Machine learning in physics

Last updated

Applying classical methods of machine learning to the study of quantum systems is the focus of an emergent area of physics research. A basic example of this is quantum state tomography, where a quantum state is learned from measurement. [1] Other examples include learning Hamiltonians, [2] [3] learning quantum phase transitions, [4] [5] and automatically generating new quantum experiments. [6] [7] [8] [9] Classical machine learning is effective at processing large amounts of experimental or calculated data in order to characterize an unknown quantum system, making its application useful in contexts including quantum information theory, quantum technologies development, and computational materials design. In this context, it can be used for example as a tool to interpolate pre-calculated interatomic potentials [10] or directly solving the Schrödinger equation with a variational method. [11]

Contents

Applications of machine learning to physics

Noisy data

The ability to experimentally control and prepare increasingly complex quantum systems brings with it a growing need to turn large and noisy data sets into meaningful information. This is a problem that has already been studied extensively in the classical setting, and consequently, many existing machine learning techniques can be naturally adapted to more efficiently address experimentally relevant problems. For example, Bayesian methods and concepts of algorithmic learning can be fruitfully applied to tackle quantum state classification, [12] Hamiltonian learning, [13] and the characterization of an unknown unitary transformation. [14] [15] Other problems that have been addressed with this approach are given in the following list:

Calculated and noise-free data

Quantum machine learning can also be applied to dramatically accelerate the prediction of quantum properties of molecules and materials. [26] This can be helpful for the computational design of new molecules or materials. Some examples include

Variational circuits

Variational circuits are a family of algorithms which utilize training based on circuit parameters and an objective function. [33] Variational circuits are generally composed of a classical device communicating input parameters (random or pre-trained parameters) into a quantum device, along with a classical Mathematical optimization function. These circuits are very heavily dependent on the architecture of the proposed quantum device because parameter adjustments are adjusted based solely on the classical components within the device. [34] Though the application is considerably infantile in the field of quantum machine learning, it has incredibly high promise for more efficiently generating efficient optimization functions.

Sign problem

Machine learning techniques can be used to find a better manifold of integration for path integrals in order to avoid the sign problem. [35]

Fluid dynamics

This section is an excerpt from Deep learning § Partial differential equations.[ edit ]
Physics informed neural networks have been used to solve partial differential equations in both forward and inverse problems in a data driven manner. [36] One example is the reconstructing fluid flow governed by the Navier-Stokes equations. Using physics informed neural networks does not require the often expensive mesh generation that conventional CFD methods relies on. [37] [38]

Physics discovery and prediction

Illustration of how an AI learns the basic fundamental physical concept of 'unchangeableness' An AI learns basic physical principles.webp
Illustration of how an AI learns the basic fundamental physical concept of 'unchangeableness'

A deep learning system was reported to learn intuitive physics from visual data (of virtual 3D environments) based on an unpublished approach inspired by studies of visual cognition in infants. [40] [39] Other researchers have developed a machine learning algorithm that could discover sets of basic variables of various physical systems and predict the systems' future dynamics from video recordings of their behavior. [41] [42] In the future, it may be possible that such can be used to automate the discovery of physical laws of complex systems. [41] Beyond discovery and prediction, "blank slate"-type of learning of fundamental aspects of the physical world may have further applications such as improving adaptive and broad artificial general intelligence.[ additional citation(s) needed ] In specific, prior machine learning models were "highly specialised and lack a general understanding of the world". [40]

See also

Related Research Articles

<span class="mw-page-title-main">Kondo effect</span> Physical phenomenon due to impurities

In physics, the Kondo effect describes the scattering of conduction electrons in a metal due to magnetic impurities, resulting in a characteristic change i.e. a minimum in electrical resistivity with temperature. The cause of the effect was first explained by Jun Kondo, who applied third-order perturbation theory to the problem to account for scattering of s-orbital conduction electrons off d-orbital electrons localized at impurities. Kondo's calculation predicted that the scattering rate and the resulting part of the resistivity should increase logarithmically as the temperature approaches 0 K. Experiments in the 1960s by Myriam Sarachik at Bell Laboratories provided the first data that confirmed the Kondo effect. Extended to a lattice of magnetic impurities, the Kondo effect likely explains the formation of heavy fermions and Kondo insulators in intermetallic compounds, especially those involving rare earth elements such as cerium, praseodymium, and ytterbium, and actinide elements such as uranium. The Kondo effect has also been observed in quantum dot systems.

