Data-driven discovery of intrinsic Lie point symmetries of differential equation systems

Ilker Gurcan; Kadir Ustun; Ramazan Gokberk Cinbis

cs.LGNov 5, 2026

Data-driven discovery of intrinsic Lie point symmetries of differential equation systems

Authors: Ilker Gurcan, Kadir Ustun, Ramazan Gokberk Cinbis

Organizations: Department of Computer Engineering, Middle East Technical University, Ankara, Türkiye · ASELSAN Inc., Ankara, Türkiye · ROMER Robotics and AI Center, Middle East Technical University, Ankara, Türkiye

Abstract

Discovering continuous symmetries directly from differential-equation data can reveal geometric structure for model reduction and scientific machine learning without symbolic determining equations or a closed-form representation of the governing system. We formulate intrinsic Lie point-symmetry discovery as an operator-theoretic problem on sampled jet-space data. Admissible generators are identified from a sampled determining operator and separated from redundant, solution-dependent, and geometrically trivial directions. The framework combines prolongation, sequential subspace restrictions, and a fixed nonlinear dictionary linear in trainable coefficients. A rank-stabilization preflight test diagnoses systems whose represented intrinsic symmetry algebra does not stabilize under dictionary refinement. On a controlled differential-equation benchmark with an analytically known 11-dimensional represented intrinsic symmetry algebra, the method recovers the complete represented generator set, reproduces its Lie-bracket structure to numerical precision, and approximates a known generator beyond the training domain. Here we show that explicit intrinsic continuous-symmetry generators can be recovered from data obtained from exact or approximate differential systems.

Figures & tables

Figure 1: Several solution curves of $\Delta^{(1)}$ in ( 1 ) related by its rotational symmetry. The remaining solution curves are obtained by rotating the particular solution shown in light blue.

Figure 2: The equation manifold

\Delta^{(1)}=\left\{(x,u;u_{x})\mid u_{x}-xu=0\right\}

J^{(1)}(\pi)

, together with a solution section

u=f(x)

and its first jet lift

x\mapsto j^{1}i_{f}(x)

Figure 3: A schematic relation between a Lie algebra, its associated Lie group, and the induced action on

E

. The exponential map generates the one-parameter subgroup

g_{v}(\epsilon)=\exp(\epsilon v)\in G

, whose action produces the local flow

\operatorname{Fl}^{z}_{v}(\epsilon)=\exp(\epsilon v)\cdot z

, tangent to

v|_{z}\in T_{z}E

\epsilon=0

Figure 4: A trivial generator (left panel) is tangent to every admissible solution graph, whereas a nontrivial generator (right panel) maps at least one admissible solution graph to a distinct solution graph.

Generator	$\bm{\dim\left(T_{i}\right)}$	Validation symmetry RMS	Speed-tail ratio
$v_{1}$	$11$	$3.33\times 10^{-15}$	$1.037$
$v_{2}$	$10$	$5.68\times 10^{-15}$	$1.011$
$v_{3}$	$9$	$2.77\times 10^{-15}$	$1.003$
$v_{4}$	$8$	$5.24\times 10^{-15}$	$1.002$
$v_{5}$	$7$	$4.92\times 10^{-14}$	$1.648$
$v_{6}$	$6$	$7.86\times 10^{-14}$	$1.661$

Table 1: Sequential recovery of the represented intrinsic symmetry directions for the two-dimensional wave equation at the nominal grid resolution

o=8

. Here,

\dim\left(T_{i}\right)

denotes the dimension of the novelty-restricted coefficient space immediately before optimization of the

i^{\mathrm{th}}

generator. The validation symmetry RMS and speed-tail ratio are evaluated for the accepted generator on the held-out validation set. Each accepted generator removes one globally novel direction; after the eleventh generator, the novelty-restricted space collapses to zero and the procedure terminates by clean represented exhaustion.

Quantity	Value
Recovered generators	$11$
Unordered generator pairs	$55$
Fitting samples	$89\,769$
Validation samples	$22\,442$
Response-design rank	$11$
Response-design condition number	$1.55\times 10^{1}$

Table 2: Post-recovery Lie-bracket closure of the recovered intrinsic symmetry generators for the two-dimensional wave equation.

	Relative RMS	Absolute RMS	Cosine similarity
Fitting	$4.12\times 10^{-10}$	$9.82\times 10^{-10}$	$1.000$
Validation	$4.09\times 10^{-10}$	$9.89\times 10^{-10}$	$1.000$

Table 3: Global constant-span reconstruction of the analytical inversion-in-

x

generator from the eleven recovered generators.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure E.1: Flowchart of Cartan’s iterative procedure for a linear Pfaffian system, including torsion compatibility, prolongation, involutivity, and the resulting differential freedom. The blue arrow indicates relabeling of the Pfaffian one-form systems for the next iteration. The prolongation index

l

starts at

0

Figure E.2: Flowchart of the sequential symmetry-discovery procedure. Starting from the DE system, its sampled datasets, and the prescribed hyperparameters, the algorithm constructs the represented determining and nonadditive admissible spaces, applies the sequential novelty restriction, initializes a candidate generator within the remaining search subspace, and optimizes the regularized nontriviality objective. Accepted generators are appended successively to the recovered intrinsic algebra and used to construct the novelty restriction for the next discovery step. Blue-bordered halt states indicate clean exhaustion of the numerically resolvable represented symmetry directions, whereas red-bordered halt states indicate recovery failure despite the presence of a represented candidate direction. The initial halt associated with a DE system outside the admissible scope of the framework is treated separately in the applicability flowchart of Section 6 .

