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We develop a statistical method to learn a molecular Hamiltonian matrix from a time-series of electron density matrices. We extend our previous method to larger molecular systems by incorporating physical properties to reduce dimensionality, while also exploiting regularization techniques like ridge regression for addressing multicollinearity. With the learned Hamiltonian we can solve the Time-Dependent Hartree-Fock (TDHF) equation to propagate the electron density in time, and predict its dynamics for field-free and field-on scenarios. We observe close quantitative agreement between the predicted dynamics and ground truth for both field-off trajectories similar to the training data, and field-on trajectories outside of the training data.

We have used the Arecibo Telescope and the Green Bank Telescope to carry out a deep search for H{\sc i}~21\,cm emission from a large sample of Green Pea galaxies, yielding 19 detections, and 21 upper limits on the H{\sc i} mass. We obtain H{\sc i} masses of $\rm M_{HI} \approx (4-300) \times 10^8 \, \rm M_\odot$ for the detections, with a median H{\sc i} mass of $\approx 2.6 \times 10^9 \, \rm M_\odot$; for the non-detections, the median $3\sigma$ upper limit on the H{\sc i} mass is $\approx 5.5 \times 10^8 \, \rm M_\odot$. These are the first estimates of the atomic gas content of Green Pea galaxies. We find that the H{\sc i}-to-stellar mass ratio in Green Peas is consistent with trends identified in star-forming galaxies in the local Universe. However, the median H{\sc i} depletion timescale in Green Peas is $\approx 0.6$~Gyr, an order of magnitude lower than that obtained in local star-forming galaxies. This implies that Green Peas consume their atomic gas on very short timescales. A significant fraction of the Green Peas of our sample lie $\gtrsim 0.6$~dex ($2\sigma$) above the local $\rm M_{HI} - M_B$ relation, suggesting recent gas accretion. Further, $\approx 30$\% of the Green Peas are more than $\pm 2\sigma$ deviant from this relation, suggesting possible bimodality in the Green Pea population. We obtain a low H{\sc i}~21\,cm detection rate in the Green Peas with the highest O32~$\equiv$~[O{\sc iii}]$\lambda$5007/[O{\sc ii}]$\lambda$3727 luminosity ratios, O32~$> 10$, consistent with the high expected Lyman-continuum leakage from these galaxies., Comment: 8 pages, 3 pages, Astrophysical Journal Letters, in press

While there has been a surge of recent interest in learning differential equation models from time series, methods in this area typically cannot cope with highly noisy data. We break this problem into two parts: (i) approximating the unknown vector field (or right-hand side) of the differential equation, and (ii) dealing with noise. To deal with (i), we describe a neural network architecture consisting of tensor products of one-dimensional neural shape functions. For (ii), we propose an alternating minimization scheme that switches between vector field training and filtering steps, together with multiple trajectories of training data. We find that the neural shape function architecture retains the approximation properties of dense neural networks, enables effective computation of vector field error, and allows for graphical interpretability, all for data/systems in any finite dimension $d$. We also study the combination of either our neural shape function method or existing differential equation learning methods with alternating minimization and multiple trajectories. We find that retrofitting any learning method in this way boosts the method's robustness to noise. While in their raw form the methods struggle with 1% Gaussian noise, after retrofitting, they learn accurate vector fields from data with 10% Gaussian noise., Comment: Accepted for publication in the Proceedings of the 54th Asilomar Conference on Signals, Systems, and Computers

Certain neural network architectures, in the infinite-layer limit, lead to systems of nonlinear differential equations. Motivated by this idea, we develop a framework for analyzing time signals based on non-autonomous dynamical equations. We view the time signal as a forcing function for a dynamical system that governs a time-evolving hidden variable. As in equation discovery, the dynamical system is represented using a dictionary of functions and the coefficients are learned from data. This framework is applied to the time signal classification problem. We show how gradients can be efficiently computed using the adjoint method, and we apply methods from dynamical systems to establish stability of the classifier. Through a variety of experiments, on both synthetic and real datasets, we show that the proposed method uses orders of magnitude fewer parameters than competing methods, while achieving comparable accuracy. We created the synthetic datasets using dynamical systems of increasing complexity; though the ground truth vector fields are often polynomials, we find consistently that a Fourier dictionary yields the best results. We also demonstrate how the proposed method yields graphical interpretability in the form of phase portraits., Comment: 25 pages, 2 figures

We develop a computational method to learn a molecular Hamiltonian matrix from matrix-valued time series of the electron density. As we demonstrate for three small molecules, the resulting Hamiltonians can be used for electron density evolution, producing highly accurate results even when propagating 1000 time steps beyond the training data. As a more rigorous test, we use the learned Hamiltonians to simulate electron dynamics in the presence of an applied electric field, extrapolating to a problem that is beyond the field-free training data. We find that the resulting electron dynamics predicted by our learned Hamiltonian are in close quantitative agreement with the ground truth. Our method relies on combining a reduced-dimensional, linear statistical model of the Hamiltonian with a time-discretization of the quantum Liouville equation within time-dependent Hartree Fock theory. We train the model using a least-squares solver, avoiding numerous, CPU-intensive optimization steps. For both field-free and field-on problems, we quantify training and propagation errors, highlighting areas for future development.

We consider the problem of learning an interpretable potential energy function from a Hamiltonian system's trajectories. We address this problem for classical, separable Hamiltonian systems. Our approach first constructs a neural network model of the potential and then applies an equation discovery technique to extract from the neural potential a closed-form algebraic expression. We demonstrate this approach for several systems, including oscillators, a central force problem, and a problem of two charged particles in a classical Coulomb potential. Through these test problems, we show close agreement between learned neural potentials, the interpreted potentials we obtain after training, and the ground truth. In particular, for the central force problem, we show that our approach learns the correct effective potential, a reduced-order model of the system., Comment: 12 pages