PaperlayerView exact revision on arXiv

Source-linked AI summary

From Flow-Box Invariants to Global Conservation Laws: Machine Learning-Assisted Detection of Singularities and Near-Integrability

Jimmie Adriazola, Ioannis G. Kevrekidis, Panayotis G. Kevrekidis, Wei Zhu

arXiv:2610.23760v1nlin.CDcs.LG

TL;DR

Fixed-region conservation-law discovery leaves unresolved how locally learned laws persist as phase-space domains expand. This paper applies neural deflation over nested domains to track independent-law counts; experiments recover local laws and detect their successive breakdown, while persistence scales distinguish dynamical regimes.

  • Problem

    Fixed-region data-driven methods do not reveal how conservation laws persist or break down as the phase-space domain expands.

    Source: §I · p. 3

  • Method

    The framework applies neural deflation over nested expanding domains and uses conservation and deflation losses to track the number of independent laws.

    Source: §I · p. 4

  • Results

    The experiments recover flow-box local laws and detect successive breakdown as domains expand, with the second candidate’s persistence scale differentiating Hénon–Heiles regimes.

    Source: §2 · p. 30, §V · p. 39

  • Takeaways & Limitations

    The framework distinguishes local laws from persistent global structure and uses domain-scale persistence to study proximity to integrable behavior.

    Source: §I · p. 4

    For nearby non-integrable cases ε = −0.3 and ε = −0.2, the second candidate’s eventual breakdown was not numerically resolved over the tested box sizes.

    Source: §1 · p. 26

Abstract

from arXiv · show

Conservation laws reveal invariant structure in dynamical systems and constrain their evolution to lower-dimensional level sets. Locally, however, per the flow-box theorem, every $d$-dimensional system generically admits $d-1$ functionally independent conservation laws near a regular point even though, generically, only a few such laws persist globally. Data-driven methods typically learn conservation laws on a prescribed region of phase space from which training samples are drawn, leaving their behavior beyond that region unresolved. We formulate this local-to-global continuation problem and develop an {expanding-domain neural deflation} framework that tracks the number of independent conservation laws over nested domains. The method recovers the expected $d-1$ local invariants and detects their successive breakdown as the domain expands, relaxing to the expected global conservation law limit. It distinguishes the local invariants (associated with Frobenius integrability) from persistent global structure and shows that the domain scale over which additional laws survive reflects proximity to Liouville integrability. Pointwise losses can localize isolated obstructions, whereas in more complicated dynamics the breakdown of our loss functions is gradual and spatially diffuse. The framework thus provides a data-driven approach for studying the persistence and breakdown of (so-far-learned) conservation laws across phase space.

I. INTRODUCTION

The introduction frames conservation-law discovery as a local-to-global problem: fixed-region methods cannot show how learned laws persist as phase-space domains grow. The flow-box theorem guarantees d −1 independent local laws near regular points, but these need not represent global structure.

I. INTRODUCTION

  • Fixed-region data-driven methods identify laws only where training samples provide coverage, leaving their persistence and breakdown on larger domains unresolved.The introduction asks whether a complete local family can be recovered and tracked as the domain expands, including where individual laws first fail.Source: §I · pp. 3–4

B. The flow-box theorem and local conservation laws

  • Near every regular point, the flow-box theorem guarantees d −1 functionally independent local conservation laws, regardless of Hamiltonian structure or global integrability.These laws follow from the local geometry of a nonvanishing vector field and do not by themselves indicate global structure.Source: §I · p. 3, B. The flow-box theorem and local… · pp. 6–7

I. INTRODUCTION

  • A local law may fail to continue smoothly, become dependent on other laws, or encounter an equilibrium, singular set, recurrence, or other global obstruction as the domain grows.Conversely, some laws may persist over larger domains and extend globally.Source: §I · p. 3
  • The proposed expanding-domain neural deflation framework tracks independent-law counts, distinguishing the local laws from those persisting over larger domains.Its experiments report recovery of flow-box local laws and changes in persistence scale across integrable, near-integrable, and non-integrable regimes.Source: §I · p. 4

C. The local-to-global continuation problem

The paper formulates continuation as tracking the maximum number of independent conservation laws over nested domains. It extends neural deflation across those domains to estimate this count and study where local laws cease to persist.