Quantum metrology is the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems, particularly exploiting quantum entanglement and quantum squeezing. This field promises to develop measurement techniques that give better precision than the same measurement performed in a classical framework. Together with quantum hypothesis testing, it represents an important theoretical model at the basis of quantum sensing.

The Bose–Hubbard model gives a description of the physics of interacting spinless bosons on a lattice. It is closely related to the Hubbard model that originated in solid-state physics as an approximate description of superconducting systems and the motion of electrons between the atoms of a crystalline solid. The model was introduced by Gersch and Knollman in 1963 in the context of granular superconductors. The model rose to prominence in the 1980s after it was found to capture the essence of the superfluid-insulator transition in a way that was much more mathematically tractable than fermionic metal-insulator models.

Reservoir computing is a framework for computation derived from recurrent neural network theory that maps input signals into higher dimensional computational spaces through the dynamics of a fixed, non-linear system called a reservoir. After the input signal is fed into the reservoir, which is treated as a "black box," a simple readout mechanism is trained to read the state of the reservoir and map it to the desired output. The first key benefit of this framework is that training is performed only at the readout stage, as the reservoir dynamics are fixed. The second is that the computational power of naturally available systems, both classical and quantum mechanical, can be used to reduce the effective computational cost.

<span class="mw-page-title-main">Landau–Zener formula</span> Formula for the probability that a system will change between two energy states.

The Landau–Zener formula is an analytic solution to the equations of motion governing the transition dynamics of a two-state quantum system, with a time-dependent Hamiltonian varying such that the energy separation of the two states is a linear function of time. The formula, giving the probability of a diabatic transition between the two energy states, was published separately by Lev Landau, Clarence Zener, Ernst Stueckelberg, and Ettore Majorana, in 1932.

Within quantum technology, a quantum sensor utilizes properties of quantum mechanics, such as quantum entanglement, quantum interference, and quantum state squeezing, which have optimized precision and beat current limits in sensor technology. The field of quantum sensing deals with the design and engineering of quantum sources and quantum measurements that are able to beat the performance of any classical strategy in a number of technological applications. This can be done with photonic systems or solid state systems.

<span class="mw-page-title-main">Topological insulator</span> State of matter with insulating bulk but conductive boundary

A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material.

The toric code is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice. It is the simplest and most well studied of the quantum double models. It is also the simplest example of topological order—Z2 topological order (first studied in the context of Z2 spin liquid in 1991). The toric code can also be considered to be a Z2 lattice gauge theory in a particular limit. It was introduced by Alexei Kitaev.

In quantum many-body physics, topological degeneracy is a phenomenon in which the ground state of a gapped many-body Hamiltonian becomes degenerate in the limit of large system size such that the degeneracy cannot be lifted by any local perturbations.

<span class="mw-page-title-main">Time crystal</span> Structure that repeats in time; a novel type or phase of non-equilibrium matter

In condensed matter physics, a time crystal is a quantum system of particles whose lowest-energy state is one in which the particles are in repetitive motion. The system cannot lose energy to the environment and come to rest because it is already in its quantum ground state. Because of this, the motion of the particles does not really represent kinetic energy like other motion; it has "motion without energy". Time crystals were first proposed theoretically by Frank Wilczek in 2012 as a time-based analogue to common crystals – whereas the atoms in crystals are arranged periodically in space, the atoms in a time crystal are arranged periodically in both space and time. Several different groups have demonstrated matter with stable periodic evolution in systems that are periodically driven. In terms of practical use, time crystals may one day be used as quantum computer memory.

<span class="mw-page-title-main">Quantum machine learning</span> Interdisciplinary research area at the intersection of quantum physics and machine learning

Quantum machine learning is the integration of quantum algorithms within machine learning programs.

<span class="mw-page-title-main">Interatomic potential</span> Functions for calculating potential energy

Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space. Interatomic potentials are widely used as the physical basis of molecular mechanics and molecular dynamics simulations in computational chemistry, computational physics and computational materials science to explain and predict materials properties. Examples of quantitative properties and qualitative phenomena that are explored with interatomic potentials include lattice parameters, surface energies, interfacial energies, adsorption, cohesion, thermal expansion, and elastic and plastic material behavior, as well as chemical reactions.

