PDE-LEARN: Using deep learning to discover partial differential equations from noisy, limited data.

In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational Neural Network, U, to approximate the system response function and a sparse, trainable vector, ξ, to characterize the hidden PDE that the system response function satisfies. Our approach couples the training of U and ξ using a loss function that (1) makes U approximate the system response function, (2) encapsulates the fact that U satisfies a hidden PDE that ξ characterizes, and (3) promotes sparsity in ξ using ideas from iteratively reweighted least-squares. Further, PDE-LEARN can simultaneously learn from several data sets, allowing it to incorporate results from multiple experiments. This approach yields a robust algorithm to discover PDEs directly from realistic scientific data. We demonstrate the efficacy of PDE-LEARN by identifying several PDEs from noisy and limited measurements.

PDE-READ: Human-readable partial differential equation discovery using deep learning.

PDE discovery shows promise for uncovering predictive models of complex physical systems but has difficulty when measurements are noisy and limited. We introduce a new approach for PDE discovery that uses two Rational Neural Networks and a principled sparse regression algorithm to identify the hidden dynamics that govern a system's response. The first network learns the system response function, while the second learns a hidden PDE describing the system's evolution. We then use a parameter-free sparse regression algorithm to extract a human-readable form of the hidden PDE from the second network. We implement our approach in an open-source library called PDE-READ. Our approach successfully identifies the governing PDE in six benchmark examples. We demonstrate that our approach is robust to both sparsity and noise and it, therefore, holds promise for application to real-world observational data.

Data-driven discovery of Green's functions with human-understandable deep learning.

There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner. To do this, we develop a novel data-driven approach for creating a human-machine partnership to accelerate scientific discovery. By collecting physical system responses under excitations drawn from a Gaussian process, we train rational neural networks to learn Green's functions of hidden linear partial differential equations. These functions reveal human-understandable properties and features, such as linear conservation laws and symmetries, along with shock and singularity locations, boundary effects, and dominant modes. We illustrate the technique on several examples and capture a range of physics, including advection-diffusion, viscous shocks, and Stokes flow in a lid-driven cavity.

Housing Trajectories of Forensic Psychiatric Patients.

The objectives of this study were to describe the disposition and housing trajectories of individuals found Not Criminally Responsible on account of Mental Disorder (NCRMD), and the factors that predict different trajectories. To do so, disposition and housing status were coded for 934 NCRMD patients over a 36-month follow-up period. Sequential data analysis resulted in four distinct trajectories: detention in hospital, conditional discharge in supportive housing, conditional discharge in independent housing, and absolute discharge to unknown housing. The likelihood of a placement in supportive housing compared with detention significantly decreased for individuals with a higher index offense severity. Less restrictive trajectories were significantly predicted by clinical factors. The results revealed little change in the disposition and housing trajectories of NCRMD patients. Furthermore, decisions about disposition and housing placement reflect a knowledge-practice gap between risk factors known to be predictive of community resources use in the forensic population. Copyright © 2016 John Wiley & Sons, Ltd.

A case study of preferential bestiality.

In a previous article, we presented phallometric data to illustrate a case of preferential bestiality or zoophilia (Earls & Lalumière, Sex Abuse: J Res Treat, 14:83-88, 2002). Based on the available literature, we argued that a marked preference for having sex with animals over sex with humans is extremely rare. In the present article, we describe a second case of zoophilia that challenges the widely held assumptions that men who have sex with animals are generally of below average intelligence and come from rural areas. In addition, we provide a brief review of a burgeoning quantitative literature using large groups of zoophiles recruited from internet sources. Although estimates of the prevalence of zoophilia are not possible at this time, it appears that zoophilia is not as rare as once thought and shares many features with other atypical sexual interests.

Birth order, birth interval, and deviant sexual preferences among sex offenders.

We investigated whether birth order and birth interval are associated with phallometric responding among 168 sex offenders. Based on the immunoreactivity hypothesis, we predicted that a larger number of older brothers and a shorter birth interval between the offenders and their next older brothers would be associated with more deviant responding. Results showed that a larger number of older brothers (but not older sisters, younger brothers, or younger sisters) was associated with more deviant responding on the Most Deviant Index, a general measure of sexual deviance. A larger number of older brothers was associated with arousal to nonsexual violence against women and children, but contrary to expectation, the same association was also found for the number of younger brothers. A short birth interval with the next older brother was associated with greater relative arousal to nonsexual violence, but the correlation with the Most Deviant Index did not reach statistical significance. Overall, results were consistent with the main predictions, but the effects of birth order and birth interval were mostly restricted to arousal to nonsexual violence. Birth order and birth interval were not associated with pedophilic interests or gender preferences.

A case study of preferential bestiality (zoophilia).

Humans show a wide array of sexual preferences and behaviors. Although most humans prefer and have sex with consenting adults of the opposite sex, some individuals have unconventional preferences with regard to the sex or age of sexual partners, or with regard to the nature of sexual activities. In this paper, we describe a rare case of preferential bestiality, or zoophilia. The client meets the most stringent criteria for the diagnosis of zoophilia. In particular, his phallometrically measured arousal pattern shows a sexual preference for horses over other species, including humans.