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

nlin.CDOct 16, 2026

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

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

Organizations: School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA · Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218, USA · Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003-4515, USA · School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Pointwise conservation loss for the 2D node (Example 1 ) on $U_{r}^{(2)}=[2-r,2+r]\times[-r,r]$ , with $r=1.5,1.9,2.1,2.5$ from left to right. The loss remains small for $r<2$ but jumps at the origin once $r>2$ , where the local conservation law $y/x$ can no longer be continued smoothly.

Panel of Figure 2 — Figure 2: Pointwise losses for the 3D degenerate node (Example 2 ) on the slice $z=0$ of $U_{r}^{(3)}=[2-r,2+r]\times[-r,r]^{2}$ , with $r=1.5,1.9,2.1,2.5$ from left to right. Panels (a)–(d) show the conservation loss, equivalently the deflated loss, for $I_{1}$ , while panels (e)–(h) show the deflated loss for $I_{2}$ . The first conservation law persists for all $r$ , whereas the loss for the second jumps near the equilibrium line once $r>2$ .

Panel of Figure 3 — Figure 3: Pointwise conservation loss for the 2D center (Example 3 ) on $U_{r}^{(2)}=[2-r,2+r]\times[-r,r]$ , with $r=1.5,1.9,2.1,2.5$ from left to right. Unlike the 2D node, the conservation law $x^{2}+y^{2}$ persists after the domain contains the origin.

Panel of Figure 4 — Figure 4: Pointwise losses for the 3D degenerate center (Example 4 ) on the slice $z=0$ of $U_{r}^{(3)}=[2-r,2+r]\times[-r,r]^{2}$ , with $r=1.5,1.9,2.1,2.5$ from left to right. Panels (a)–(d) show the conservation loss, equivalently the deflated loss, for $I_{1}$ , while panels (e)–(h) show the deflated loss for $I_{2}$ . Unlike the 3D degenerate node, both conservation laws persist after the box intersects the equilibrium line.

Panel of Figure 5 — Figure 5: Domain-averaged deflated losses of the three learned candidates for the Hénon–Heiles system on the expanding boxes $U_{r}^{\mathrm{rec}}=(1,1,0,0)+[-r,r]^{4}$ . Panels (a)–(f) correspond to $\epsilon=-1/3,-0.3,-0.2,-0.1,0,$ and $0.1$ , respectively. Three conservation laws are recovered before the box reaches the equilibrium at $r=1$ , after which the third is lost. The two remaining laws persist for the integrable case $\epsilon=-1/3$ . For $\epsilon=-0.3$ and $\epsilon=-0.2$ , no clear loss of the second candidate is observed over the tested range of $r$ , while a pronounced separation between the first and second losses emerges from $\epsilon=-0.1$ onward.

Panel of Figure 6 — Figure 6: Pointwise deflated loss of the third learned conservation law $I_{3}$ for the Hénon–Heiles system on the $p_{x}$ – $p_{y}$ slice $x=y=1-r$ of the boundary of $U_{r}^{\mathrm{rec}}$ . Panels (a) and (b) correspond to $\epsilon=-1/3$ with $r=0.9$ and $r=1.0$ , respectively, while panels (c) and (d) show the corresponding results for $\epsilon=-0.2$ . For both parameter values, the loss is small at $r=0.9$ but increases sharply at $r=1$ , when the boundary first reaches the equilibrium at the origin, marking the breakdown of the third local conservation law.

Panel of Figure 7 — Figure 7: Domain-averaged deflated losses of the three learned candidates for the Hénon–Heiles system on the energy sublevel sets $U_{r(1+\epsilon)}^{\mathrm{en}}(\epsilon)$ growing from the origin. Panels (a)–(i) correspond to $\epsilon=-1/3,-0.3,-0.27,-0.25,-0.24,-0.21,-0.2,-0.1,$ and $0$ , respectively. The horizontal axis is the normalized energy parameter $r$ , with energy level $h=r(1+\epsilon)$ . Two candidates persist throughout the tested range in the integrable case. The breakdown of the second candidate first becomes visible at $\epsilon=-0.25$ and occurs at progressively smaller values of $r$ as $\epsilon$ moves away from $-1/3$ ; by $\epsilon=-0.1$ , it is lost from the smallest tested domain.

