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The nonlinear Schrödinger equation (NLSE) in one spatial dimension has stationary solutions similar to those of the linear Schrödinger equation (LSE) as well as more exotic solutions such as solitary waves and quantum droplets. Here, we present a newly discovered conformal duality which unifies the stationary and time-dependent traveling-wave solutions of the one-dimensional cubic-quintic NLSE, the cubic NLSE and LSE. Any two systems that are classified by the same single number called the cross ratio are related by this symmetry. Notably, the conformal duality can also be adapted in Newtonian mechanics and serves as a powerful tool for investigating physical systems that otherwise cannot be directly accessed in experiments. Further, we show that the conformal symmetry is a valuable resource to substantially improve NLSE parameter estimation from noisy empirical data by introducing an optimization afterburner. The new method therefore has far reaching practical applications for nonlinear physical systems.

Solving the generalized eigenvalue problem is a useful method for finding energy eigenstates of large quantum systems. It uses projection onto a set of basis states which are typically not orthogonal. One needs to invert a matrix whose entries are inner products of the basis states, and the process is unfortunately susceptible to even small errors. The problem is especially bad when matrix elements are evaluated using stochastic methods and have significant error bars. In this work, we introduce the trimmed sampling algorithm in order to solve this problem. Using the framework of Bayesian inference, we sample prior probability distributions determined by uncertainty estimates of the various matrix elements and likelihood functions composed of physics-informed constraints. The result is a probability distribution for the eigenvectors and observables which automatically comes with a reliable estimate of the error and performs far better than standard regularization methods. The method should have immediate use for a wide range of applications involving classical and quantum computing calculations of large quantum systems.

Emulators that can bypass computationally expensive scientific calculations with high accuracy and speed can enable new studies of fundamental science as well as more potential applications. In this work we discuss solving a system of constraint equations efficiently using a self-learning emulator. A self-learning emulator is an active learning protocol that can be used with any emulator that faithfully reproduces the exact solution at selected training points. The key ingredient is a fast estimate of the emulator error that becomes progressively more accurate as the emulator is improved, and the accuracy of the error estimate can be corrected using machine learning. We illustrate with three examples. The first uses cubic spline interpolation to find the solution of a transcendental equation with variable coefficients. The second example compares a spline emulator and a reduced basis method emulator to find solutions of a parametrized differential equation. The third example uses eigenvector continuation to find the eigenvectors and eigenvalues of a large Hamiltonian matrix that depends on several control parameters.

We investigate the impact of the QCD vacuum at nonzero θ on the properties of light nuclei, Big Bang nucleosynthesis, and stellar nucleosynthesis. Our analysis starts with a calculation of the θ-dependence of the neutron-proton mass difference and neutron decay using chiral perturbation theory. We then discuss the θ-dependence of the nucleon-nucleon interaction using a one-boson-exchange model and compute the properties of the two-nucleon system. Using the universal properties of four-component fermions at large scattering length, we then deduce the binding energies of the three-nucleon and four-nucleon systems. Based on these results, we discuss the implications for primordial abundances of light nuclei, the production of nuclei in stellar environments, and implications for an anthropic view of the universe.