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Abstract Classical molecular dynamics simulations are based on solving Newton’s equations of motion. Using a small timestep, numerical integrators such as Verlet generate trajectories of particles as solutions to Newton’s equations. We introduce operators derived using recurrent neural networks that accurately solve Newton’s equations utilizing sequences of past trajectory data, and produce energy-conserving dynamics of particles using timesteps up to 4000 times larger compared to the Verlet timestep. We demonstrate significant speedup in many example problems including 3D systems of up to 16 particles.
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SOLVING DIFFERENCE EQUATIONS IN SEQUENCES: UNIVERSALITY AND UNDECIDABILITY
We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assumption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable:
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- PAR ID:
- 10169850
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 8
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fms.2020.14
