@article{holliday_lindner_ditto_2023, title={Solving quantum billiard eigenvalue problems with physics-informed machine learning}, volume={13}, ISSN={["2158-3226"]}, url={https://doi.org/10.1063/5.0161067}, DOI={10.1063/5.0161067}, abstractNote={A particle confined to an impassable box is a paradigmatic and exactly solvable one-dimensional quantum system modeled by an infinite square well potential. Here, we explore some of its infinitely many generalizations to two dimensions, including particles confined to rectangle-, ellipse-, triangle-, and cardioid-shaped boxes using physics-informed neural networks. In particular, we generalize an unsupervised learning algorithm to find the particles’ eigenvalues and eigenfunctions, even in cases where the eigenvalues are degenerate. During training, the neural network adjusts its weights and biases, one of which is the energy eigenvalue, so that its output approximately solves the stationary Schrödinger equation with normalized and mutually orthogonal eigenfunctions. The same procedure solves the Helmholtz equation for the harmonics and vibration modes of waves on drumheads or transverse magnetic modes of electromagnetic cavities. Related applications include quantum billiards, quantum chaos, and Laplacian spectra.}, number={8}, journal={AIP ADVANCES}, author={Holliday, Elliott G. G. and Lindner, John F. F. and Ditto, William L. L.}, year={2023}, month={Aug} }
 @article{choudhary_lindner_holliday_miller_sinha_ditto_2021, title={Forecasting Hamiltonian dynamics without canonical coordinates}, volume={103}, ISSN={["1573-269X"]}, DOI={10.1007/s11071-020-06185-2}, abstractNote={Conventional neural networks are universal function approximators, but they may need impractically many training data to approximate nonlinear dynamics. Recently introduced Hamiltonian neural networks can efficiently learn and forecast dynamical systems that conserve energy, but they require special inputs called canonical coordinates, which may be hard to infer from data. Here, we prepend a conventional neural network to a Hamiltonian neural network and show that the combination accurately forecasts Hamiltonian dynamics from generalised noncanonical coordinates. Examples include a predator–prey competition model where the canonical coordinates are nonlinear functions of the predator and prey populations, an elastic pendulum characterised by nontrivial coupling of radial and angular motion, a double pendulum each of whose canonical momenta are intricate nonlinear combinations of angular positions and velocities, and real-world video of a compound pendulum clock.}, number={2}, journal={NONLINEAR DYNAMICS}, author={Choudhary, Anshul and Lindner, John F. and Holliday, Elliott G. and Miller, Scott T. and Sinha, Sudeshna and Ditto, William L.}, year={2021}, month={Jan}, pages={1553–1562} }