# HongKun Zhang

Professor

Prof. Hongkun Zhang is an expert in the field of hyperbolic dynamical systems including chaotic billiards. One major trend of Modern Dynamical Systems is to investigate the stochastic properties for stochastic processes generated by chaotic dynamical systems, including the decay rates of correlations, Central Limit Theory and other probability limiting theorems. Her research field also extends to general stochastic processes and probability theory, with applications in financial mathematics using SDE and statistics, risk management using Extreme Value Theory, networks using stochastic analysis.

Most recently, her interest has shift to machine learning of dynamical systems, including data-driven deep learning of chaotic systems, understanding dynamical properties of neural networks, leaning using Graph Neural Networks, as well as deep learning of financial assets.

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RESEARCH INTERESTS

Machine Learning of Dynamical systems, Chaotic Billiards and ergodic theory; Deep learning using Graph Neural Network; Financial Mathematics

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MAJOR AWARDS AND GRANTS

- 023-2026. NSF Grant (DMS - 2220211) "ATD: Deep Learning on Anomaly Detection for Human Dynamics and Hazard Response"
- 2020 Simons Foundation:Mathematics and Physical Sciences-Collaboration Grants for Mathematicians
- 2018 AMS China Exchange Program Ky and Yu-Fen Fan Fund Travel Grant
- 2015 “Simons Fellowship” in Mathematics and Physics;
- 2012-2017 NSF CAREER Award (DMS-1151762) ;
- 2009-2012 NSF Grant (DMS-0901448);
- 2008 AWM Mathematics Travel Grant;
- 2008 AWM Mentor Grant.

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Selected Publications

**More Recent Research Work in Machine Learning and Deep Neural Networks:**

19. α-SGHN: A Robust Model for Learning Particle Interactions in Lattice Systems, Yixian Gao Ru Geng, Panayotis Kevrekidis, Hong-Kun Zhang and Jian Zu, Submitted, 2024.

18. CauchyNet: Integrating Complex Analysis with Machine Learning for Enhanced Predictive Modeling, (with Xin Li, Zhihong Xia), submitted.

17. Symbolic Extended Dynamic Mode Decomposition C. Kennedy, J. Kaushagen, and H-K Zhang, accepted by Chaos, 2024.

16. Data-driven Learning Chaotic Dynamical System using Discrete-Temporal Sobolev Networks, Connor Kennedy, Trace Crowdis, Haoran Hu, Sankaran Vaidyanathan, Hong-Kun Zhang, 2024, (173) 106152, **Neural Networks**.

15. GAHNet: A Lattice Structure Learning Model Based on Graph Neural Networks, R. Geng, Y. Gao, H. Zhang and J. Zu, submitted

14. Separable Graph Hamiltonian Network: A Graph Deep Learning Model for Lattice Systems, R. Geng, Y. Gao, H. Zhang and J. Zu, **Physical Review Research**. 6, 013176,2024

13. Machine learning of independent conservation laws through neural deflation, Wei Zhu, Hong-Kun Zhang, P. G. Kevrekidis, Physical Review E. 2023

12. Graph Signal Processing on Dynamic Graphs Based on Temporal-attention Product, Applied and Computational Harmonic Analysis, 67 (2023) 101579, (with Ru Geng, Yixian Gao, Jian Zu).

11. A Modified PINN Approach for Identifiable Compartmental Models in Epidemiology with Application to COVID-19, Haoran Hu, Connor M. Kennedy, Panayotis G. Kevrekidis and Hong-Kun Zhang, Viruses, 2022 Nov 7;14(11):2464.

10. Analysis of the Spatio-Temporal Dynamics of COVID-19 in Massachusetts via Spectral Graph Wavelet Theory, IEEE Transactions on Signal and Information Processing over Networks, R. Geng, Y. Gao, H. Zhang and J. Zu, vol. 8, 670-683, 2022,

9. Euler Difference Neural Network for Deep learning in chaotic dynamical systems, (with Xuejin Zhang, Jian Zu). Comm Math Sci., 2024

7. Perturbed Hamiltonian Neural Networks, Submitted (with Yuting Li, Yong Li)

6. Inverse Interior Scattering Problems For Perturbed Bunimovich Billiards, Submitted, (with Qi Li, Yixian Gao)

5. Statistical analysis and deep learning on Copper/Gold Ratio time series, Submitted (with Hong Cao, Yulong Lian, Hao Chen)

4. Solitary wave billiards, J Cuevas-Maraver, PG Kevrekidis, HK Zhang, Physical Review E 107 (3), 034217, 2023

3. Superdiffusions for certain nonuniformly hyperbolic systems (with Luke Mohr), submitted;

2. Optimal bounds on correlation decay rates for nonuniform hyperbolic systems (with Sandro Vaienti), submitted.

1. Length Spectrum Rigidity for Piecewise Analytic Bunimovich Billiards, Jianyu Chen, Vadim Kaloshin, Hong-Kun Zhang, Communication in Mathematical Physics, 2023.

