Finding discrete symmetry groups via Machine Learning - September 10, 2024 - presentation

Aug 2, 2026

The work introduces the Symmetry Seeker Neural Network (SSNN), an automated machine-learning approach that discovers discrete symmetry groups from input-output data of physical systems. The method infers the finite set of parameter transformations that preserve measurable physical magnitudes, without prior knowledge of the symmetry group or the mathematical relations between parameters and properties. Using a tailored multi-branch neural architecture and a symmetry-consistent loss, the study demonstrates versatility across examples in mathematics, nanophotonics, and quantum chemistry.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages14

Reading Time35 min

AccessFree to download

Introduction

Symmetry in physics and ML-driven symmetry discovery

Symmetry Seeker Neural Network (SSNN)

Problem formulation and symmetry constraint

SSNN architecture and training objective

Applications and demonstrations

Mathematical example

Nanophotonics and quantum chemistry examples

Finding discrete symmetry groups via Machine Learning Pablo Calvo-Barl´e􀀓, 1, 2 Ser􀀐io 􀀎􀀍 􀀌o􀀋ri􀀐o,1, 3 E􀀋uar􀀋o S´anchez-Burillo,4 an􀀋 Lui􀀓 Mart´􀀁n-Moreno1, 2, ∗ 1 Instituto de Nanociencia 􀀓 Materia􀀐es de Arag´on (INMA), CSIC-Universidad de Zaragoza, 50009 Zaragoza, Spain 2 Departamento de F´ısica de 􀀐a Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain 3 Departamento de F´ısica Ap􀀐icada, Universidad de Zaragoza, 50009 Zaragoza, Spain 4 PredictLand S. L. , 50001 Zaragoza, Spain (Dated: September 10, 2024􀀎 􀀍e 􀀌􀀋troduce a mach􀀌􀀋e-lear􀀋􀀌􀀋g approach (de􀀋oted Symmetry Seeker Neural Network􀀎 capable of automat􀀌cally d􀀌scover􀀌􀀋g d􀀌screte symmetry groups 􀀌􀀋 phys􀀌cal systems. Th􀀌s method 􀀌de􀀋t􀀌ﬁes the ﬁ􀀋􀀌te set of parameter tra􀀋sformat􀀌o􀀋s that preserve the system’s phys􀀌cal propert􀀌es. Remarkably, the method accompl􀀌shes th􀀌s w􀀌thout pr􀀌or k􀀋owledge of the system’s symmetry or the mathemat-􀀌cal relat􀀌o􀀋sh􀀌ps betwee􀀋 parameters a􀀋d propert􀀌es. Demo􀀋strat􀀌􀀋g 􀀌ts versat􀀌l􀀌ty, we showcase examples from mathemat􀀌cs, 􀀋a􀀋ophoto􀀋􀀌cs, a􀀋d qua􀀋tum chem􀀌stry. Symmetrie􀀓 play an e􀀓􀀓ential role in the un􀀋er􀀓tan􀀋in􀀐 of phy􀀓ical theorie􀀓􀀍 For example, continuou􀀓 􀀓ymmetrie􀀓 lea􀀋 to con􀀓ervation law􀀓 [1], an􀀋 􀀋i􀀓crete 􀀓ymmetrie􀀓 (􀀓uch a􀀓 parity or time rever􀀓al) are key in 􀀓tu􀀋yin􀀐􀀓pectral 􀀋e􀀐eneracie􀀓 in quantum mechanical 􀀓y􀀓tem􀀓 [2]􀀍 Unveilin􀀐 hi􀀋􀀋en 􀀓ymmetrie􀀓 i􀀓 thu􀀓 of immen􀀓e intere􀀓t in many ﬁel􀀋􀀓 of phy􀀓ic􀀓􀀍 Until recently, thi􀀓 ta􀀓krelie􀀋 􀀓olely on human intuition􀀍 However, recent a􀀋 -vance􀀓 in Machine Learnin􀀐 (ML), an􀀋 e􀀓pecially in Neural Network􀀓 (NN) [3–7], have opene􀀋 up new an􀀋 eﬃcient way􀀓 of analyzin􀀐 􀀋ata􀀍 NN􀀓 have alrea􀀋y been 􀀓ucce􀀓􀀓fully employe􀀋 in problem􀀓 􀀓uch a􀀓 learnin􀀐 con􀀓ervation law􀀓 an􀀋 phy􀀓ically relevant parameter􀀓 of 􀀋ynamical 􀀓y􀀓tem􀀓 [8–16]; ﬁn􀀋in􀀐 coor􀀋inate tran􀀓formation􀀓 that reveal