In nonlinear science, it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and Painlev\'e waves. Actually, even if for the most well-known prototypical models such as the Kortewet\char21{}de Vries (KdV) equation and the Kadomtsev-Petviashvili (KP) equation, this kind of problem has not yet been solved. In this paper, the explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetries which are related to Darboux transformation for the well-known KdV equation. The same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves, rational solutions, Bessel function solutions, and/or general Painlev\'e II solutions.

We propose a one-shot, privacy-preserving distributed algorithm to perform logistic regression (ODAL) across multiple clinical sites. ODAL effectively utilizes the information from the local site (where the patient-level data are accessible) and incorporates the first-order (ODAL1) and second-order (ODAL2) gradients of the likelihood function from other sites to construct an estimator without requiring iterative communication across sites or transferring patient-level data. We evaluated ODAL via extensive simulation studies and an application to a dataset from the University of Pennsylvania Health System. The estimation accuracy was evaluated by comparing it with the estimator based on the combined individual participant data or pooled data (ie, gold standard). Our simulation studies revealed that the relative estimation bias of ODAL1 compared with the pooled estimates was <3%, and the ratio of standard errors was <1.25 for all scenarios. ODAL2 achieved higher accuracy (with relative bias <0.1% and ratio of standard errors <1.05). In real data analysis, we investigated the associations of 100 medications with fetal loss during pregnancy. We found that ODAL1 provided estimates with relative bias <10% for 85% of medications, and ODAL2 has relative bias <10% for 99% of medications. For communication cost, ODAL1 requires transferring $p$ numbers from each site to the local site and ODAL2 requires transferring $(p×p+p)$ numbers from each site to the local site, where $p$ is the number of parameters in the regression model. This study demonstrates that ODAL is privacy-preserving and communication-efficient with small bias and high statistical efficiency.

The nonlinear Schrödinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas. However, due to the difficulty of solving this equation, in particular in high dimensions, lots of methods are proposed to effectively obtain different kinds of solutions, such as neural networks among others. Recently, a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly. Compared with traditional neural networks, this method can obtain remarkably accurate solution with extraordinarily less data. Meanwhile, this method also provides a better physical explanation and generalization. In this paper, based on the above method, we present an improved deep learning method to recover the soliton solutions, breather solution, and rogue wave solutions of the nonlinear Schrödinger equation. In particular, the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time. Moreover, the effects of different numbers of initial points sampled, collocation points sampled, network layers, neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions. Numerical experiments show that the dynamical behaviors of soliton solutions, breather solution, and rogue wave solutions of the integrable nonlinear Schrödinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.