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
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
Figure 1: Pointwise conservation loss for the 2D node (Example 1 ) on Ur(2)=[2−r,2+r]×[−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.
Figure 2: Pointwise losses for the 3D degenerate node (Example 2 ) on the slice z=0 of Ur(3)=[2−r,2+r]×[−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 I1 , while panels (e)–(h) show the deflated loss for I2 . The first conservation law persists for all r , whereas the loss for the second jumps near the equilibrium line once r>2 .
Figure 3: Pointwise conservation loss for the 2D center (Example 3 ) on Ur(2)=[2−r,2+r]×[−r,r] , with r=1.5,1.9,2.1,2.5 from left to right. Unlike the 2D node, the conservation law x2+y2 persists after the domain contains the origin.
Figure 4: Pointwise losses for the 3D degenerate center (Example 4 ) on the slice z=0 of Ur(3)=[2−r,2+r]×[−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 I1 , while panels (e)–(h) show the deflated loss for I2 . Unlike the 3D degenerate node, both conservation laws persist after the box intersects the equilibrium line.
Figure 5: Domain-averaged deflated losses of the three learned candidates for the Hénon–Heiles system on the expanding boxes Urrec=(1,1,0,0)+[−r,r]4 . Panels (a)–(f) correspond to ϵ=−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 ϵ=−1/3 . For ϵ=−0.3 and ϵ=−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 ϵ=−0.1 onward.
Figure 6: Pointwise deflated loss of the third learned conservation law I3 for the Hénon–Heiles system on the px – py slice x=y=1−r of the boundary of Urrec . Panels (a) and (b) correspond to ϵ=−1/3 with r=0.9 and r=1.0 , respectively, while panels (c) and (d) show the corresponding results for ϵ=−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.
Figure 7: Domain-averaged deflated losses of the three learned candidates for the Hénon–Heiles system on the energy sublevel sets Ur(1+ϵ)en(ϵ) growing from the origin. Panels (a)–(i) correspond to ϵ=−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+ϵ) . Two candidates persist throughout the tested range in the integrable case. The breakdown of the second candidate first becomes visible at ϵ=−0.25 and occurs at progressively smaller values of r as ϵ moves away from −1/3 ; by ϵ=−0.1 , it is lost from the smallest tested domain.
Figure 8: Dependence of Λ(r;ϵ) on the normalized energy r for the Hénon–Heiles system. At fixed r , the variation with ϵ is modest, while Λ 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 Urrec centered at x0=(0,0,0.2,0) . The vertical dashed lines mark r1=0.2 , when the box first contains the stable equilibrium, and r2≈0.6325 , when it first intersects the separatrix of the saddle energy −1 . The third candidate is lost before r1 , while the second remains accurate through r2 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 Uhen . 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.
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,py>0}∩{H⋆=0.04} , with T=100 . The panels correspond to ϵ=−0.330,−0.300,−0.250,−0.200 from top left to bottom right. Colors show log10C2 on the perturbed flow, with a separate color scale in each panel. White curves are contours of log10(μ2g) computed from the integrable flow. Black curves mark saddle-level contours of g .
Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.
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.
Rahul D Ray
Department of Electronics and Electrical Engineering, BITS Pilani, Hyderabad Campus
We ask a representation-learning question about physical world models: when does a conservation law remain certifiable after a model learns a latent representation? A certified horizon bounds -- in advance, from measurable model defects -- how many steps a rollout provably stays on a physical invariant's level set. The key design choice is what is certified: not a learned latent Hamiltonian or a learned scalar witness (a model can conserve either while drifting in true energy), but the decoded physical invariant obtained by decoding the latent state and evaluating the known invariant. Around this object we derive shell-horizon certificates whose budget decomposes into representation, readout, and latent-dynamics defects, with a monotone alignment bridge through which a soft learned witness yields a certified horizon for the decoded invariant, and test them across state, learned-lift, and pixel observations on conservative systems. Conservation certificates can survive learned representation, but not all geometric priors survive equally. Hard canonical symplectic structure yields the longest horizons in known phase coordinates yet does not cross a learned chart, whereas a controlled-Lipschitz-aligned soft invariant survives in the nonlinear learned-representation settings we test -- two lift systems, with the gain growing with nonlinearity, and pixels. Pixel certification is recovered on a readout-stable sub-tube, and the Kepler problem exposes a geometric boundary. The central object is therefore not a latent Hamiltonian, but a decoded physical invariant whose robustness to representation learning can be measured, certified, and falsified.
Hongbo Wang
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA
We explore whether intrinsic symmetries of the training data lead to conserved quantities during gradient-flow training of neural networks. Under the assumption that the loss function is analytic and non-polynomial, we prove that data symmetries generically do not induce any additional integrals of motion. For mean squared error (MSE) loss, on the other hand, there are situations in which data augmentation yields extra conserved quantities. We build a framework, utilizing tensorizable networks to describe this phenomenon. Tensorizable networks are a family of architectures whose dependence on parameters and inputs can be separated using an intermediate representation. They include linear and polynomial networks, as well as Lightning Attention.