**Research work related to dynamical systems:**

52. Inducing Schemes with Finite Weighted Complexity. Chen, J., Wang, F. & Zhang, HK. J Stat Phys 190, 195 (2023)

51. Lyapunov Exponents and Nonadapted Measures for Dispersing Billiards, V. Climenhaga, F. Demers, Y. Lima, H. Zhang, Communications in mathematical Physics, 2023

50. Improved Young Tower and Thermodynamic Formalism for Hyperbolic Systems with Singularities (with Jianyu Chen and Fang Wang), submitted;

49 Necessary and sufficient condition for M_2-convergence to a Lévy process for billiards with cusps at flat points, Paul Jung, Ian Melbourne, Françoise Pène, Paulo Varandas, Hong-Kun Zhang, DCDS,2021

48. Topological entropy of Bunimovich stadium billiards, Michal Misiurewicz, Hong-Kun Zhang, Pure and Applied Functional Analysis, 221-229, 2021

47. Decay of correlations for unbounded observables, Fang Wang, Hong-Kun Zhang and Pengfei Zhang, Nonlinearity, Volume 34, Number 4, 2021

46. Wei Z., Zhang Z., Zhang H., Wang T. (2021) Conditional Dependence Among Oil, Gold and U.S. Dollar Exchange Rates: A Copula-GARCH Approach. In: Sriboonchitta S., Kreinovich V., Yamaka W. (eds) . Behavioral

Predictive Modeling in Economics. Studies in Computational Intelligence, vol 897. Springer, Cham.

45. Local limit theorem for randomly deforming billiards, (with Mark F Demers, Françoise Pène), Communications in Mathematical Physics, Commun. Math. Phys. 375, 2281–2334 (2020).

44 Central Limit Theorem for Billiards With Flat Points (with Kien Nguyen), Springer Conference Proceedings Differential Equations and Dynamical Systems, 2018.

843 Stable laws for chaotic billiards with cusps at flat points (with Paul Jung), Annales Henri Poincare, 2019.42

42. Convergence to a -stable Lévy motion for chaotic billiards with several cusps at flat points. , (with Paul Jung, Franccoise Pene), Nonlinearity. 2020, 33 807.

41. Statistical Properties for 1-d Expanding Maps with Singularities of Low Regularity, (with Jianyu Chen), DCDS, 2019.

40. Stability of periodic orbits in no-slip billiards (with Christopher Lee Cox, Renato Feres), Nonlinearity, 31 (2018) 4443–4471;

39. Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities, (with J. Chen, Y.Yang) J. Statistical Phys, 2018, 1-26.

38. Fluctuation of the entropy production for the Lorentz gas under small external forces (with Luc-Rey Bellet, Mark Demers). Commun. Math. Phys.363(2) (2018), 699–740.

37. Decay of correlations for billiards with flat points I: channel effect, Contemporary Mathematics, Vol 689 (2017).

36. Decay of correlations for billiards with flat points II: cusp effect, Contemporary Mathematics, Vol 689 (2017).

35. Stability of periodic orbits in no-slip billiards, (with Christopher Lee Cox, Fatima Correia), New Horizons in Mathematical Physics, (2017).

34 Compound Poisson law for hitting times to periodic orbits in two-dimensional hyperbolic systems, (with Matthew Nicol and Meagan Carney), J. Statistical Physics, (2017);

33. The dynamics of precious metal markets VaR: A GARCHEVT approach, (with Zhijing Zhang) Journal of Commodity Markets Volume 4, Issue 1, December 2016, Pages 14–27

32. On Another Edge of Defocusing: Hyperbolicity of Asymmetric Lemon Billiards,( Bunimovich, H.K.Zhang &P.Zhang), Communications in Mathematical Physics, 341, 2016, 781-803

31. Calendar Effects in AAPL Value-at-Risk, Hong-Kun Zhang , Zijing Zhang, Journal of Mathematics and System Science, 2016.

30. Forecast and backtesting of VAR models in Crude Oil Market, (with Yue-Xian Li, Jin-Guo Lian), Research & Reviews: Journal of Statistics and Mathematical Sciences, 2016.

29. Diffusivity in multiple scattering systems,(Timothy Chumley, Renato Feres and H.K. Zhang, ) Trans. Amer. Math. Soc. 368 (2016), 109-148

28. Predicting the Risk of Bankruptcy for ARO Stock (with Yuexian Li, Jinguo Lian); International Journal of Engineering Research & Science; (2015) 1:9.

27. Stability and ergodicity of moon billiards (with Fatima Correia) Chaos 25, 083110 (2015)

25. Spectral analysis of hyperbolic systems with singularities (with Mark Demers) Nonlinearity 27, 2014.

25. Current for dispersing billiards under general forces,( N. Chernov, H.-K, Zhang and P. Zhang, ) J. Stat. Physics, 2013.

24. A functional analytic approach to perturbations of the Lorentz gas, (Mark Demers, Hong-Kun Zhang),

Communications in Mathematical Physics, 2013, 767-830.

