{"type":"ScholarlyArticle","title":"Stochastic Neural Simulator for Generalizing Dynamical Systems across Environments","authors":[{"name":"Liu Jiaqi","slug":"liu-jiaqi","given":"Liu","family":"Jiaqi","affiliation":"Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education , China; College of Computer Science and Technology , Jilin University , China"},{"name":"Jiaxu Cui","slug":"jiaxu-cui","given":"Jiaxu","family":"Cui","affiliation":"Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education , China; College of Computer Science and Technology , Jilin University , China"},{"name":"Jiayi Yang","slug":"jiayi-yang","given":"Jiayi","family":"Yang","affiliation":"Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education , China; College of Computer Science and Technology , Jilin University , China"},{"name":"Bo Yang","slug":"bo-yang","given":"Bo","family":"Yang","affiliation":"Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education , China; College of Computer Science and Technology , Jilin University , China"}],"abstract":"Neural simulators for modeling complex dynamical systems have been extensively studied for various real-world applications, such as weather forecasting, ocean current prediction, and computational fluid dynamics simulation. Although they have demonstrated powerful fitting and predicting, most existing models are only built to learn single-system dynamics. Several advanced researches have considered learning dynamics across environments, which can exploit the potential commonalities among the dynamics across environments and adapt to new environments. However, these methods still are prone to scarcity problems where per-environment data is sparse or limited. Therefore, we propose a novel CoNDP (Context-Informed Neural ODE Processes) to achieve learning system dynamics from sparse observations across environments. It can fully use contextual information of each environment to better capture the intrinsic commonalities across environments and distinguishable differences among environments while modeling uncertainty of system evolution, producing more accurate predictions. Intensive experiments are conducted on five complex dynamical systems in various fields. Results show that the proposed CoNDP can achieve optimal results compared with common neural simulators and state-of-the-art cross-environmental models.","doi":"10.24963/ijcai.2024/653","url":"https://ijcai.org/proceedings/2024/653","pdf":"https://ijcai.org/proceedings/2024/0653.pdf","markdown":"https://ijcai.org/proceedings/2024/653.md","bibtex":"https://ijcai.org/proceedings/2024/bibtex/653","volume":{"slug":"2024","title":"Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2024","url":"https://ijcai.org/proceedings/2024/","year":2024,"location":"Jeju, Korea","dates":"3-9 August 2024","editors":"Kate Larson","isbn":"978-1-956792-04-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2024"},"section":"Multidisciplinary Topics and Applications","track":"Main Track","number":653,"pages":{"first":5909,"last":5917},"keywords":["Multidisciplinary Topics and Applications: MTA: Physical sciences","Machine Learning: ML: Time series and data streams"],"videos":[],"references":[{"position":1,"raw":"[Baradel et al., 2020] Fabien Baradel, Natalia Neverova, Julien Mille, Greg Mori, and Christian Wolf. Cophy: Counterfactual learning of physical dynamics. In Inter- national Conference on Learning Representations, 2020.","title":"Cophy: Counterfactual learning of physical dynamics","authors":"Fabien Baradel, Natalia Neverova, Julien Mille, Greg Mori, Christian Wolf","year":"2020","venue":"International Conference on Learning Representations","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Böttcher et al., 2022] Lucas Böttcher, Nino Antulov- Fantulin, and Thomas Asikis. Ai pontryagin or how artificial neural networks learn to control dynamical systems. Nature communications, 13(1):333, 2022.","title":"Ai pontryagin or how artificial neural networks learn to control dynamical systems","authors":"Lucas Böttcher, Nino