Error analysis of a first-order IMEX scheme for the logarithmic Schr\"odinger equation

The logarithmic Schr\"odinger equation (LogSE) has a logarithmic nonlinearity $f(u)=u\ln |u|^2$ that is not differentiable at $u=0.$ Compared with its counterpart with a regular nonlinear term, it possesses richer and unusual dynamics, though the low regularity of the nonlinearity brings about significant challenges in both analysis and computation. Among very limited numerical studies, the semi-implicit regularized method via regularising $f(u)$ as $u^{\varepsilon}\ln ({\varepsilon}+ |u^{\varepsilon}|)^2$ to overcome the blowup of $\ln |u|^2$ at $u=0$ has been investigated recently in literature. With the understanding of $f(0)=0,$ we analyze the non-regularized first-order Implicit-Explicit (IMEX) scheme for the LogSE. We introduce some new tools for the error analysis that include the characterization of the H\"older continuity of the logarithmic term, and a nonlinear Gr\"{o}nwall's inequality. We provide ample numerical results to demonstrate the expected convergence. We position this work as the first one to study the direct linearized scheme for the LogSE as far as we can tell.Comment: 19 pages, 5 figure

Toward a Brain-Inspired System: Deep Recurrent Reinforcement Learning for a Simulated Self-Driving Agent

An effective way to achieve intelligence is to simulate various intelligent behaviors in the human brain. In recent years, bio-inspired learning methods have emerged, and they are different from the classical mathematical programming principle. From the perspective of brain inspiration, reinforcement learning has gained additional interest in solving decision-making tasks as increasing neuroscientific research demonstrates that significant links exist between reinforcement learning and specific neural substrates. Because of the tremendous research that focuses on human brains and reinforcement learning, scientists have investigated how robots can autonomously tackle complex tasks in the form of making a self-driving agent control in a human-like way. In this study, we propose an end-to-end architecture using novel deep-Q-network architecture in conjunction with a recurrence to resolve the problem in the field of simulated self-driving. The main contribution of this study is that we trained the driving agent using a brain-inspired trial-and-error technique, which was in line with the real world situation. Besides, there are three innovations in the proposed learning network: raw screen outputs are the only information which the driving agent can rely on, a weighted layer that enhances the differences of the lengthy episode, and a modified replay mechanism that overcomes the problem of sparsity and accelerates learning. The proposed network was trained and tested under a third-party OpenAI Gym environment. After training for several episodes, the resulting driving agent performed advanced behaviors in the given scene. We hope that in the future, the proposed brain-inspired learning system would inspire practicable self-driving control solutions

Application of Model-Based Time Series Prediction of Infrared Long-Wave Radiation Data for Exploring the Precursory Patterns Associated with the 2021 Madoi Earthquake

Taking the Madoi MS 7.4 earthquake of 21 May 2021 as an example, this paper proposes using time series prediction models to predict the outgoing long-wave radiation (OLR) anomalies and study short-term pre-earthquake signals. Five time series prediction models, including autoregressive integrated moving average (ARIMA) and long short-term memory (LSTM), were trained with the OLR time series data of the aseismic moments in the 5° × 5° spatial range around the epicenter. The model with the highest prediction accuracy was selected to retrospectively predict the OLR values during the aseismic period and before the earthquake in the area. It was found, by comparing the predicted time series values with the actual time series value, that the similarity indexes of the two time series before the earthquake were lower than the index of the aseismic period, indicating that the predicted time series before the earthquake significantly differed from the actual time series. Meanwhile, the temporal and spatial distribution characteristics of the anomalies in the 90 days before the earthquake were analyzed with a 95% confidence interval as the criterion of the anomalies, and the following was found: out of 25 grids, 18 grids showed anomalies—the anomalies of the different grids appeared on similar dates, and the anomalies of high values appeared centrally at the time of the earthquake, which supports the hypothesis that pre-earthquake signals may be associated with the earthquake

Efficient and Direct Inference of Heart Rate Variability using Both Signal Processing and Machine Learning

Heart Rate Variability (HRV) measures the variation of the time between consecutive heartbeats and is a major indicator of physical and mental health. Recent research has demonstrated that photoplethysmography (PPG) sensors can be used to infer HRV. However, many prior studies had high errors because they only employed signal processing or machine learning (ML), or because they indirectly inferred HRV, or because there lacks large training datasets. Many prior studies may also require large ML models. The low accuracy and large model sizes limit their applications to small embedded devices and potential future use in healthcare. To address the above issues, we first collected a large dataset of PPG signals and HRV ground truth. With this dataset, we developed HRV models that combine signal processing and ML to directly infer HRV. Evaluation results show that our method had errors between 3.5% to 25.7% and outperformed signal-processing-only and ML-only methods. We also explored different ML models, which showed that Decision Trees and Multi-level Perceptrons have 13.0% and 9.1% errors on average with models at most hundreds of KB and inference time less than 1ms. Hence, they are more suitable for small embedded devices and potentially enable the future use of PPG-based HRV monitoring in healthcare