hi􀀋􀀋en 􀀓ymmetrie􀀓 [17, 18]; learnin􀀐 Lie al-􀀐ebra􀀓 [19–23]; 􀀋i􀀓coverin􀀐 tran􀀓formation􀀓 that pre􀀓erve the 􀀓tati􀀓tical 􀀋i􀀓tribution of 􀀋ata 􀀓et􀀓 [22, 24]; reco􀀐nizin􀀐 􀀓ymmetrie􀀓 from NN embe􀀋􀀋in􀀐 layer􀀓 [21, 25] an􀀋 cla􀀓􀀓ifyin􀀐 whether pair􀀓 of event􀀓 are relate􀀋 by 􀀓ymmetry or not [12, 26]􀀍 A common 􀀓cenario not a􀀋􀀋re􀀓􀀓e􀀋 in prior 􀀓tu􀀋ie􀀓 ari􀀓e􀀓 when the phy􀀓ical ma􀀐nitu􀀋e􀀓 of a 􀀓y􀀓tem 􀀋epen􀀋 on parameter􀀓 po􀀓􀀓e􀀓􀀓in􀀐 a 􀀋i􀀓crete 􀀓et of 􀀓ymmetry tran􀀓formation􀀓 that leave the􀀓e ma􀀐nitu􀀋e􀀓 invariant􀀍 To ﬁn􀀋 the􀀓e tran􀀓formation􀀓, we intro􀀋uce the Symmetry Seeker Neural Network (SSNN), an NN architecture capable of 􀀋i􀀓coverin􀀐 the matrix repre􀀓entation of all 􀀓ymmetry 􀀐roup element􀀓 􀀓olely from a 􀀋ata􀀓et of phy􀀓ical parameter􀀓 an􀀋 their corre􀀓pon􀀋in􀀐 mea􀀓urable ma􀀐 -nitu􀀋e􀀓􀀍 Importantly, our approach 􀀋eman􀀋􀀓 no prior knowle􀀋􀀐e of the 􀀓ymmetry 􀀐roup or the mathematical relation􀀓hip􀀓 between parameter􀀓 an􀀋 phy􀀓ical propertie􀀓􀀍 A􀀋􀀋itionally, the SSNN can tackle inver􀀓e 􀀋e􀀓i􀀐n problem􀀓 with 􀀓ymmetry-relate􀀋 multivalue􀀋 􀀓olution􀀓􀀍 The text i􀀓 􀀓tructure􀀋 a􀀓 follow􀀓: we ﬁr􀀓t intro􀀋uce the SSNN al􀀐orithm an􀀋 􀀓howca􀀓e it􀀓 eﬀectivene􀀓􀀓 on a 􀀓imple mathematical problem􀀍 Then, we apply thi􀀓 metho􀀋 to example􀀓 in nanophotonic􀀓 an􀀋 quantum chemi􀀓try􀀍 To 􀀋eﬁne the problem in 􀀐eneral term􀀓, we con􀀓i􀀋era function y (x), where the “ma􀀐nitu􀀋e􀀓” y ∈ Rm 􀀋epen􀀋 on the “parameter􀀓” x ∈ Rn 􀀍 The “􀀓y􀀓tem” may FIG. 1: Schemat􀀌c represe􀀋tat􀀌o􀀋 of the SSNN: Mag􀀋􀀌tudes y e􀀋ter a sta􀀋dard NN, wh􀀌ch y􀀌elds a vector x(0NN) . Th􀀌s NN 􀀌s compleme􀀋ted w􀀌th a set of M bra􀀋ches. The α-th bra􀀋ch performs a l􀀌􀀋ear tra􀀋sformat􀀌o􀀋 ˆWα x(0NN) , prov􀀌d􀀌􀀋g a pred􀀌ct􀀌o􀀋 xN(αN) for the phys􀀌cal parameters. 􀀍􀀌thout the bra􀀋ches, the prese􀀋ce of symmetr􀀌es would lead to 􀀌􀀋correct pred􀀌ct􀀌o􀀋s. I􀀋 co􀀋trast, the SSNN, tra􀀌􀀋ed w􀀌th the custom􀀌zed loss fu􀀋ct􀀌o􀀋 g􀀌ve􀀋 by Eq. (2􀀎, pred􀀌cts parameters accurately from ava􀀌lable data. After tra􀀌􀀋􀀌􀀋g, the bra􀀋ches ˆWα prov􀀌dea represe􀀋tat􀀌o􀀋 of the system’s symmetry group. pre􀀓ent 􀀋i􀀓crete 􀀓ymmetrie􀀓, 􀀓o 􀀓ymmetry-relate􀀋 parameter􀀓 lea􀀋 to the 􀀓ame ma􀀐nitu􀀋e􀀓􀀍 If 􀀓o, the 􀀓ymmetry tran􀀓formation􀀓 form a ﬁ

FAQs

What problem does the Symmetry Seeker Neural Network (SSNN) address?

SSNN automatically discovers discrete symmetry groups by learning the finite set of parameter transformations that keep physical magnitudes invariant.

Does SSNN require prior knowledge of the system’s symmetry or equations?

No. The approach is designed to work without prior information about the symmetry group or the mathematical relations between parameters and physical properties.

How is the SSNN structured to infer symmetry group elements?

SSNN combines a neural network that produces a latent vector with multiple trainable linear transformation branches that generate multiple parameter predictions corresponding to candidate symmetry actions.

Aug 2, 2026

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