A quantum heat engine is a device that generates power from the heat flow between hot and cold reservoirs. The operation mechanism of the engine can be described by the laws of quantum mechanics. The first realization of a quantum heat engine was pointed out by Scovil and Schulz-DuBois in 1959, showing the connection of efficiency of the Carnot engine and the 3-level maser. Quantum refrigerators share the structure of quantum heat engines with the purpose of pumping heat from a cold to a hot bath consuming power first suggested by Geusic, Schulz-DuBois, De Grasse and Scovil. When the power is supplied by a laser the process is termed optical pumping or laser cooling, suggested by Wineland and Hänsch. Surprisingly heat engines and refrigerators can operate up to the scale of a single particle thus justifying the need for a quantum theory termed quantum thermodynamics.

In physics, non-Hermitian quantum mechanics, describes quantum mechanical systems where Hamiltonians are not Hermitian.

Continuous-variable (CV) quantum information is the area of quantum information science that makes use of physical observables, like the strength of an electromagnetic field, whose numerical values belong to continuous intervals. One primary application is quantum computing. In a sense, continuous-variable quantum computation is "analog", while quantum computation using qubits is "digital." In more technical terms, the former makes use of Hilbert spaces that are infinite-dimensional, while the Hilbert spaces for systems comprising collections of qubits are finite-dimensional. One motivation for studying continuous-variable quantum computation is to understand what resources are necessary to make quantum computers more powerful than classical ones.

Many-body localization (MBL) is a dynamical phenomenon occurring in isolated many-body quantum systems. It is characterized by the system failing to reach thermal equilibrium, and retaining a memory of its initial condition in local observables for infinite times.

Shortcuts to adiabaticity (STA) are fast control protocols to drive the dynamics of system without relying on the adiabatic theorem. The concept of STA was introduced in a 2010 paper by Xi Chen et al. Their design can be achieved using a variety of techniques. A universal approach is provided by counterdiabatic driving, also known as transitionless quantum driving. Motivated by one of authors systematic study of dissipative Landau-Zener transition, the key idea was demonstrated earlier by a group of scientists from China, Greece and USA in 2000, as steering an eigenstate to destination. Counterdiabatic driving has been demonstrated in the laboratory using a time-dependent quantum oscillator.

Randomized benchmarking is an experimental method for measuring the average error rates of quantum computing hardware platforms. The protocol estimates the average error rates by implementing long sequences of randomly sampled quantum gate operations. Randomized benchmarking is the industry-standard protocol used by quantum hardware developers such as IBM and Google to test the performance of the quantum operations.

<span class="mw-page-title-main">Giuseppe Carleo</span> Italian physicist

Giuseppe Carleo is an Italian physicist. He is a professor of computational physics at EPFL and the head of the Laboratory of Computational Quantum Science.

Robert W. Spekkens is a Canadian theoretical quantum physicist working in the fields of quantum foundations and quantum information.

