Machine learning conservation laws from differential equations

Liu, Ziming; Madhavan, Varun; Tegmark, Max

doi:10.1103/physreve.106.045307

Cited by 24 publications

(10 citation statements)

References 16 publications

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“…The key here is all: software which will nominally find a single conserved quantity does exist (but does not always perform in a timely manner). A very recent article Liu et al (2022) appeared once the work here was substantially complete and may suggest a solution. Given the problem domain in which we are operating (galaxies), we are able to provide support to any symbolic regression packages trying to find algebraic formulae from trajectory data.…”

Section: Discussionmentioning

confidence: 85%

“…The reason we say "appears to" above is that no robust mechanism yet exists for determining the algebraic formulae for multiple conserved quantities from a single trajectory in a timely manner, though it may be that the investigations of Liu et al (2022) will suggest a viable solution. Addressing this symbolic regression issue needs to be tackled with some urgency, and this or a similar investigation repeated.…”

Section: Discussionmentioning

confidence: 99%

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Stellar Dynamical Modeling—Counting Conserved Quantities

Long

Mao

Wang

2023

Res. Astron. Astrophys.

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Knowing the conserved quantities that a galaxy's stellar orbits conform to is important in helping us understand the stellar distribution and structures within the galaxy. Isolating integrals of motion and resonances are particularly important, non-isolating integrals less so. We compare the behavior and results of two methods for counting the number of conserved quantities, one based on the correlation integral approach and the other a more recent method using machine learning. Both methods use stellar orbit trajectories in phase space as their only input, and we create such trajectories from theoretical spherical, axisymmetric and triaxial model galaxies. The orbits have known isolating integrals and resonances. We find that neither method is fully effective in recovering the numbers of these quantities, nor in determining the number of non-isolating integrals. From a computer performance perspective, we find the correlation integral approach to be faster. Determining the algebraic formulae of (multiple) conserved quantities from the trajectories has not been possible due to the lack of an appropriate symbolic regression capability. Notwithstanding the shortcomings we have noted, it may that the methods are usable as part of a trajectory analysis tool kit.

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Section: Discussionmentioning

confidence: 85%

Section: Discussionmentioning

confidence: 99%

Stellar Dynamical Modeling—Counting Conserved Quantities

Long

Mao

Wang

2023

Res. Astron. Astrophys.

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“…In one of these works mentioned, the Schrödinger equation is solved for a particle inside a Pöschl-Teller potential, they obtain the ground state energy and its wavefunction. We would like to highlight the work by Radu and Duque, who presented a treatment based in neural networks for solving the Schrödinger equation for arbitrary potentials [18][19][20][21][22].…”

Section: Discussionmentioning

confidence: 99%

“…For stationary states of ( ) x , y the Schrödinger equation becomes an eigenvalue problem, as seen in (19):…”

Section: About the Variational And Shooting Methodsmentioning

confidence: 99%

Solution to the schrödinger equation for bound states of polar molecules using shallow neural networks

Silva-Molina,

Jimenez-Valencia,

Castellanos-Jaramillo

et al. 2024

Phys. Scr.

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The time independent Schrödinger equation can be solved analytically in very few cases. This is why scientists rely on approximate numerical methods. One that is yet to be a part of the everyday toolbox for physicists are neural networks. With it, it is possible to calculate eigenfunctions and the energy spectra of bound states. A double exponential central potential is used to describe the interaction between two electric dipoles that form through van der Waals interactions. These systems can be important when the electric dipoles are very large and differ from covalent, ionic, and other kinds of bonds, as is the case of the monomers M-C_60, where M={Cs,Li} or other alkali metals. Based in statistical physics, it has been shown that a dimer can be formed through a spontaneous and exothermic reaction. The ground state and the first two excited states of this system were calculated, also the solutions and their associated energies were plotted. The result for the ground state is compared to the shooting method and the direct variational method. The numerical value of the ground states obtained suggests that the dimers can be dissociated by electromagnetic waves with wavelengths in the far infrared regime, very close to microwaves.

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“…Thus, N I is at least 3 (in our case), and there is at least one hidden symmetry. Starting from particle trajectory data in the equilibrium electromagnetic fields, future research on this topic could make use of machine learning models to help find the number of invariants or even discover the analytic formula of these invariants (e.g., Liu & Tegmark 2021;Liu et al 2022).…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Kinetic Equilibrium of Two-dimensional Force-free Current Sheets

Artemyev

Angelopoulos

et al. 2023

ApJ

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Force-free current sheets are local plasma structures with field-aligned electric currents and approximately uniform plasma pressures. Such structures, widely found throughout the heliosphere, are sites for plasma instabilities and magnetic reconnection, the growth rate of which is controlled by the structure’s current-sheet configuration. Despite the fact that many kinetic equilibrium models have been developed for one-dimensional force-free current sheets, their two-dimensional (2D) counterparts, which have a magnetic field component normal to the current sheets, have not received sufficient attention to date. Here, using particle-in-cell simulations, we search for such 2D force-free current sheets through relaxation from an initial, magnetohydrodynamic equilibrium. Kinetic equilibria are established toward the end of our simulations, thus demonstrating the existence of kinetic force-free current sheets. Although the system currents in the late equilibrium state remain field aligned as in the initial configuration, the velocity distribution functions of both ions and electrons systematically evolve from their initial drifting Maxwellians to their final time-stationary Vlasov state. The existence of 2D force-free current sheets at kinetic equilibrium necessitates future work in discovering additional integrals of motion of the system, constructing the kinetic distribution functions, and eventually investigating their stability properties.

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Machine learning conservation laws from differential equations

Cited by 24 publications

References 16 publications

Stellar Dynamical Modeling—Counting Conserved Quantities

Stellar Dynamical Modeling—Counting Conserved Quantities

Solution to the schrödinger equation for bound states of polar molecules using shallow neural networks

Kinetic Equilibrium of Two-dimensional Force-free Current Sheets

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