Antulov-Fantulin, Thomas Asikis","year":"2022","venue":"Nature communications","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Brunton et al., 2016] Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical sys- tems. Proceedings of the national academy of sciences, 113(15):3932-3937, 2016.","title":"Discovering governing equations from data by sparse identification of nonlinear dynamical systems","authors":"Joshua L Steven L Brunton, Nathan Proctor, Kutz","year":"2016","venue":"Proceedings of the national academy of sciences","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Chang et al., 2019] Bo Chang, Minmin Chen, Eldad Haber, and Ed H. Chi. AntisymmetricRNN: A dynamical system view on recurrent neural networks. In International Con- ference on Learning Representations, 2019.","title":"AntisymmetricRNN: A dynamical system view on recurrent neural networks","authors":"Bo Chang, Minmin Chen, Eldad Haber, Ed H Chi","year":"2019","venue":"International Conference on Learning Representations","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Chen et al., 2018] Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary dif- ferential equations. Advances in neural information pro- cessing systems, 31, 2018.","title":"Neural ordinary differential equations","authors":"Yulia Ricky Tq Chen, Jesse Rubanova, David K Bettencourt, Duvenaud","year":"2018","venue":"Advances in neural information processing systems","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Cho et al., 2014] Kyunghyun Cho, Bart van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. In Proceedings of the 2014 Confer- ence on Empirical Methods in Natural Language Process- ing (EMNLP), page 1724. Association for Computational Linguistics, 2014.","title":"Learning phrase representations using rnn encoder-decoder for statistical machine translation","authors":"Cho","year":"2014","venue":"Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Constantin and Foias ¸, 1988] Peter Constantin and Ciprian Foias ¸. Navier-stokes equations. University of Chicago press, 1988.","title":"Navier-stokes equations","authors":"Peter Constantin, Ciprian Foias, ¸","year":"1988","venue":null,"doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Engelken et al., 2023] Rainer Engelken, Fred Wolf, and Larry F Abbott. Lyapunov spectra of chaotic recurrent neural networks. Physical Review Research, 5(4):043044, 2023.","title":"Lyapunov spectra of chaotic recurrent neural networks","authors":"Rainer Engelken, Fred Wolf, Larry F Abbott","year":"2023","venue":"Physical Review Research","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Fu et al., 2017] Haohuan Fu, Conghui He, Bingwei Chen, Zekun Yin, Zhenguo Zhang, Wenqiang Zhang, Tingjian Zhang, Wei Xue, Weiguo Liu, Wanwang Yin, et al. 9- pflops nonlinear earthquake simulation on sunway taihu- light: enabling depiction of 18-hz and 8-meter scenarios. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analy- sis, pages 1-12, 2017.","title":"9pflops nonlinear earthquake simulation on sunway taihulight: enabling depiction of 18-hz and 8-meter scenarios","authors":"Fu","year":"2017","venue":"Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Galea et al., 2010] Sandro Galea, Matthew Riddle, and George A Kaplan. Causal thinking and complex system approaches in epidemiology. International journal of epi- demiology, 39(1):97-106, 2010.","title":"Causal thinking and complex system approaches in epidemiology","authors":"Galea","year":"2010","venue":"International journal of epidemiology","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Garnelo et al., 2018] Marta Garnelo, Jonathan Schwarz, Dan Rosenbaum, Fabio Viola, Danilo J Rezende, SM Es- lami, and Yee Whye Teh. Neural processes. In ICML Workshop on Theoretical Foundations and Applications of Deep Generative Models, 2018.","title":"Neural processes","authors":"Marta Garnelo, Jonathan Schwarz, Dan Rosenbaum, Fabio Viola, Danilo J Rezende, S M Eslami, Yee Whye Teh","year":"2018","venue":"ICML Workshop on Theoretical Foundations and Applications of Deep Generative