References

  1. 1 2 Torlai, Giacomo; Mazzola, Guglielmo; Carrasquilla, Juan; Troyer, Matthias; Melko, Roger; Carleo, Giuseppe (May 2018). "Neural-network quantum state tomography". Nature Physics. 14 (5): 447–450. arXiv: 1703.05334 . Bibcode:2018NatPh..14..447T. doi:10.1038/s41567-018-0048-5. ISSN   1745-2481. S2CID   125415859.
  2. ↑ Cory, D. G.; Wiebe, Nathan; Ferrie, Christopher; Granade, Christopher E. (2012-07-06). "Robust Online Hamiltonian Learning". New Journal of Physics. 14 (10): 103013. arXiv: 1207.1655 . Bibcode:2012NJPh...14j3013G. doi:10.1088/1367-2630/14/10/103013. S2CID   9928389.
  3. ↑ Cao, Chenfeng; Hou, Shi-Yao; Cao, Ningping; Zeng, Bei (2020-02-10). "Supervised learning in Hamiltonian reconstruction from local measurements on eigenstates". Journal of Physics: Condensed Matter. 33 (6): 064002. arXiv: 2007.05962 . doi:10.1088/1361-648x/abc4cf. ISSN   0953-8984. PMID   33105109. S2CID   220496757.
  4. ↑ Broecker, Peter; Assaad, Fakher F.; Trebst, Simon (2017-07-03). "Quantum phase recognition via unsupervised machine learning". arXiv: 1707.00663 [cond-mat.str-el].
  5. ↑ Huembeli, Patrick; Dauphin, Alexandre; Wittek, Peter (2018). "Identifying Quantum Phase Transitions with Adversarial Neural Networks". Physical Review B. 97 (13): 134109. arXiv: 1710.08382 . Bibcode:2018PhRvB..97m4109H. doi:10.1103/PhysRevB.97.134109. ISSN   2469-9950. S2CID   125593239.
  6. 1 2 Krenn, Mario (2016-01-01). "Automated Search for new Quantum Experiments". Physical Review Letters. 116 (9): 090405. arXiv: 1509.02749 . Bibcode:2016PhRvL.116i0405K. doi:10.1103/PhysRevLett.116.090405. PMID   26991161. S2CID   20182586.
  7. 1 2 Knott, Paul (2016-03-22). "A search algorithm for quantum state engineering and metrology". New Journal of Physics. 18 (7): 073033. arXiv: 1511.05327 . Bibcode:2016NJPh...18g3033K. doi:10.1088/1367-2630/18/7/073033. S2CID   2721958.
  8. ↑ Dunjko, Vedran; Briegel, Hans J (2018-06-19). "Machine learning & artificial intelligence in the quantum domain: a review of recent progress". Reports on Progress in Physics. 81 (7): 074001. arXiv: 1709.02779 . Bibcode:2018RPPh...81g4001D. doi:10.1088/1361-6633/aab406. hdl: 1887/71084 . ISSN   0034-4885. PMID   29504942. S2CID   3681629.
  9. ↑ Melnikov, Alexey A.; Nautrup, Hendrik Poulsen; Krenn, Mario; Dunjko, Vedran; Tiersch, Markus; Zeilinger, Anton; Briegel, Hans J. (1221). "Active learning machine learns to create new quantum experiments". Proceedings of the National Academy of Sciences. 115 (6): 1221–1226. arXiv: 1706.00868 . doi: 10.1073/pnas.1714936115 . ISSN   0027-8424. PMC   5819408 . PMID   29348200.
  10. ↑ Behler, Jörg; Parrinello, Michele (2007-04-02). "Generalized Neural-Network Representation of High-Dimensional Potential-Energy Surfaces". Physical Review Letters. 98 (14): 146401. Bibcode:2007PhRvL..98n6401B. doi:10.1103/PhysRevLett.98.146401. PMID   17501293.
  11. 1 2 Carleo, Giuseppe; Troyer, Matthias (2017-02-09). "Solving the quantum many-body problem with artificial neural networks". Science. 355 (6325): 602–606. arXiv: 1606.02318 . Bibcode:2017Sci...355..602C. doi:10.1126/science.aag2302. PMID   28183973. S2CID   206651104.
  12. 1 2 Sentís, Gael; Calsamiglia, John; Muñoz-Tapia, Raúl; Bagan, Emilio (2012). "Quantum learning without quantum memory". Scientific Reports. 2: 708. arXiv: 1106.2742 . Bibcode:2012NatSR...2E.708S. doi:10.1038/srep00708. PMC   3464493 . PMID   23050092.
  13. ↑ Wiebe, Nathan; Granade, Christopher; Ferrie, Christopher; Cory, David (2014). "Quantum Hamiltonian learning using imperfect quantum resources". Physical Review A. 89 (4): 042314. arXiv: 1311.5269 . Bibcode:2014PhRvA..89d2314W. doi:10.1103/physreva.89.042314. hdl:10453/118943. S2CID   55126023.