C. The local-to-global continuation problem

  • A local invariant can fail to extend smoothly to a larger domain, and a family can lose functional independence even when its individual laws continue.The flow-box theorem establishes local existence but does not determine how far those laws continue.Source: C. The local-to-global continuation… · p. 7

1. How many functionally independent conservation laws exist on each domain Ur?

  • The continuation problem is to approximate the number of independent laws as domain size changes and investigate when decreases occur and where laws first fail.The study also asks whether critical domain sizes can be interpreted through system dynamics and which local laws persist across the phase-space region of interest.Source: C. The local-to-global continuation… · p. 8, §1 · p. 8

D. Related work

  • Prior data-driven discovery methods use fixed datasets or prescribed phase-space regions; the paper applies neural deflation over nested domains to track law persistence and breakdown.The authors state these methods do not distinguish laws that exist only locally from those extending globally.Source: D. Related work · p. 10

III. NEURAL DEFLATION OVER EXPANDING DOMAINS

  • The adapted neural deflation method uses gradient-based deflation to seek a maximal independent family on each domain, without requiring Hamiltonian or Poisson structure.It omits pairwise involution constraints, targeting the d −1 local invariants predicted by the flow-box theorem rather than a family in involution.Source: §III · pp. 10–11

A. Learning a conservation law on a fixed domain

On a fixed domain, the method learns candidate conservation laws from vector-field evaluations and uses neural deflation to discourage later candidates from reproducing earlier laws. This enables a sequential search for independent laws rather than repeated recovery of the same invariant.

A. Learning a conservation law on a fixed domain

  • The fixed-domain procedure assumes vector-field evaluations at sampled points and draws separate training and validation sets from a distribution supported on the domain.The distribution is typically uniform.Source: A. Learning a conservation law on a… · p. 11
  • A small validation loss indicates that the first trained candidate approximately satisfies the conservation equation over the domain.A conservation loss alone cannot ensure a complete family because later networks may learn functions of an already identified invariant.Source: A. Learning a conservation law on a… · p. 12

B. Functional independence through neural deflation

  • Neural deflation penalizes functional dependence by assessing whether a candidate gradient adds a component beyond the span of previously learned gradients.The pointwise independence factor vanishes when the new gradient lies in that span, including when the candidate is locally a smooth function of earlier laws.Source: B. Functional independence through… · pp. 12–13
  • The deflated objective combines conservation and independence losses, while the method omits pairwise involution constraints to target independent laws for general vector fields.The exponent α controls the strength of the dependence penalty near dependent configurations.Source: B. Functional independence through… · p. 13

C. Estimating the number of conservation laws

The method estimates how many independent conservation laws a domain supports from validation-loss increases across sequential candidates. Repeating this procedure over nested domains uses domain size as a continuation parameter, while numerical persistence remains evidence rather than proof of global validity.

C. Estimating the number of conservation laws

  • A pronounced validation-loss increase is expected once the method attempts to learn more independent laws than the domain supports, so the preceding candidate index estimates the count.The method compares both loss magnitudes and their separation across successive candidates rather than imposing a universal absolute threshold.Source: C. Estimating the number of… · pp. 14–15

D. Continuation over expanding domains

  • Training and validation on each nested domain produces an estimated independent-law count for every domain size.The domain family may use rectangular regions or Hamiltonian energy sublevel sets; the latter respects Hamiltonian geometry and can parameterize continuation by energy.Source: D. Continuation over expanding domains · p. 15
  • A decrease in the estimated count indicates that some smaller-domain law cannot extend as part of an independent family, but persistence over tested domains does not rigorously establish global validity.For sufficiently small domains centered at a regular point, the count is predicted to be d −1.Source: D. Continuation over expanding domains · pp. 15–16

E. Average, maximal, and pointwise diagnostics

Pointwise losses complement the estimated number of conservation laws by showing where conservation or independence fails. Across nested domains, the procedure estimates how many independent laws persist and examines whether their breakdown can be localized.