23. Ergodicity of the generalized lemon billiards,(J.Chen,L.Morh,H.K.Zhang&P.Zhang), Chaos, 2013, 23(4).

22. Dispersing billiards with moving scatterers (Mikko Stenlund, Lai-Sang Young, Hong-Kun Zhang),

Communications in Mathematical Physics, 2013, 909-905.

21. Multiple scattering in random mechanical systems and diffusion approximation (with Renato Feres, Jasmine Ng, Hong-Kun Zhang), Communications in Mathematical Physics,2013, 713-745

20. Evaluation of the Stochastic Modeling on Options,(with Zhijuan Mao, Zhian Liang, Jinguo Lian) International Journal of Engineering Research and Applications, 2 (2012): .2463-2473.

19. Spectral gap for a class of random billiards, (Joint with Renato Feres), Communications in Mathematical Physics, 313, 479–515 (2012)

18. Free path of billiards with flat points. D.& C. Dynamical Systems, 32, 4445 - 4466, (2012)

17. Spectral analysis of the transfer operator for the Lorentz gas (with Mark Demers), Journal of Modern Dynamical Systems, 5:4 (2011) 665 – 709.

16. Current in periodic Lorentz gases with twists. Communications in Math. Physics, 306 (2011) 747-776.

15. New approach to differential equations with countable impulses. (Joint with Jin-Guo Lian, Jiong Sun). Acta Mathematicae Application Sinica, 27: 2 (2011) 255-262.

14. Estimates for correlations in billiards with large arcs. Acta Mathematicae Application Sinica, English Series, 27: 3 (2011) 381-392.

13. The spectrum of the billiard Laplacian of a family of random billiards, (Joint with Renato Feres) J. Statis. Phys. 141:6 (2010) 1039-1054.

12. Mixing rate for rectangular billiards with diamond obstacles, Am.J.Math. Manage. Sci. 30: 53-65 (2010).

11. Decay of Correlations on Non-Holder Observables, International Journal of Nonlinear Science, 10:3, 2010, 375--385.

10. On statistical properties of hyperbolic systems with general singularities, J. Statis. Phys. 136, 2009, 615-642. (With N. Chernov)

9. Stability of the T-periodic Solution on the ES-S Model, Rocky Mountain Journal of Mathematics, 38 (2008), 1493-1504. (with J. G. Lian)

8. Improved estimates for correlations in billiards, Communications in Mathematical Physics, 277 (2008), 305-321. (With N. Chernov)

7. Stability of T-periodic Solution on the Extended Simplified Brusselator Model, International Journal of Biomathematics, Vol. 1, (2008) 19-27, (with J. G. Lian) .

6. Unique periodic solution of ES-S model, Appl. Math. E-Notes, 8 (2008), 25-29 (with J. G. Lian)

5. Hyperbolic behavior of Jacobi fields on billiard flows, Impulsive and Hybrid Dynamical Systems,2007, Waterloo, 1794-1798. (with J. G. Lian)

4. Regularity of Bunimovich’s stadia,Regular and Chaotic Dynamics,(2007) 3, 335-356 (With N. Chernov)

3. A family of chaotic billiards with variable mixing rates. Stochastics and Dynamics, (2005) 5, 535-553. (With N. Chernov)

2. Billiards with Polynomial mixing rates. Nonlinearity , (2005) 18, 1527-1553. (With N. Chernov)

1. A class of ordinary differential operators with jump boundary conditions. Lecture Notes in Pure and Appl. Math., 234, (2003), 253-274. (With R. Kauffman)

**CONFERENCE ORGANIZED/CO-ORGANIZED**

1. International Conference on Statistical Properties of Nonequilibrium Dynamical Systems, one of the main organizers,SUSTC, Shenzhen, China, July 27 - August 2, 2016;

2. AIM Workshop: Stochastic methods for nonequilibrium dynamical systems, one of the Main organizers, Palo Alto CA, June 1 - 5, 2015.

3. The Dynamical Systems, Ergodic Theory, And Probability Conference Dedicated To The Memory Of Nikolai Chernov, co-organizer, UAB, Alabama, May 18-20, 2015.

4. 2016 CMS Winter Meeting, Section: Stochastic Properties of Dynamical Systems , co- organizer, Niagara Falls, Canada, Dec.2-5, 2016.

5. Second USA-Uzbekistan Conference on Analysis and Mathematical Physics, co-organizer, Urgench State University (Uzbekistan), August 8-12, 2017.

6. Mathematical Physics Perspective of Billiards and Dominoes, on of the main organizers, UMass Amherst, September 22-23, 2017.

7. BIRS Workshop: New Developments in Open Dynamical Systems and Their Applications, co- organizer, Banff, March 18-23, 2018.

8. “INTERNATIONAL CONFERENCE on THERMODYNAMICS 2.0” , June 22 – 24, 2020 (Zoom) Co-organizer.

9. “2nd Northeastern Dynamical Systems Conference”, Co-organizer, UMass Amherst, September 22-23, 2019.

10. “International Conference on Thermodynamics 2.0” July 18-20, 2022, Boone, North Carolina, Co-organizer.