  14. 1 2 Bisio, Alessandro; Chiribella, Giulio; D'Ariano, Giacomo Mauro; Facchini, Stefano; Perinotti, Paolo (2010). "Optimal quantum learning of a unitary transformation". Physical Review A. 81 (3): 032324. arXiv: 0903.0543 . Bibcode:2010PhRvA..81c2324B. doi:10.1103/PhysRevA.81.032324. S2CID   119289138.
  15. 1 2 Jeongho; Junghee Ryu, Bang; Yoo, Seokwon; Pawłowski, Marcin; Lee, Jinhyoung (2014). "A strategy for quantum algorithm design assisted by machine learning". New Journal of Physics. 16 (1): 073017. arXiv: 1304.2169 . Bibcode:2014NJPh...16a3017K. doi:10.1088/1367-2630/16/1/013017. S2CID   54494244.
  16. ↑ Granade, Christopher E.; Ferrie, Christopher; Wiebe, Nathan; Cory, D. G. (2012-10-03). "Robust Online Hamiltonian Learning". New Journal of Physics. 14 (10): 103013. arXiv: 1207.1655 . Bibcode:2012NJPh...14j3013G. doi:10.1088/1367-2630/14/10/103013. ISSN   1367-2630. S2CID   9928389.
  17. ↑ Wiebe, Nathan; Granade, Christopher; Ferrie, Christopher; Cory, D. G. (2014). "Hamiltonian Learning and Certification Using Quantum Resources". Physical Review Letters. 112 (19): 190501. arXiv: 1309.0876 . Bibcode:2014PhRvL.112s0501W. doi:10.1103/PhysRevLett.112.190501. ISSN   0031-9007. PMID   24877920. S2CID   39126228.
  18. ↑ Wiebe, Nathan; Granade, Christopher; Ferrie, Christopher; Cory, David G. (2014-04-17). "Quantum Hamiltonian Learning Using Imperfect Quantum Resources". Physical Review A. 89 (4): 042314. arXiv: 1311.5269 . Bibcode:2014PhRvA..89d2314W. doi:10.1103/PhysRevA.89.042314. hdl:10453/118943. ISSN   1050-2947. S2CID   55126023.
  19. ↑ Sasaki, Madahide; Carlini, Alberto; Jozsa, Richard (2001). "Quantum Template Matching". Physical Review A. 64 (2): 022317. arXiv: quant-ph/0102020 . Bibcode:2001PhRvA..64b2317S. doi:10.1103/PhysRevA.64.022317. S2CID   43413485.
  20. ↑ Sasaki, Masahide (2002). "Quantum learning and universal quantum matching machine". Physical Review A. 66 (2): 022303. arXiv: quant-ph/0202173 . Bibcode:2002PhRvA..66b2303S. doi:10.1103/PhysRevA.66.022303. S2CID   119383508.
  21. ↑ Sentís, Gael; Guţă, Mădălin; Adesso, Gerardo (2015-07-09). "Quantum learning of coherent states". EPJ Quantum Technology. 2 (1): 17. arXiv: 1410.8700 . doi:10.1140/epjqt/s40507-015-0030-4. ISSN   2196-0763. S2CID   6980007.
  22. ↑ Lee, Sang Min; Lee, Jinhyoung; Bang, Jeongho (2018-11-02). "Learning unknown pure quantum states". Physical Review A. 98 (5): 052302. arXiv: 1805.06580 . Bibcode:2018PhRvA..98e2302L. doi:10.1103/PhysRevA.98.052302. S2CID   119095806.
  23. ↑ Zahedinejad, Ehsan; Ghosh, Joydip; Sanders, Barry C. (2016-11-16). "Designing High-Fidelity Single-Shot Three-Qubit Gates: A Machine Learning Approach". Physical Review Applied. 6 (5): 054005. arXiv: 1511.08862 . Bibcode:2016PhRvP...6e4005Z. doi:10.1103/PhysRevApplied.6.054005. ISSN   2331-7019. S2CID   7299645.
  24. ↑ Banchi, Leonardo; Pancotti, Nicola; Bose, Sougato (2016-07-19). "Quantum gate learning in qubit networks: Toffoli gate without time-dependent control". npj Quantum Information . 2: 16019. Bibcode:2016npjQI...216019B. doi: 10.1038/npjqi.2016.19 . hdl: 11858/00-001M-0000-002C-AA64-F .
  25. ↑ Ness, Gal; Vainbaum, Anastasiya; Shkedrov, Constantine; Florshaim, Yanay; Sagi, Yoav (2020-07-06). "Single-exposure absorption imaging of ultracold atoms using deep learning". Physical Review Applied. 14 (1): 014011. arXiv: 2003.01643 . Bibcode:2020PhRvP..14a4011N. doi:10.1103/PhysRevApplied.14.014011. S2CID   211817864.
  26. ↑ von Lilienfeld, O. Anatole (2018-04-09). "Quantum Machine Learning in Chemical Compound Space". Angewandte Chemie International Edition. 57 (16): 4164–4169. doi:10.1002/anie.201709686. PMID   29216413.
