Machine learning conservation laws from differential equations
Open accessAbstract
We present a machine learning algorithm that discovers conservation laws from differential equations, both numerically (parametrized as neural networks) and symbolically, ensuring their functional independence (a nonlinear generalization of linear independence). Our independence module can be viewed as a nonlinear generalization of singular value decomposition. Our method can readily handle inductive biases for conservation laws. We validate it with examples including the three-body problem, the KdV equation, and nonlinear Schrödinger equation.
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Symbolic representations may remain crucial for efficient computation, but algorithmic-assisted discovery could lessen the need for their transmission, as these representations could be easily regenerated.
“While symbolic representations may remain crucial for efficient computation, algorithmic-assisted discovery could lessen the need for their transmission, as these representations could be easily regenerated 113 .”
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Physics-informed machine learning
Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021) · Nature Reviews Physics · doi:10.1038/s42254-021-00314-5 · Cited by 8,718 · see citing works
More Is Different: Broken symmetry and the nature of the hierarchical structure of science.
Anderson, P. W. (1972) · Science · doi:10.1126/science.177.4047.393 · Cited by 4,018 · see citing works
AI Feynman: A physics-inspired method for symbolic regression
Udrescu, S., & Tegmark, M. (2019) · Science Advances · doi:10.1126/sciadv.aay2631 · Cited by 1,470 · see citing works
Self-citationMiura, R. M., Gardner, C. S., & Kruskal, M. D. (1968) · Journal of Mathematical Physics · doi:10.1063/1.1664701 · Cited by 1,016 · see citing works
Machine Learning Conservation Laws from Trajectories
Liu, Z., & Tegmark, M. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.180604 · Cited by 130 · see citing works
Self-citationIntegrable Systems: Twistors, Loop Groups, and Riemann Surfaces
Hitchin, N. J., Segal, G. B., & Ward, R. S. (1999) · CERN Document Server (European Organization for Nuclear Research) · Cited by 76 · see citing works
Discovering symmetry invariants and conserved quantities by interpreting siamese neural networks
Wetzel, S. J., Melko, R. G., Scott, J., Panju, M., & Ganesh, V. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.033499 · Cited by 75 · see citing works
Machine Learning Hidden Symmetries
Liu, Z., & Tegmark, M. (2022) · Physical Review Letters · doi:10.1103/physrevlett.128.180201 · Cited by 59 · see citing works
Self-citationOn the Degeneracy of the Two-Dimensional Harmonic Oscillator
Dulock, V. A., & McIntosh, H. V. (1965) · American Journal of Physics · doi:10.1119/1.1971258 · Cited by 48 · see citing works
Sturm, P. O., & Wexler, A. S. (2022) · Geoscientific model development · doi:10.5194/gmd-15-3417-2022 · Cited by 48 · see citing works
Elements of Classical and Quantum Integrable Systems
Arutyunov, G. E. (2019) · UNITEXT for physics · doi:10.1007/978-3-030-24198-8 · Cited by 32 · see citing works
Discovering invariants via machine learning
Ha, S., & Jeong, H. (2021) · Physical Review Research · doi:10.1103/physrevresearch.3.l042035 · Cited by 29 · see citing works
Mototake, Y.-I. (2021) · Physical review. E · doi:10.1103/physreve.103.033303 · Cited by 20 · see citing works
Vickers, J. A. (2001) · Bulletin of the London Mathematical Society · doi:10.1112/blms/33.1.117 · Cited by 6 · see citing works
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Research works printed but not linked: 5
- [13]S.-M. Udrescu, A. Tan, J. Feng, O. Neto, T. Wu, and M. Tegmark, Ai feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity, Advances in Neural Information Processing Systems 33 , 4860 (2020).
- [14]L. Landau and E. Lifshitz, Mechanics third edition (1976) Chap. 2.
- [15]D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.6980 (2014). · arXiv
- [16]V. Sitzmann, J. Martel, A. Bergman, D. Lindell, and G. Wetzstein, Implicit neural representations with periodic activation functions, Advances in Neural Information Processing Systems 33 , 7462 (2020).
- [27]Z. Liu, Y. Chen, Y. Du, and M. Tegmark, Physicsaugmented learning: A new paradigm beyond physicsinformed learning, arXiv preprint arXiv:2109.13901 (2021). · arXiv
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Brinkmann, L., Baumann, F., Bonnefon, J.-F., Derex, M., Müller, T. F., Nussberger, A.-M., Czaplicka, A., Acerbi, A., Griffiths, T. L., Henrich, J., et al. (2023) · Nature Human Behaviour · doi:10.1038/s41562-023-01742-2 · Cited by 132 · see citing works
- Authors
- Liu, Z., Madhavan, V., & Tegmark, M.
- Affiliations
- The NSF AI Institute for Artificial Intelligence and Fundamental Interactions (US)
- Indian Institute of Technology Kharagpur (IN)
- Venue
- Physical Review E
- Publisher
- American Physical Society (APS)
- Published
- 2022-10-21
- Volume
- 106
- Issue
- 4
- Pages
- 045307-045307
- ISSN
- 2470-0045 (ISSN-L)
- 2470-0053
- OpenAlex
- W4307046645
- Cited by
- 77
- References
- 14
- Citations by year
- 2026: 5 · 2025: 13 · 2024: 50 · 2023: 8 · 2022: 1
- FWCI
- 6.79
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Physical review. E (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University) (cc-by)
- Funders
- Companhia Brasileira de Metalurgia e Mineração
- Foundational Questions Institute
- National Science Foundation (PHY-2019786)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Model Reduction and Neural Networks
- Physical Sciences › Mathematics › Numerical Analysis › Numerical methods for differential equations
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Nonlinear Waves and Solitons
- Keywords
- conservation laws, differential equations, artificial neural networks, symbolic regression, functional independence, singular value decomposition, inductive bias, three-body problem, Korteweg-de Vries equation, nonlinear Schrödinger equation
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