E. Average, maximal, and pointwise diagnostics

  • Large conservation loss marks where a candidate ceases to be constant along the vector field, while a small independence factor marks nearly dependent gradients.The deflated loss combines these effects; inspecting them separately can clarify a breakdown.Source: E. Average, maximal, and pointwise… · p. 16
  • High-loss regions can concentrate near an equilibrium or isolated singularity, but oscillatory recurrences may leave the breakdown location unresolved along the critical r-shell.The critical domain radius may still be identified even when the specific location cannot.Source: E. Average, maximal, and pointwise… · p. 16

F. Summary of the expanding-domain procedure

  • For each nested domain, the procedure samples training and validation points, learns candidates sequentially with deflated loss, and estimates the law count from the first candidate with a pronounced validation-loss increase.Comparisons across domain sizes detect decreases in the estimated count and may help localize an obstruction.Source: F. Summary of the expanding-domain… · pp. 16–17

IV. NUMERICAL EXPERIMENTS

  • The experiments progress from elementary systems with analytically known laws to integrable and non-integrable Hénon–Heiles regimes and the non-integrable double pendulum.On each domain, validation losses estimate the supported law count, with pointwise values examined when a loss increases.Source: §IV · p. 17

A. Network architecture and training setup

Candidate conservation laws are represented by fully connected neural networks and trained independently on sampled nested domains. Validation losses assess candidate validity and independence, while maximal and pointwise losses are used to detect localized failures.

A. Network architecture and training setup

  • Each candidate is a scalar-output network with four hidden layers of width 64, SiLU activations, Xavier-uniform weight initialization, and zero biases.The output layer is linear.Source: A. Network architecture and training… · p. 17
  • For each domain U_r ⊂ R^d, independent training and validation sets each contain 50,000 uniformly drawn points; candidates train for 1,000 epochs with Adam at learning rate 10^-3.Network gradients with respect to phase variables are evaluated using automatic differentiation.Source: A. Network architecture and training… · p. 18
  • The deflation exponent is α = 1 in all experiments, with α = 0.5 reported to yield qualitatively similar results.Validation averages measure overall validity, while maximal and pointwise values help detect localized failures.Source: A. Network architecture and training… · p. 18
  • Because continuous sampling does not hit isolated equilibria or other measure-zero sets exactly, empirical losses assess conservation and independence almost everywhere.This is stated to be consistent with Theorem 2.Source: A. Network architecture and training… · p. 18

B. Elementary two- and three-dimensional systems

Elementary examples test whether expanding-domain neural deflation recovers local laws and distinguishes their failure to continue from globally valid laws that pass through equilibria. The examples show that an equilibrium can obstruct a local law in node systems while global invariants persist across equilibria in center systems.

B. Elementary two- and three-dimensional systems

  • The elementary systems are used to test recovery of flow-box local laws, detection of their loss as domains expand, and discrimination between continuation obstructions and equilibria across which a global law remains valid.The systems' relevant local and global laws are stated to be analytically known.Source: B. Elementary two- and… · p. 18
  • The 2D node has no nonconstant global conservation law on R^2, although a smooth local law can be expressed as a function of polar angle on a simply connected domain excluding the origin.The pointwise loss remains small for r < 2 and jumps near the origin once r > 2.Source: B. Elementary two- and… · pp. 18, 21
  • The 2D center's energy x^2 + y^2 remains a global conservation law across the origin, despite the origin being an equilibrium and a critical point of the invariant.Its pointwise loss remains small after the expanding box contains the origin.Source: B. Elementary two- and… · pp. 19–20
  • For the 2D node, the local law y/x has small loss for r = 1.5 and r = 1.9, then develops high loss near the origin after the box expands past r = 2.The origin is identified as the obstruction to continuing this law over the enlarged domain.Source: B. Elementary two- and… · p. 20, Figure 1 · p. 21
  • For the 3D degenerate node, the global law z persists past r = 2, while the additional local law y/x breaks down near the equilibrium line and the independent-law count falls from two to one.The pointwise deflated losses show the contrast on the z = 0 slice.Source: B. Elementary two- and… · p. 20, Figure 2 · p. 22
  • For the 3D degenerate center, both x^2 + y^2 and z remain globally defined, and both learned candidates retain small losses when the box intersects the equilibrium line.The gradients of the two global laws are independent away from that line and hence almost everywhere.Source: B. Elementary two- and… · pp. 19–21
  • Together, the node and center examples show that an equilibrium can obstruct continuation of an angular law without obstructing a global energy law or the global law z.In three dimensions, the additional law breaks down for the degenerate node but persists for the degenerate center.Source: B. Elementary two- and… · pp. 21–22

C. Hénon–Heiles system

The Hénon–Heiles experiments use expanding rectangular boxes and energy sublevel sets to track when learned conservation laws cease to persist. The loss patterns distinguish the local three-law regime from longer-lived structure, with persistence depending on proximity to the integrable parameter value.

