@article{holliday_lindner_ditto_2023, title={Solving quantum billiard eigenvalue problems with physics-informed machine learning}, volume={13}, ISSN={["2158-3226"]}, 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{xie_bae_lindner_2022, title={Alien suns reversing in exoplanet skies}, volume={12}, ISSN={["2045-2322"]}, DOI={10.1038/s41598-022-11527-8}, abstractNote={Earth's rapid spin, modest tilt, and nearly circular orbit ensure that the sun always appears to move forward, rising in the east and setting in the west. However, for some exoplanets, solar motion can reverse causing alien suns to apparently move backward. Indeed, this dramatic motion marginally occurs for Mercury in our own solar system. For exoplanetary observers, we study the scope of solar motion as a function of eccentricity, spin-orbit ratio, obliquity, and nodal longitude, and we visualize the motion in spatial and spacetime plots. For zero obliquity, reversals occur when a planet's spin angular speed is between its maximum and minimum orbital angular speeds, and we derive exact nonlinear equations for eccentricity and spin-orbit to bound reversing and non-reversing motion. We generalize the notion of solar day to gracefully handle the most common reversals.}, number={1}, journal={SCIENTIFIC REPORTS}, author={Xie, Xinchen and Bae, Hwan and Lindner, John F.}, year={2022}, month={May} }
 @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} }
 @article{miller_lindner_choudhary_sinha_ditto_2021, title={Negotiating the separatrix with machine learning}, volume={12}, ISSN={["2185-4106"]}, url={https://doi.org/10.1587/nolta.12.134}, DOI={10.1587/nolta.12.134}, abstractNote={Physics-informed machine learning has recently been shown to efficiently learn complex trajectories of nonlinear dynamical systems, even when order and chaos coexist. However, care must be taken when one or more variables are unbounded, such as in rotations. Here we use the framework of Hamiltonian Neural Networks (HNN) to learn the complex dynamics of nonlinear single and double pendulums, which can both librate and rotate, by mapping the unbounded phase space onto a compact cylinder. We clearly demonstrate that our approach can successfully forecast the motion of these challenging systems, capable of both bounded and unbounded motion. It is also evident that HNN can yield an energy surface that closely matches the surface generated by the true Hamiltonian function. Further we observe that the relative energy error for HNN decreases as a power law with number of training pairs, with HNN clearly outperforming conventional neural networks quantitatively.}, number={2}, journal={IEICE NONLINEAR THEORY AND ITS APPLICATIONS}, author={Miller, Scott T. and Lindner, John F. and Choudhary, Anshul and Sinha, Sudeshna and Ditto, William L.}, year={2021}, pages={134–142} }