment the case-study section with verification success rates computed on dense grids (distinct from the training samples) and will include direct numerical comparisons against sum-of-squares and neural-Lyapunov baselines for the reported examples. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method is a novel constrained optimization formulation

full rationale

The paper defines Lyapunov discovery as a constrained self-supervised symbolic regression task on fixed-depth expression trees, with stability conditions encoded directly as constraints in the objective and solved via a custom branch-and-bound-and-check procedure. No derivation step reduces the claimed discovery of valid Lyapunov functions to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz smuggled from prior work by the same authors. The central claim rests on the solver's ability to produce symbolic expressions satisfying the encoded inequalities at the modeled points, which is independent of the target stability properties and does not presuppose the functional form or the final result. This is a standard new-method presentation with no load-bearing self-referential reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on the assumption that a fixed-depth expression tree can represent a valid Lyapunov function for the target systems and that the stability conditions can be turned into tractable constraints; these are domain assumptions rather than derived quantities.

free parameters (1)
  • expression tree depth
    The depth is fixed in advance; its specific value is a modeling choice that determines expressivity and search cost.
axioms (1)
  • domain assumption The dynamical system is autonomous and continuous
    Required for the Lyapunov conditions to be well-defined as stated in the abstract.

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Forward citations

Cited by 1 Pith paper

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