Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning

Conservation laws are considered to be fundamental laws of nature. It has broad applications in many fields, including physics, chemistry, biology, geology, and engineering. Solving the differential equations associated with conservation laws is a major branch in computational mathematics. The recent success of machine learning, especially deep learning in areas such as computer vision and natural language processing, has attracted a lot of attention from the community of computational mathematics and inspired many intriguing works in combining machine learning with traditional methods. In this paper, we are the first to view numerical PDE solvers as an MDP and to use (deep) RL to learn new solvers. As proof of concept, we focus on 1-dimensional scalar conservation laws. We deploy the machinery of deep reinforcement learning to train a policy network that can decide on how the numerical solutions should be approximated in a sequential and spatial-temporal adaptive manner. We will show that the problem of solving conservation laws can be naturally viewed as a sequential decision-making process, and the numerical schemes learned in such a way can easily enforce long-term accuracy. Furthermore, the learned policy network is carefully designed to determine a good local discrete approximation based on the current state of the solution, which essentially makes the proposed method a meta-learning approach. In other words, the proposed method is capable of learning how to discretize for a given situation mimicking human experts. Finally, we will provide details on how the policy network is trained, how well it performs compared with some state-of-the-art numerical solvers such as WENO schemes, and supervised learning based approach L3D and PINN, and how well it generalizes.     

Psychosocial mechanisms of a behavioral treatment for urinary incontinence of prostate cancer survivors

Abstract

Purpose: We examined underlying psychosocial processes of a behavioral treatment for urinary incontinence (UI) of prostate cancer survivors.

Design: Secondary analysis of data collected from a clinical trial.

Sample: Two hundred forty-four prostate cancer survivors who participated in a clinical trial of behavioral intervention to UI as intervention or control subjects.

Methods: The participants had a 3-month behavioral intervention or usual care and were followed up for an additional 3?months. They were assessed at baseline, 3, and 6?months. Latent growth curve models were performed to examine trajectories of each study variable and relationships among the variables.

Findings: Increasing self-efficacy and social support were significantly and independently associated with more reduction of urinary leakage frequency over time.

Implications for psychosocial oncology: Providing problem-solving skills and social support, including peer support, are essential for empowering patients to reduce UI.