D₄-PINN: Hard-Constraint Group-Invariant Physics-Informed Neural Networks with Aerospace Thermal-Analysis Applications

D₄-PINN: Hard-Constraint Group-Invariant Physics-Informed Neural Networks with Aerospace Thermal-Analysis Applications | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article D₄-PINN: Hard-Constraint Group-Invariant Physics-Informed Neural Networks with Aerospace Thermal-Analysis Applications Chuyao Gong This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9719695/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract D₄-PINN enforces dihedral- symmetry of the solution at machine precision ( ) by construction, rather than through a soft penalty in the loss. The network is the Reynolds average of a free multilayer perceptron over the eight-element orbit of the input; the resulting forward map satisfies for every regardless of training. The core contribution is a certified symmetry guarantee—not an accuracy or efficiency claim—whose value is demonstrated on steady-state thermal analysis of -symmetric aerospace composite panels. A generalization analysis based on the Mishra–Molinaro PINN total-error decomposition and the quotient-space Rademacher complexity construction of Sannai, Imaizumi, and Kawano quantifies the sample-complexity benefit of the -invariant hypothesis class. An intermediate lemma establishes that the Rademacher complexity of the PDE residual under the Laplacian is bounded by a constant multiple of that of the hypothesis class, with the amplification factor polynomially controlled in the network depth and spectral-norm bound. -PINN is benchmarked against the unconstrained PINN, a soft-constraint variant, an algebraic-invariant PINN using the fundamental invariants and as input, two equivariant architectures, and a finite element method. Under equalised wall-clock time, the per-epoch overhead of Reynolds averaging offsets the sample-efficiency gain from the invariant hypothesis class; on smooth forward problems the two architectures achieve comparable accuracy with neither holding a systematic advantage. -PINN’s accuracy advantage is task-dependent: lower error on cubic semilinear problems, lower parameter recovery error on inverse problems, and a – advantage on the Allen–Cahn equation across . The machine-precision -invariance guarantee ( ), in contrast, is unconditional—it holds across all seeds, architectures, and tasks. The value of -PINN resides in this certified symmetry guarantee and the task-dependent regularisation it provides, not in uniform accuracy or throughput. The practical necessity of certified symmetry is demonstrated on two aerospace thermal-analysis scenarios: (i) steady-state heat conduction on a square composite panel with -symmetric heat sources, where -PINN’s symmetry deviation serves as an anomaly detector for manufacturing defects that break symmetry; and (ii) joint identification of thermal conductivity and source amplitude from sparse boundary temperature measurements, where the hard symmetry constraint acts as a structural prior that stabilises parameter recovery when observations are limited to . In both cases the soft-constraint alternative, whose symmetry error saturates at , fails to provide the certified guarantee required for anomaly detection and ill-posed inverse problems. On the Allen–Cahn equation , a variational (Deep Ritz) formulation is used to circumvent the trivial-solution trap. -PINN’s effectiveness decreases as shrinks, but the non-trivial -symmetric solution remains the global energy minimiser for all tested; whether spontaneous symmetry breaking occurs for yet smaller is left as an open question, quantified via the eigenvalue of the linearised operator about . All code, experimental data, and a one-command reproducibility pipeline are released. A self-contained C++17 inference implementation enables zero-dependency deployment. Computational Physics Physics-informed neural networks Group invariance Dihedral symmetry D4 Reynolds operator Semilinear elliptic equations Allen–Cahn equation Algebraic invariants Reproducible scientific computing Full Text Additional Declarations The authors declare no competing interests. Supplementary Files file.gitignore bootstrap.py genarch.py escnnbaseline.py generateallfigures.py generatearchitecturefigure.py generatetikzfigures.py README.md requirements.txt runall.py Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eD₄-PINN enforces dihedral- \u0026nbsp;\u0026nbsp;symmetry of the solution at machine precision ( \u0026nbsp;) by construction, rather than through a soft penalty in the loss. The network is the Reynolds average of a free multilayer perceptron over the eight-element \u0026nbsp;\u0026nbsp;\u0026nbsp;orbit of the input; the resulting forward map satisfies \u0026nbsp;\u0026nbsp;\u0026nbsp;for every \u0026nbsp;\u0026nbsp;\u0026nbsp;regardless of training. The core contribution is a \u003cem\u003ecertified\u003c/em\u003e symmetry guarantee—not an accuracy or efficiency claim—whose value is demonstrated on steady-state thermal analysis of \u0026nbsp;\u0026nbsp;-symmetric aerospace composite panels.