Models","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Glass et al., 2021] David S Glass, Xiaofan Jin, and Ing- mar H Riedel-Kruse. Nonlinear delay differential equa- tions and their application to modeling biological network motifs. Nature communications, 12(1):1788, 2021.","title":"Nonlinear delay differential equations and their application to modeling biological network motifs","authors":"Xiaofan David S Glass, Ingmar H Jin, Riedel-Kruse","year":"2021","venue":"Nature communications","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Guglielmi et al., 2022] Nicola Guglielmi, Elisa Iacomini, and Alex Viguerie. Delay differential equations for the spatially resolved simulation of epidemics with specific application to covid-19. Mathematical Methods in the Ap- plied Sciences, 45(8):4752-4771, 2022.","title":"Delay differential equations for the spatially resolved simulation of epidemics with specific application to covid-19","authors":"Nicola Guglielmi, Elisa Iacomini, Alex Viguerie","year":"2022","venue":"Mathematical Methods in the Applied Sciences","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Gupta and Lermusiaux, 2021] Abhinav Gupta and Pierre FJ Lermusiaux. Neural closure models for dynami- cal systems. Proceedings of the Royal Society A, 477(2252):20201004, 2021.","title":"Neural closure models for dynamical systems","authors":"Abhinav Gupta, Pierre Fj Lermusiaux","year":"2021","venue":"Proceedings of the Royal Society A","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Hochreiter and Schmidhuber, 1997] Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8):1735-1780, 1997.","title":"Long short-term memory","authors":"Schmidhuber Hochreiter, Sepp Hochreiter, Jürgen Schmidhuber","year":"1997","venue":"Neural computation","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Horváth et al., 2009] Judit Horváth, István Szalai, and Patrick De Kepper. An experimental design method lead- ing to chemical turing patterns. Science, 324(5928):772- 775, 2009.","title":"An experimental design method leading to chemical turing patterns","authors":"Judit Horváth, István Szalai, Patrick De Kepper","year":"2009","venue":"Science","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Huang et al., 2020] Zijie Huang, Yizhou Sun, and Wei Wang. Learning continuous system dynamics from irregularly-sampled partial observations. Advances in Neural Information Processing Systems, 33:16177-16187, 2020.","title":"Learning continuous system dynamics from irregularly-sampled partial observations","authors":"Huang","year":"2020","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Huang et al., 2023] Zijie Huang, Yizhou Sun, and Wei Wang. Generalizing graph ode for learning complex sys- tem dynamics across environments. In Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pages 798-809, 2023.","title":"Generalizing graph ode for learning complex system dynamics across environments","authors":"Huang","year":"2023","venue":"Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Kazemi et al., 2020] Seyed Mehran Kazemi, Rishab Goel, Sepehr Eghbali, Janahan Ramanan, Jaspreet Sahota, San- jay Thakur, Stella Wu, Cathal Smyth, Pascal Poupart, and Marcus Brubaker. Time2vec: Learning a vector represen- tation of time, 2020.","title":"Time2vec: Learning a vector representation of time","authors":"Rishab Seyed Mehran Kazemi, Sepehr Goel, Janahan Eghbali, Jaspreet Ramanan, Sanjay Sahota, Stella Thakur, Cathal Wu, Pascal Smyth, Marcus Poupart, Brubaker","year":"2020","venue":null,"doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Kidger et al., 2020] Patrick Kidger, James Morrill, James Foster, and Terry Lyons. Neural controlled differential equations for irregular time series. Advances in Neural Information Processing Systems, 33:6696-6707, 2020.","title":"Neural controlled differential equations for irregular time series","authors":"Patrick Kidger, James Morrill, James Foster, Terry Lyons","year":"2020","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Kirchmeyer et al., 2022] Matthieu Kirchmeyer, Yuan