Figure 8: Dependence of $\Lambda(r;\epsilon)$ on the normalized energy $r$ for the Hénon–Heiles system. At fixed $r$ , the variation with $\epsilon$ is modest, while $\Lambda$ increases substantially as the energy domain expands.

Figure 9: Domain-averaged deflated losses of the three learned candidates for the double pendulum on the expanding boxes

U_{r}^{\mathrm{rec}}

centered at

\mathbf{x}_{0}=(0,0,0.2,0)

. The vertical dashed lines mark

r_{1}=0.2

, when the box first contains the stable equilibrium, and

r_{2}\approx 0.6325

, when it first intersects the separatrix of the saddle energy

-1

. The third candidate is lost before

r_{1}

, while the second remains accurate through

r_{2}

and then gradually breaks down.

Figure 10: Domain-averaged conservation losses of the learned candidates for the double pendulum on the expanding energy sublevel sets

U_{h}^{\mathrm{en}}

. The third candidate has a large loss from the outset because every domain contains the equilibrium. The energy candidate remains accurate, while the loss of the second candidate increases gradually as the energy level grows, indicating a transition from two numerically recovered conservation laws to one.

Panel of Figure 11 — Figure 11: SALI maps and pointwise conservation losses of the second conservation law for the double pendulum at $E=-2.2$ (top row) and $E=-2.0$ (bottom row). In each row, the first panel shows the SALI on the Poincaré section, while the second and third show conservation losses from two independent training runs. The SALI structures do not consistently align with regions of large loss, and the loss patterns vary between runs.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 12: Finite-time conservation defect of $F$ on the section $\{y=0,\ p_{y}>0\}\cap\{H_{\star}=0.04\}$ , with $T=100$ . The panels correspond to $\epsilon=-0.330,-0.300,-0.250,-0.200$ from top left to bottom right. Colors show $\log_{10}\mathcal{C}_{2}$ on the perturbed flow, with a separate color scale in each panel. White curves are contours of $\log_{10}(\mu^{2}g)$ computed from the integrable flow. Black curves mark saddle-level contours of $g$ .

Dynamical Systems

Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.

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From Data to Laws: Neural Discovery of Conservation Laws Without False Positives

Rahul D Ray

Department of Electronics and Electrical Engineering, BITS Pilani, Hyderabad Campus

Scientific Discovery Dynamical Systems Neuro-Symbolic Reasoning

Open paper AskPDF arXiv

Conservation laws are fundamental to understanding dynamical systems, but discovering them from data remains challenging due to parameter variation, non-polynomial invariants, local minima, and false positives on chaotic systems. We introduce NGCG, a neural-symbolic pipeline that decouples dynamics learning from invariant discovery and systematically addresses these challenges. A multi-restart variance minimiser learns a near-constant latent representation; system-specific symbolic extraction (polynomial Lasso, log-basis Lasso, explicit PDE candidates, and PySR) yields closed-form expressions; a strict constancy gate and diversity filter eliminate spurious laws. On a benchmark of nine diverse systems including Hamiltonian and dissipative ODEs, chaos, and PDEs, NGCG achieves consistent discovery (DR=1.0, FDR=0.0, F1=1.0) on all four systems with true conservation laws, with constancy two to three orders of magnitude lower than the best baseline. It is the only method that succeeds on the Lotka--Volterra system, and it correctly outputs no law on all five systems without invariants. Extensive experiments demonstrate robustness to noise (

σ= 0.1

), sample efficiency (50--100 trajectories), insensitivity to hyperparameters, and runtime under one minute per system. A Pareto analysis shows that the method provides a range of candidate expressions, allowing users to trade complexity for constancy. NGCG achieves strong performance relative to prior methods for data-driven conservation-law discovery, combining high accuracy with interpretability.

Abstract

Dynamical Systems

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