Figure E.3: Applicability and admissible-generator realization of the proposed framework. Starting from the DE system

\Delta^{(s)}=\mathbf{0}

and the associated on- and off-shell datasets, the flowchart first distinguishes ODEs from PDEs, then branches according to differential order and linearity. First-order ODEs are excluded from the present scope because arbitrary nontrivial functional freedom remains after removal of the characteristic-zero solution-tangent sector. For regular, determined ODE systems with

s>1

, the local point-symmetry algebra is finite-dimensional; linear ODEs with

s\geq 3

use the projectable fiber-affine (PFA) realization, whereas the remaining admissible ODE classes use the regular generator realization. For PDEs, linear systems with

s\geq 2

use the PFA realization, while nonlinear PDEs and first-order linear PDEs use the regular realization in the present implementation. All retained branches are subjected to the RSP test. A stable represented geometric-rank plateau over the prescribed refinement window permits entry into sequential recovery, whereas failure to stabilize places the DE outside the completeness-certified scope of the current framework. Here, “Admissible” refers to admissibility within the stabilized finite represented and sampled setting; it does not by itself establish exact formal closure or exclude unresolved arbitrary functional freedom in the underlying determining system.

Panel of Figure 11 — Figure F.1: Representative integral curves of three recovered generators selected from the low-, intermediate-, and high-speed-tail regimes reported in Table 1 . The trajectories are shown through their projections onto the $x$ – $y$ , $x$ – $t$ , $y$ – $t$ , $x$ – $u$ , $y$ – $u$ , and $t$ – $u$ coordinate planes, with the black markers indicating the initial points. The selected generators are (a) $v_{4}$ , with $\rho_{99}=1.002$ and $\varepsilon_{\max}=0.15$ ; (b) $v_{7}$ , with $\rho_{99}=1.654$ and $\varepsilon_{\max}=0.05$ ; and (c) $v_{10}$ , with $\rho_{99}=1.993$ and $\varepsilon_{\max}=0.03$ . The integration horizons are chosen separately so that the displayed trajectories remain within, or sufficiently close to, the region in which the corresponding locally learned vector fields remain reliable; the behavior outside the training support is examined separately in 5.3 Approximation of a Known Symmetry Generator and Locality of the Representation .

Figure F.2: Integral-curve comparison between the analytical inversion-in-

x

generator

v_{\mathrm{inv},x}

and its constant-span reconstruction

\widetilde{v}_{\mathrm{inv},x}

, integrated from the same initial points up to

\varepsilon_{\max}=0.1

using

100

steps.

Figure F.3: Integral-curve comparison between the analytical inversion-in-

x

generator

v_{\mathrm{inv},x}

and its constant-span reconstruction

\widetilde{v}_{\mathrm{inv},x}

after shifting the initial

x

interval to

\left[0.8,0.86\right]

, with the flows integrated up to

\varepsilon_{\max}=0.04

Symbol	Principal meaning	Context
Jet-space and differential-equation geometry
$\mathbf{X}$	Base space of the independent variables.	Jet geometry
$E,\;\pi:E\to\mathbf{X}$	Total space and fiber-bundle projection over the base space.	Jet geometry
$\mathbf{x},\;\mathbf{u}$	Independent- and dependent-variable coordinate vectors, respectively.	Jet geometry
$p,\;q,\;m,\;s$	Numbers of independent variables, dependent variables, and constituent differential equations, and the differential order of the DE system, respectively.	DE system
$J^{(s)}\left(\pi\right)$	Order- $s$ jet bundle of local sections of $\pi$ .	Jet geometry

Appendix

Table 14

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Discovering underlying symmetries from data has emerged as a crucial challenge in scientific discovery. Existing data-driven methods for symmetry discovery fail to determine the exact number and mathematical form of unknown infinitesimal generators. Recent explicit methods represent generators using a predefined function library and identify them through algebraic optimization, but they often struggle to capture complex symmetries involving high-order polynomials or transcendental functions. To address this limitation, we formulate symmetry discovery as a joint optimization problem over the function library and coefficients. We propose a novel framework that leverages an encoder-decoder architecture to dynamically generate symbolic expressions and expand the library. This generation process is optimized via reinforcement learning, which accelerates the exploration of the symbolic search space through step-wise rewards. Experiments demonstrate that LieDiscover can successfully uncover open-form infinitesimal generators involving high-order polynomials or transcendental functions, which remain intractable for existing methods. The discovered symmetries also improve performance in downstream PDE solving and discovery tasks.

Abstract

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