Yin, Jérémie Donà, Nicolas Baskiotis, Alain Rakotomamonjy, and Patrick Gallinari. Generalizing to new physical sys- tems via context-informed dynamics model. In Interna- tional Conference on Machine Learning, pages 11283- 11301. PMLR, 2022.","title":"Generalizing to new physical systems via context-informed dynamics model","authors":"Matthieu Kirchmeyer, Yuan Yin, Jérémie Donà, Nicolas Baskiotis, Alain Rakotomamonjy, Patrick Gallinari","year":"2022","venue":"International Conference on Machine Learning","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Kuang, 1993] Yang Kuang. Delay differential equations: with applications in population dynamics. Academic press, 1993.","title":"Delay differential equations: with applications in population dynamics","authors":"Yang Kuang","year":"1993","venue":null,"doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Legaard et al., 2023] Christian Legaard, Thomas Schranz, Gerald Schweiger, Ján Drgoňa, Basak Falay, Cláudio Gomes, Alexandros Iosifidis, Mahdi Abkar, and Peter Larsen. Constructing neural network based models for simulating dynamical systems. ACM Computing Surveys, 55(11):1-34, 2023.","title":"Constructing neural network based models for simulating dynamical systems","authors":"Legaard","year":"2023","venue":"ACM Computing Surveys","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Li et al., 2021] Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Burigede liu, Kaushik Bhat- tacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equa- tions. In International Conference on Learning Represen- tations, 2021.","title":"Fourier neural operator for parametric partial differential equations","authors":"Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, Anima Anandkumar","year":"2021","venue":"International Conference on Learning Representations","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Linot et al., 2023] Alec J Linot, Joshua W Burby, Qi Tang, Prasanna Balaprakash, Michael D Graham, and Romit Maulik. Stabilized neural ordinary differential equations for long-time forecasting of dynamical systems. Journal of Computational Physics, 474:111838, 2023.","title":"Stabilized neural ordinary differential equations for long-time forecasting of dynamical systems","authors":"Alec J Linot, Joshua W Burby, Qi Tang, Prasanna Balaprakash, Michael D Graham, Romit Maulik","year":"2023","venue":"Journal of Computational Physics","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Liu et al., 2021] Yong Liu, Xin Liu, Fang Li, Haohuan Fu, Yuling Yang, Jiawei Song, Pengpeng Zhao, Zhen Wang, Dajia Peng, Huarong Chen, et al. Closing the\" quantum supremacy\" gap: achieving real-time simulation of a ran- dom quantum circuit using a new sunway supercomputer. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analy- sis, pages 1-12, 2021.","title":"Closing the\" quantum supremacy\" gap: achieving real-time simulation of a random quantum circuit using a new sunway supercomputer","authors":"Yong Liu, Xin Liu, Fang Li, Haohuan Fu, Yuling Yang, Jiawei Song, Pengpeng Zhao, Zhen Wang, Dajia Peng, Huarong Chen","year":"2021","venue":"Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Lotka, 1925] Alfred James Lotka. Elements of physical bi- ology. Williams & Wilkins, 1925.","title":"Alfred James Lotka. Elements of physical biology","authors":"Lotka","year":"1925","venue":null,"doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Morrill et al., 2021] James Morrill, Cristopher Salvi, Patrick Kidger, and James Foster. Neural rough differ- ential equations for long time series. In International Conference on Machine Learning, pages 7829-7838.","title":"Neural rough differential equations for long time series","authors":"James Morrill, Cristopher Salvi, Patrick Kidger, James Foster","year":null,"venue":"International Conference on Machine Learning","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"PMLR, 2021.","title":null,"authors":"Pmlr","year":"2021","venue":null,"doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Norcliffe et al., 2020] Alexander