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These systems are characterized by high sensitivity to initial conditions, making it challenging to predict their future behavior confidently. In this study, we propose a novel deep-learning framework that addresses this challenge by directly exploiting the long-term compounding of local prediction errors during model training, aiming to extend the time horizon for reliable predictions of chaotic systems. Our approach observes the future trajectories of initial errors at a time horizon, modeling the evolution of the loss to that point through the use of two major components: (1) a recurrent architecture (Error Trajectory Tracing) designed to trace the trajectories of predictive errors through phase space, and (2) a training regime, Horizon Forcing, that pushes the model\u2019s focus out to a predetermined time horizon. We validate our method on three classic chaotic systems and six real-world time series prediction tasks with chaotic characteristics. The results show that our approach outperforms the state-of-the-art methods.<\/jats:p>","DOI":"10.1145\/3718090","type":"journal-article","created":{"date-parts":[[2025,6,25]],"date-time":"2025-06-25T11:14:43Z","timestamp":1750850083000},"page":"1-22","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["Horizon Forcing: Improving the Recurrent Forecasting of Chaotic Systems"],"prefix":"10.1145","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4456-4556","authenticated-orcid":false,"given":"Yong","family":"Zhuang","sequence":"first","affiliation":[{"name":"Computer Science, Grand Valley State University, Allendale Charter Township, Michigan, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8298-6023","authenticated-orcid":false,"given":"Matthew","family":"Almeida","sequence":"additional","affiliation":[{"name":"Computer Science, University of Massachusetts Boston, Boston, Massachusetts, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3383-551X","authenticated-orcid":false,"given":"Wei","family":"Ding","sequence":"additional","affiliation":[{"name":"Computer Science, University of Massachusetts Boston, Boston, Massachusetts, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7493-5245","authenticated-orcid":false,"given":"Shafiqul","family":"Islam","sequence":"additional","affiliation":[{"name":"Tufts University, Medford, Massachusetts, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2840-6745","authenticated-orcid":false,"given":"Zihan","family":"Li","sequence":"additional","affiliation":[{"name":"Computer Science, University of Massachusetts Boston, Boston, Massachusetts, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3789-7686","authenticated-orcid":false,"given":"Ping","family":"Chen","sequence":"additional","affiliation":[{"name":"University of Massachusetts Boston, Boston, Massachusetts, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2025,10,16]]},"reference":[{"key":"e_1_3_3_2_2","unstructured":"R. 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