Rectangular boxes growing from a regular point

  • At r = 1, rectangular boxes first meet the equilibrium and the third candidate’s loss rises, reducing the recovered conservation laws from three to two in both integrable and non-integrable cases.Before this transition, all three candidates have small, comparable losses; the transition is essentially independent of ε.Source: Figure 5 · p. 25, §1 · p. 26
  • For ε = −1/3, the two remaining candidates retain small losses as rectangular boxes expand, while near-integrable cases can retain the second candidate beyond the tested range.For ε = −0.1, the second candidate’s loss rises substantially while the Hamiltonian remains accurate; the loss separation supports non-integrability but is not a precise detector.Source: §1 · pp. 26–28

Energy sublevel sets growing from the origin

  • For ε = −1/3, two candidates remain accurate across the tested energy sublevels, but at ε = −0.3 and −0.27 their eventual breakdown is not resolved.Thus, energy sublevels do not sharply detect loss of integrability near ε = −1/3.Source: §2 · p. 29, Figure 7 · p. 30

Domain-expansion comparison

  • Rectangular boxes recover three local laws before the equilibrium transition, whereas energy sublevels from the equilibrium directly expose how second-candidate persistence depends on proximity to integrability.The second candidate’s breakdown need not coincide with the first hyperbolic equilibrium or occur within the tested domain range.Source: C. Hénon–Heiles system · p. 24, §2 · p. 29

Energy sublevel sets growing from the origin

  • Energy sublevel sets show the second candidate’s loss first becoming clear at ε = −0.25, with breakdown at progressively smaller normalized energies farther from ε = −1/3.At ε = −0.1, the second candidate already has a large loss at the smallest tested domains; the third candidate has a large loss from the outset because the domains contain the origin.Source: §2 · pp. 28–29, Figure 7 · p. 30

Conservation loss near the integrable limit

  • The energy-domain loss benchmark Λ(r;ε) varies by less than a factor of 1.5 across tested ε through r = 0.1, but increases about tenfold from r = 0.02 to r = 0.12.The explicit second-integral benchmark’s loss qualitatively parallels the learned second candidate’s behavior, consistent with near-integrable structure.Source: Figure 8 · p. 31, §3 · p. 32

D. Double pendulum

The double-pendulum experiments track three local conservation-law candidates across expanding boxes and energy sublevel sets. The candidates successively break down toward the system’s single global energy law, but the second law’s loss is gradual and does not consistently align with SALI structures.

Rectangular boxes growing from a regular point

  • The Hamiltonian-associated candidate stays near the numerical loss floor, while the third candidate deteriorates rapidly before the box reaches the stable equilibrium at r1 = 0.2.The flow-box theorem guarantees three independent laws only on some sufficiently small neighborhood, not throughout every larger box avoiding an equilibrium.Source: D. Double pendulum · p. 34, Figure 9 · p. 35
  • The three candidates transition to two and eventually one as rectangular boxes expand; the first loss occurs before the stable equilibrium, and the second becomes clearer after the box intersects the separatrix.The second candidate remains accurate through approximately r2 ≈ 0.6325, then gradually deteriorates; neither transition is sharply localized.Source: D. Double pendulum · p. 35

Energy sublevel sets growing from the origin

  • The energy-sublevel experiment starts with a large third-candidate loss because every domain contains the equilibrium, where the flow-box theorem does not predict three independent laws.Source: D. Double pendulum · p. 36
  • On expanding energy sublevel sets, two laws are numerically resolved at the smallest levels, but only the Hamiltonian remains accurate on larger domains.The second candidate’s loss gradually separates from the first; the transition does not identify a sharp energy level for breakdown.Source: D. Double pendulum · p. 36, Figure 10 · p. 37