  27. ↑ Bartok, Albert P.; Payne, Mike C.; Risi, Kondor; Csanyi, Gabor (2010). "Gaussian approximation potentials: The accuracy of quantum mechanics, without the electrons" (PDF). Physical Review Letters. 104 (13): 136403. arXiv: 0910.1019 . Bibcode:2010PhRvL.104m6403B. doi:10.1103/PhysRevLett.104.136403. PMID   20481899. S2CID   15918457.
  28. ↑ Rupp, Matthias; Tkatchenko, Alexandre; Müller, Klaus-Robert; von Lilienfeld, O. Anatole (2012-01-31). "Fast and Accurate Modeling of Molecular Atomization Energies With Machine Learning". Physical Review Letters. 355 (6325): 602. arXiv: 1109.2618 . Bibcode:2012PhRvL.108e8301R. doi:10.1103/PhysRevLett.108.058301. PMID   22400967. S2CID   321566.
  29. ↑ Xia, Rongxin; Kais, Sabre (2018-10-10). "Quantum machine learning for electronic structure calculations". Nature Communications. 9 (1): 4195. arXiv: 1803.10296 . Bibcode:2018NatCo...9.4195X. doi:10.1038/s41467-018-06598-z. PMC   6180079 . PMID   30305624.
  30. ↑ van Nieuwenburg, Evert; Liu, Ye-Hua; Huber, Sebastian (2017). "Learning phase transitions by confusion". Nature Physics. 13 (5): 435. arXiv: 1610.02048 . Bibcode:2017NatPh..13..435V. doi:10.1038/nphys4037. S2CID   119285403.
  31. ↑ Hentschel, Alexander (2010-01-01). "Machine Learning for Precise Quantum Measurement". Physical Review Letters. 104 (6): 063603. arXiv: 0910.0762 . Bibcode:2010PhRvL.104f3603H. doi:10.1103/PhysRevLett.104.063603. PMID   20366821. S2CID   14689659.
  32. ↑ Quek, Yihui; Fort, Stanislav; Ng, Hui Khoon (2018-12-17). "Adaptive Quantum State Tomography with Neural Networks". arXiv: 1812.06693 [quant-ph].
  33. ↑ "Variational Circuits — Quantum Machine Learning Toolbox 0.7.1 documentation". qmlt.readthedocs.io. Retrieved 2018-12-06.
  34. ↑ Schuld, Maria (2018-06-12). "Quantum Machine Learning 1.0". XanaduAI. Retrieved 2018-12-07.
  35. ↑ Alexandru, Andrei; Bedaque, Paulo F.; Lamm, Henry; Lawrence, Scott (2017). "Deep Learning Beyond Lefschetz Thimbles". Physical Review D. 96 (9): 094505. arXiv: 1709.01971 . Bibcode:2017PhRvD..96i4505A. doi:10.1103/PhysRevD.96.094505. S2CID   119074823.
  36. ↑ Raissi, M.; Perdikaris, P.; Karniadakis, G. E. (2019-02-01). "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations". Journal of Computational Physics. 378: 686–707. Bibcode:2019JCoPh.378..686R. doi: 10.1016/j.jcp.2018.10.045 . ISSN   0021-9991. OSTI   1595805. S2CID   57379996.
  37. ↑ Mao, Zhiping; Jagtap, Ameya D.; Karniadakis, George Em (2020-03-01). "Physics-informed neural networks for high-speed flows". Computer Methods in Applied Mechanics and Engineering. 360: 112789. Bibcode:2020CMAME.360k2789M. doi: 10.1016/j.cma.2019.112789 . ISSN   0045-7825. S2CID   212755458.
  38. ↑ Raissi, Maziar; Yazdani, Alireza; Karniadakis, George Em (2020-02-28). "Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations". Science. 367 (6481): 1026–1030. Bibcode:2020Sci...367.1026R. doi:10.1126/science.aaw4741. PMC   7219083 . PMID   32001523.
  39. 1 2 Piloto, Luis S.; Weinstein, Ari; Battaglia, Peter; Botvinick, Matthew (11 July 2022). "Intuitive physics learning in a deep-learning model inspired by developmental psychology". Nature Human Behaviour. 6 (9): 1257–1267. doi: 10.1038/s41562-022-01394-8 . ISSN   2397-3374. PMC   9489531 . PMID   35817932.
  40. 1 2 "DeepMind AI learns physics by watching videos that don't make sense". New Scientist. Retrieved 21 August 2022.
  41. 1 2 Feldman, Andrey (11 August 2022). "Artificial physicist to unravel the laws of nature". Advanced Science News. Retrieved 21 August 2022.
  42. ↑ Chen, Boyuan; Huang, Kuang; Raghupathi, Sunand; Chandratreya, Ishaan; Du, Qiang; Lipson, Hod (July 2022). "Automated discovery of fundamental variables hidden in experimental data". Nature Computational Science. 2 (7): 433–442. doi:10.1038/s43588-022-00281-6. ISSN   2662-8457. S2CID   251087119.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.