[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-126865-en":3,"doc-seo-126865-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},126865,1099523885336,"Violet","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Rare events in a stochastic vegetation-water dynamical system based on machine learning - Abstract","Stochastic vegetation-water dynamical systems are central to ecological stability, biodiversity, water management, and adaptation to climate change. This study introduces a machine learning approach to analyze rare events under multiplicative Gaussian noise. Using Freidlin-Wentzell large deviation theory, the work derives asymptotic formulas for the quasipotential and the mean first exit time. A neural network based on vector-field decomposition computes most probable transition paths and mean first exit times for both non-characteristic and characteristic boundary cases, enabling early warnings of vegetation degradation.","arXiv :2402 . 18315v1 [math .DS] 28 Feb 2024  \nRare events in a stochastic vegetation-water dynamical system based on machine learning  \nYang Lia , Shenglan Yuanb,c,∗, Shengyuan Xua  \na School of Automation, Nanjing University of Science and Technology, 200 Xiaolingwei  \nStreet, Nanjing 210094, Jiangsu Province, China  \nb School of Sciences, Great Bay University, Songshan Lake International Innovation Entrepreneurship Community A5, Dongguan 523000, Guangdong Province, China c Great Bay Institute for Advanced Study, Songshan Lake International Innovation Entrepreneurship Community A5, Dongguan 523000, Guangdong Province, China  \nAbstract  \nStochastic vegetation-water dynamical systems play a pivotal role in ecological stability, biodiversity, water resource management, and adaptation to climate change. This research proposes a machine learning-based method for analyzing rare events in stochastic vegetation-water dynamical systems with multiplicative Gaussian noise. Utilizing the Freidlin-Wentzell large deviation theory, we derive the asymptotic expressions for the quasipotential and the mean first exit time. Based on the decomposition of vector field, we design a neural network architecture to compute the most probable transition pathsand the mean first exit time for both non-characteristic and characteristic boundary scenarios. The results indicate that this method can effectively predict early warnings of vegetation degradation, providing new theoretical foundations and mathematical tools for ecological management and conservation. Moreover, the method offers new possibilities for exploring more complex and higher-dimensional stochastic dynamical systems.  \nKeywords: Stochastic vegetation-water system, Rare events, Machine learning, Most probable path, Mean first exit time  \n􀀃 Corresponding author  \nEmail addresses: [liyangbx5433@163.com](liyangbx5433@163.com) (Yang Li), [shenglanyuan@gbu.edu.cn](shenglanyuan@gbu.edu.cn)  \n(Shenglan Yuan), [syxu@njust.edu.cn](syxu@njust.edu.cn) (Shengyuan Xu)  \nPreprint submitted to Applied Mathematics and Computation February 29, 2024  \n1. Introduction  \nThe stochastic noise is indeed very common in ecosystems, which refers to random fluctuations in ecological systems that are not predictable or explainable by deterministic factors [1, 2, 3] . These fluctuations can arise from a variety of sources, including environmental conditions, dispersal patterns and interactions among species, etc. Under random fluctuations, rare events [4], i.e., transition from one stable state to another, frequently occur, even for weak noise. Thus they are well worth investigating, such as vegetation degradation and species extinction [5] .  \nThe Freidlin-Wentzell large deviation theory [6] is actually a principle that is mainly applied to study rare events of dynamical systems with small random perturbations. As the noise intensity decreases, the convergence rate at which the sample trajectories converge to the reference orbit is exponential in terms of noise. Large deviation techniques are widely used, which have become an extremely active branch of applied probability. They can estimate the escape probability of stochastic systems, reckon the probability of deviation from the reference orbit, and quantify the asymptotic probability of errors in hypothesis testing.  \nIn large deviation theory, the action functional is an important concept that builds the relationship between the stationary probability distribution and the path distribution of stochastic dynamical systems [6] . It is defined as the integral of a certain Lagrangian, similar to classical mechanics. The minimization of the action functional leads to the most probable path connecting given initial and final states, along which the stochastic system moves with highest probability than other paths [7, 8] . Although variables in statistical mechanics do not move along a definite trajectory, the most probable path offers us a very intuitive way to comprehend the stoc","cbCaiaNQq9NqHt1m","https://ap.wps.com/l/cbCaiaNQq9NqHt1m","pdf",636244,1,27,"English","en",105,"# Introduction\n## Stochastic noise and rare events\n## Freidlin-Wentzell large deviation theory\n## Action functional, most probable path, and applications\n## Quasipotential and mean first exit time","[{\"question\":\"What rare events are studied in this stochastic vegetation-water system?\",\"answer\":\"The focus is on rare transitions between stable states, such as vegetation degradation and species extinction, occurring even under weak stochastic noise.\"},{\"question\":\"Which theoretical framework supports the asymptotic results?\",\"answer\":\"Freidlin-Wentzell large deviation theory is used to derive asymptotic expressions for the quasipotential and the mean first exit time.\"},{\"question\":\"How is machine learning used to compute transition behavior?\",\"answer\":\"A neural network architecture based on the decomposition of the vector field computes the most probable transition paths and the mean first exit time for different boundary scenarios.\"}]","Rare events in a stochastic vegetation-water dynamical system based on machine learning - 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