\u003c/p\u003e\n\u003cp\u003eA generalization analysis based on the Mishra–Molinaro PINN total-error decomposition and the quotient-space Rademacher complexity construction of Sannai, Imaizumi, and Kawano quantifies the sample-complexity benefit of the \u0026nbsp;\u0026nbsp;-invariant hypothesis class. An intermediate lemma establishes that the Rademacher complexity of the PDE residual under the Laplacian is bounded by a constant multiple of that of the hypothesis class, with the amplification factor polynomially controlled in the network depth and spectral-norm bound.\u003c/p\u003e\n\u003cp\u003e-PINN is benchmarked against the unconstrained PINN, a soft-constraint variant, an algebraic-invariant PINN using the \u0026nbsp;\u0026nbsp;\u0026nbsp;fundamental invariants \u0026nbsp;\u0026nbsp;\u0026nbsp;and \u0026nbsp;\u0026nbsp;\u0026nbsp;as input, two equivariant architectures, and a finite element method. Under equalised wall-clock time, the \u0026nbsp;\u0026nbsp;\u0026nbsp;per-epoch overhead of Reynolds averaging offsets the sample-efficiency gain from the invariant hypothesis class; on smooth forward problems the two architectures achieve comparable accuracy with neither holding a systematic advantage. \u0026nbsp;\u0026nbsp;-PINN’s accuracy advantage is task-dependent: \u0026nbsp;\u0026nbsp;\u0026nbsp;lower \u0026nbsp;\u0026nbsp;\u0026nbsp;error on cubic semilinear problems, \u0026nbsp;\u0026nbsp;\u0026nbsp;lower parameter recovery error on inverse problems, and a \u0026nbsp;\u0026nbsp;– \u0026nbsp;\u0026nbsp;advantage on the Allen–Cahn equation across \u0026nbsp;\u0026nbsp;. The machine-precision \u0026nbsp;\u0026nbsp;-invariance guarantee ( \u0026nbsp;), in contrast, is unconditional—it holds across all seeds, architectures, and tasks. The value of \u0026nbsp;\u0026nbsp;-PINN resides in this certified symmetry guarantee and the task-dependent regularisation it provides, not in uniform accuracy or throughput.\u003c/p\u003e\n\u003cp\u003eThe practical necessity of certified symmetry is demonstrated on two aerospace thermal-analysis scenarios: (i) steady-state heat conduction on a square composite panel with \u0026nbsp;\u0026nbsp;-symmetric heat sources, where \u0026nbsp;\u0026nbsp;-PINN’s symmetry deviation serves as an anomaly detector for manufacturing defects that break \u0026nbsp;\u0026nbsp;\u0026nbsp;symmetry; and (ii) joint identification of thermal conductivity and source amplitude from sparse boundary temperature measurements, where the hard symmetry constraint acts as a structural prior that stabilises parameter recovery when observations are limited to \u0026nbsp;\u0026nbsp;. In both cases the soft-constraint alternative, whose symmetry error saturates at \u0026nbsp;\u0026nbsp;, fails to provide the certified guarantee required for anomaly detection and ill-posed inverse problems.\u003c/p\u003e\n\u003cp\u003eOn the Allen–Cahn equation \u0026nbsp;\u0026nbsp;, a variational (Deep Ritz) formulation is used to circumvent the trivial-solution trap. \u0026nbsp;\u0026nbsp;-PINN’s effectiveness decreases as \u0026nbsp;\u0026nbsp;\u0026nbsp;shrinks, but the non-trivial \u0026nbsp;\u0026nbsp;-symmetric solution remains the global energy minimiser for all \u0026nbsp;\u0026nbsp;\u0026nbsp;tested; whether spontaneous symmetry breaking occurs for yet smaller \u0026nbsp;\u0026nbsp;\u0026nbsp;is left as an open question, quantified via the eigenvalue \u0026nbsp;\u0026nbsp;\u0026nbsp;of the linearised operator about \u0026nbsp;\u0026nbsp;.\u003c/p\u003e\n\u003cp\u003eAll code, experimental data, and a one-command reproducibility pipeline are released. A self-contained C++17 inference implementation enables zero-dependency deployment.\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e","manuscriptTitle":"D₄-PINN: Hard-Constraint Group-Invariant Physics-Informed Neural Networks with Aerospace Thermal-Analysis Applications","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-18 02:43:06","doi":"10.21203/rs.3.rs-9719695/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"6e34acf8-2fe7-4231-8eeb-60f0a180148b","owner":[],"postedDate":"May 18th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":68167672,"name":"Computational Physics"}],"tags":[],"updatedAt":"2026-05-18T02:43:06+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-18 02:43:06","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9719695","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9719695","identity":"rs-9719695","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}