Norcliffe, Cristian Bod- nar, Ben Day, Jacob Moss, and Pietro Liò. Neural ode processes. In International Conference on Learning Rep- resentations, 2020.","title":"Neural ode processes","authors":"Alexander Norcliffe, Cristian Bodnar, Ben Day, Jacob Moss, Pietro Liò","year":"2020","venue":"International Conference on Learning Representations","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Oksendal, 2013] Bernt Oksendal. Stochastic differential equations: an introduction with applications. Springer Science & Business Media, 2013.","title":"Stochastic differential equations: an introduction with applications","authors":"Bernt Oksendal","year":"2013","venue":null,"doi":null,"url":null,"internal_url":null},{"position":32,"raw":"John E Pearson. Complex patterns in a sim- ple system. Science, 261(5118):189-192, 1993.","title":"Complex patterns in a simple system","authors":"John E Pearson","year":"1993","venue":"Science","doi":null,"url":null,"internal_url":null},{"position":33,"raw":"[Perrone and Kao, 1975] Nicholas Perrone and Robert Kao. A general finite difference method for arbitrary meshes. Computers & Structures, 5(1):45-57, 1975.","title":"A general finite difference method for arbitrary meshes","authors":"Nicholas Perrone, Robert Kao","year":"1975","venue":"Computers & Structures","doi":null,"url":null,"internal_url":null},{"position":34,"raw":"[Rajchakit et al., 2021] G Rajchakit, R Sriraman, N Boon- satit, P Hammachukiattikul, Chee Peng Lim, and P Agar- wal. Exponential stability in the lagrange sense for clifford-valued recurrent neural networks with time delays. Advances in Difference Equations, 2021(1):1-21, 2021.","title":"Exponential stability in the lagrange sense for clifford-valued recurrent neural networks with time delays","authors":"Rajchakit, N Sriraman, P Boonsatit, Chee Hammachukiattikul, Peng Lim","year":"2021","venue":"Advances in Difference Equations","doi":null,"url":null,"internal_url":null},{"position":35,"raw":"[Reichstein et al., 2019] Markus Reichstein, Gustau Camps- Valls, Bjorn Stevens, Martin Jung, Joachim Denzler, Nuno Carvalhais, and fnm Prabhat. Deep learning and process understanding for data-driven earth system science. Na- ture, 566(7743):195-204, 2019.","title":"Deep learning and process understanding for data-driven earth system science","authors":"Markus Reichstein, Gustau Camps-Valls, Bjorn Stevens, Martin Jung, Joachim Denzler, Nuno Carvalhais, Prabhat","year":"2019","venue":"Nature","doi":null,"url":null,"internal_url":null},{"position":36,"raw":"[Rubanova et al., 2019] Yulia Rubanova, Ricky TQ Chen, and David K Duvenaud. Latent ordinary differential equa- tions for irregularly-sampled time series. Advances in neu- ral information processing systems, 32, 2019.","title":"Latent ordinary differential equations for irregularly-sampled time series","authors":"Yulia Rubanova, Ricky Tq Chen, David K Duvenaud","year":"2019","venue":"Advances in neural information processing systems","doi":null,"url":null,"internal_url":null},{"position":37,"raw":"[Ruoff et al., 2003] Peter Ruoff, Melinda K Christensen, Jana Wolf, and Reinhart Heinrich. Temperature de- pendency and temperature compensation in a model of yeast glycolytic oscillations. Biophysical chemistry, 106(2):179-192, 2003.","title":"Temperature dependency and temperature compensation in a model of yeast glycolytic oscillations","authors":"Peter Ruoff, Melinda K Christensen, Jana Wolf, Reinhart Heinrich","year":"2003","venue":"Biophysical chemistry","doi":null,"url":null,"internal_url":null},{"position":38,"raw":"[Samelson and Wiggins, 2006] Roger M Samelson and Stephen Wiggins. Lagrangian transport in geophysi- cal jets and waves: The dynamical systems approach, volume 31. Springer Science & Business Media, 2006.","title":"Lagrangian transport in geophysical jets and waves: The dynamical systems approach","authors":"Wiggins ; Samelson, M Roger, Stephen Samelson, Wiggins","year":"2006","venue":null,"doi":null,"url":null,"internal_url":null},{"position":39,"raw":"[Sanchez-Gonzalez et al., 2020] Alvaro