Pointwise losses and SALI

  • SALI maps reveal increasingly complex regular and chaotic structures, but these structures do not consistently correspond to the second candidate’s pointwise loss patterns.The loss patterns also vary across independent training runs.Source: D. Double pendulum · p. 37, Figure 11 · p. 38
  • The SALI comparison does not locate a sharp boundary for the second law’s failure; domain-averaged losses instead show gradual deterioration toward the global energy law.The breakdown was not sharply localized at the investigation’s resolution.Source: D. Double pendulum · pp. 37–38

V. CONCLUSION AND FUTURE WORK

The expanding-domain framework estimates how the number of independent conservation laws changes as domains grow, connecting local laws to persistent global structure. Experiments recover successive breakdowns, while also showing limits to sharp detection and motivating improved numerical and theoretical tools.

Conclusion

  • Expanding-domain neural deflation uses validation losses to estimate how the number of independent conservation laws changes with domain size.The problem is framed as continuation from the d −1 laws near a regular point to the subset that extends over larger regions.Source: §V · p. 39
  • Experiments recover local laws and detect their successive breakdown, interpolating between the d −1 local count and the asymptotic count for integrable or non-integrable systems.Source: §V · p. 39
  • In the examples, pointwise losses sharply identify some equilibrium obstructions, while the double pendulum shows diffuse breakdown uncorrelated with SALI; the method is not a sharp integrability detector.For Hénon–Heiles, persistence scale differentiates integrable, near-integrable, and non-integrable regimes.Source: §V · p. 39
  • The diagnostics depend on approximation, sampling, and optimization errors, require comparison with an empirical loss floor, and do not prove global extension from persistence on tested domains.Pointwise loss patterns may also vary across independent training runs.Source: §V · p. 39

Future work

  • Future work targets adaptive sampling, more reliable domain expansion, uncertainty estimates, geometric links to dynamical obstructions, and generalization to higher dimensions.Source: §V · pp. 39–40

DATA AVAILABILITY STATEMENT

The study used no external datasets and makes the code for its numerical results openly available on GitHub.

Data availability statement

Appendix A: Proof of Theorem 2

The proof fragment uses nonnegativity of the integrand and conditions on the numerator and denominator to establish a vanishing loss.

Proof of Theorem 2

  • The proof invokes the integrand’s nonnegativity to infer that the loss vanishes when it is zero on a set of full measure.Source: Appendix A · p. 41
  • The proof fragment connects this condition to a conservation law and functional independence of the full collection.Source: Appendix A · p. 41
  • When the numerator vanishes and the denominator is positive almost everywhere, the integrand vanishes almost everywhere.Source: Appendix A · p. 41

Appendix B: Finite-time conservation defect

Appendix B gives a compact-set, fixed-time guarantee for perturbed trajectories at sufficiently small parameter values and visualizes finite-time drift of F across a Poincaré section.

Appendix B: Finite-time conservation defect

  • The squared residual accumulates without cancellation between positive and negative drift.The finite-time defect is evaluated on the section {y = 0, py > 0}∩{H⋆ = 0.04} with T = 100.Source: Appendix B · p. 42, Figure 12 · p. 43
  • For compact Z and fixed T > 0, Theorem 3 guarantees perturbed trajectories exist on [0,T] when |µ| is sufficiently small, with constants uniform over Z.The allowable parameter threshold and constants depend on Z and T.Source: Appendix B · p. 42
  • The displayed calculations use Störmer–Verlet with step size h = 0.02, centered differences for Ḟ, trapezoidal quadrature, and a uniform initial grid in (x0, px,0).Because P = 0 on the section and E = 0.04 < 1/12, these initial conditions have energy E for every perturbed Hamiltonian.Source: Appendix B · p. 43
  • As ε increases across the four panels, low-defect regions move relative to the fixed predictor contours, which visualize where finite-time drift of the second integral is small or large.The panels use ε = −0.330, −0.300, −0.250, and −0.200, and colors show log10 C2 on the perturbed flow.Source: Appendix B · pp. 43–44, Figure 12 · p. 43
Loading 2610.23760v1…