Machine learning for discovery of governing equations in physics: From sparse regression to neural operators and Koopman methods
PHYSICA D-NONLINEAR PHENOMENA, cilt.499, 2027 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 499
- Basım Tarihi: 2027
- Doi Numarası: 10.1016/j.physd.2026.135473
- Dergi Adı: PHYSICA D-NONLINEAR PHENOMENA
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Compendex, INSPEC, MathSciNet, zbMATH, Academic Search Ultimate (EBSCO), Engineering Source (EBSCO)
- Hatay Mustafa Kemal Üniversitesi Adresli: Evet
Özet
The data-driven discovery of governing equations-the programme of extracting interpretable physical laws directly from measurement data-has matured from a proof-of-concept curiosity into a rapidly expanding research field. This survey provides a unified, critical review of the four principal method families that drive this programme: (i) sparse regression and symbolic regression, which search for parsimonious closed-form equations; (ii) neural ordinary differential equations and their structure-preserving variants, which parameterise unknown dynamics with neural networks; (iii) operator-learning architectures such as DeepONet and the Fourier Neural Operator, which approximate infinite-dimensional solution maps; and (iv) Koopman-operator methods and reservoir computing, which lift nonlinear dynamics into approximately linear spaces. For each family we present the mathematical foundations, survey algorithmic advances through 2026, and assess performance on canonical benchmarks. Cutting across these families, we examine shared challenges-noise robustness, derivative estimation, scalability, uncertainty quantification, and the incorporation of physical constraints-and identify a persistent interpretability-scalability trade-off that no single method has yet resolved. A practitioner's decision guide, a software-ecosystem overview, and a curated benchmark suite are provided to lower the entry barrier for domain scientists. We conclude by mapping open problems-including causal discovery, foundation-model-assisted symbolic search, and multi-scale equation learning- and by reflecting on the epistemological boundaries that separate curve-fitting from genuine scientific discovery.