Sanchez-Gonzalez, Jonathan Godwin, Tobias Pfaff, Rex Ying, Jure Leskovec, and Peter Battaglia. Learning to simulate complex physics with graph networks. In International conference on ma- chine learning, pages 8459-8468. PMLR, 2020.","title":"Learning to simulate complex physics with graph networks","authors":"Alvaro Sanchez-Gonzalez, Jonathan Godwin, Tobias Pfaff, Rex Ying, Jure Leskovec, Peter Battaglia","year":"2020","venue":"International conference on machine learning","doi":null,"url":null,"internal_url":null},{"position":40,"raw":"[Sankar et al., 2020] Aravind Sankar, Yanhong Wu, Liang Gou, Wei Zhang, and Hao Yang. Dysat: Deep neural rep- resentation learning on dynamic graphs via self-attention networks. In Proceedings of the 13th international con- ference on web search and data mining, pages 519-527, 2020.","title":"Dysat: Deep neural representation learning on dynamic graphs via self-attention networks","authors":"Aravind Sankar, Yanhong Wu, Liang Gou, Wei Zhang, Hao Yang","year":"2020","venue":"Proceedings of the 13th international conference on web search and data mining","doi":null,"url":null,"internal_url":null},{"position":41,"raw":"[Smith, 2011] Hal L Smith. An introduction to delay differ- ential equations with applications to the life sciences, vol- ume 57. springer New York, 2011.","title":"An introduction to delay differential equations with applications to the life sciences","authors":"Hal L Smith, Smith","year":"2011","venue":null,"doi":null,"url":null,"internal_url":null},{"position":42,"raw":"[Tang et al., 2020] Xianfeng Tang, Huaxiu Yao, Yiwei Sun, Charu Aggarwal, Prasenjit Mitra, and Suhang Wang. Joint modeling of local and global temporal dynamics for mul- tivariate time series forecasting with missing values. In Proceedings of the AAAI Conference on Artificial Intelli- gence, volume 34, pages 5956-5963, 2020.","title":"Joint modeling of local and global temporal dynamics for multivariate time series forecasting with missing values","authors":"Xianfeng Tang, Huaxiu Yao, Yiwei Sun, Charu Aggarwal, Prasenjit Mitra, Suhang Wang","year":"2020","venue":"Proceedings of the AAAI Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":43,"raw":"Hal R Varian. Dynamical systems with ap- plications to economics. Handbook of mathematical eco- nomics, 1:93-110, 1981.","title":"Dynamical systems with applications to economics","authors":"Hal R Varian","year":"1981","venue":"Handbook of mathematical economics","doi":null,"url":null,"internal_url":null},{"position":44,"raw":"[Vaswani et al., 2017] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.","title":"Attention is all you need","authors":"Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, Illia Polosukhin","year":"2017","venue":"Advances in neural information processing systems","doi":null,"url":null,"internal_url":null},{"position":45,"raw":"[Weerakody et al., 2021] Philip B Weerakody, Kok Wai Wong, Guanjin Wang, and Wendell Ela. A review of irreg- ular time series data handling with gated recurrent neural networks. Neurocomputing, 441:161-178, 2021.","title":"A review of irregular time series data handling with gated recurrent neural networks","authors":"Philip B Weerakody, Wai Kok, Guanjin Wong, Wendell Wang, Ela","year":"2021","venue":"Neurocomputing","doi":null,"url":null,"internal_url":null},{"position":46,"raw":"[Widder, 1976] David Vernon Widder. The heat equation, volume 67. Academic Press, 1976.","title":"The heat equation","authors":"David Vernon, Widder","year":"1976","venue":null,"doi":null,"url":null,"internal_url":null},{"position":47,"raw":"[Yin et al., 2021] Yuan Yin, Ibrahim Ayed, Emmanuel de Bézenac, Nicolas Baskiotis, and Patrick Gallinari. Leads: Learning dynamical systems that generalize across environments. Advances in Neural Information Processing Systems, 34:7561-7573, 2021.","title":"Leads: Learning dynamical systems that generalize across environments","authors":"Ibrahim Yuan Yin, Emmanuel Ayed, Nicolas De Bézenac, Patrick Baskiotis, Gallinari","year":"2021","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}