## 2021 |

Mauro Maggioni Jason Miller, Hongda Qiu Ming Zhong Learning Interaction Kernels for Agent Systems on Riemannian Manifolds Online 2021. Abstract | Links | BibTeX | Tags: agent-based models, interacting particle systems, Machine learning, model reduction, statistics @online{AgentSystemsManifolds, title = { Learning Interaction Kernels for Agent Systems on Riemannian Manifolds}, author = {Mauro Maggioni, Jason Miller, Hongda Qiu, Ming Zhong}, url = {https://arxiv.org/abs/2102.00327}, year = {2021}, date = {2021-01-30}, abstract = {Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive bounds on the trajectory estimation error, on the manifold, between the observed and estimated dynamics. We demonstrate the performance of our estimator on two classical first order interacting systems: Opinion Dynamics and a Predator-Swarm system, with each system constrained on two prototypical manifolds, the 2-dimensional sphere and the Poincaré disk model of hyperbolic space.}, keywords = {agent-based models, interacting particle systems, Machine learning, model reduction, statistics}, pubstate = {published}, tppubtype = {online} } Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive bounds on the trajectory estimation error, on the manifold, between the observed and estimated dynamics. We demonstrate the performance of our estimator on two classical first order interacting systems: Opinion Dynamics and a Predator-Swarm system, with each system constrained on two prototypical manifolds, the 2-dimensional sphere and the Poincaré disk model of hyperbolic space. |

Fei Lu Mauro Maggioni, Sui Tang Learning interaction kernels in heterogeneous systems of agents from multiple trajectories Journal Article Journ. Mach. Learn. res., 2 (32), pp. 1–67, 2021. Abstract | Links | BibTeX | Tags: Active Learning, interacting particle systems, inverse problems, Machine learning @article{LuMMTang21, title = {Learning interaction kernels in heterogeneous systems of agents from multiple trajectories}, author = {Fei Lu, Mauro Maggioni, Sui Tang}, url = {https://jmlr.csail.mit.edu/papers/v22/19-861.html}, year = {2021}, date = {2021-01-01}, journal = {Journ. Mach. Learn. res.}, volume = {2}, number = {32}, pages = {1–67}, abstract = {Systems of interacting particles, or agents, have wide applications in many disciplines, including Physics, Chemistry, Biology and Economics. These systems are governed by interaction laws, which are often unknown: estimating them from observation data is a fundamental task that can provide meaningful insights and accurate predictions of the behaviour of the agents. In this paper, we consider the inverse problem of learning interaction laws given data from multiple trajectories, in a nonparametric fashion, when the interaction kernels depend on pairwise distances. We establish a condition for learnability of interaction kernels, and construct an estimator based on the minimization of a suitably regularized least squares functional, that is guaranteed to converge, in a suitable L^2 space, at the optimal min-max rate for 1-dimensional nonparametric regression. We propose an efficient learning algorithm to construct such estimator, which can be implemented in parallel for multiple trajectories and is therefore well-suited for the high dimensional, big data regime. Numerical simulations on a variety examples, including opinion dynamics, predator-prey and swarm dynamics and heterogeneous particle dynamics, suggest that the learnability condition is satisfied in models used in practice, and the rate of convergence of our estimator is consistent with the theory. These simulations also suggest that our estimators are robust to noise in the observations, and can produce accurate predictions of trajectories in large time intervals, even when they are learned from observations in short time intervals.}, keywords = {Active Learning, interacting particle systems, inverse problems, Machine learning}, pubstate = {published}, tppubtype = {article} } Systems of interacting particles, or agents, have wide applications in many disciplines, including Physics, Chemistry, Biology and Economics. These systems are governed by interaction laws, which are often unknown: estimating them from observation data is a fundamental task that can provide meaningful insights and accurate predictions of the behaviour of the agents. In this paper, we consider the inverse problem of learning interaction laws given data from multiple trajectories, in a nonparametric fashion, when the interaction kernels depend on pairwise distances. We establish a condition for learnability of interaction kernels, and construct an estimator based on the minimization of a suitably regularized least squares functional, that is guaranteed to converge, in a suitable L^2 space, at the optimal min-max rate for 1-dimensional nonparametric regression. We propose an efficient learning algorithm to construct such estimator, which can be implemented in parallel for multiple trajectories and is therefore well-suited for the high dimensional, big data regime. Numerical simulations on a variety examples, including opinion dynamics, predator-prey and swarm dynamics and heterogeneous particle dynamics, suggest that the learnability condition is satisfied in models used in practice, and the rate of convergence of our estimator is consistent with the theory. These simulations also suggest that our estimators are robust to noise in the observations, and can produce accurate predictions of trajectories in large time intervals, even when they are learned from observations in short time intervals. |

## 2020 |

Lu, Fei; Li, Zhongyang; Maggioni, Mauro; Tang, Sui; Zhang, Cheng On the identifiability of interaction functions in systems of interacting particles Journal Article Forthcoming to appear in Stochastic Processes and their Applications, Forthcoming. Links | BibTeX | Tags: agent-based models, interacting particle systems, inverse problems, Machine learning, model reduction, statistics @article{IdentifiabilityInteractionFunctions, title = {On the identifiability of interaction functions in systems of interacting particles}, author = {Fei Lu and Zhongyang Li and Mauro Maggioni and Sui Tang and Cheng Zhang}, url = {https://arxiv.org/abs/1912.11965}, year = {2020}, date = {2020-10-09}, journal = {to appear in Stochastic Processes and their Applications}, keywords = {agent-based models, interacting particle systems, inverse problems, Machine learning, model reduction, statistics}, pubstate = {forthcoming}, tppubtype = {article} } |

Jason Miller Sui Tang, Ming Zhong Mauro Maggioni Learning Theory for Inferring Interaction Kernels in Second-Order Interacting Agent Systems Online Forthcoming Forthcoming. Links | BibTeX | Tags: agent-based models, interacting particle systems, inverse problems, Machine learning @online{LearningInteractionkernels2ndorder, title = {Learning Theory for Inferring Interaction Kernels in Second-Order Interacting Agent Systems}, author = {Jason Miller, Sui Tang, Ming Zhong, Mauro Maggioni}, url = {https://arxiv.org/abs/2010.03729}, year = {2020}, date = {2020-10-08}, keywords = {agent-based models, interacting particle systems, inverse problems, Machine learning}, pubstate = {forthcoming}, tppubtype = {online} } |

Fei Lu Mauro Maggioni, Sui Tang Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories Journal Article Forthcoming Foundation of Computational Mathematics, Forthcoming. Abstract | Links | BibTeX | Tags: agent-based models, interacting particle systems, Machine learning, statistics, stochastic systems @article{learningStochasticInteracting, title = {Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories}, author = {Fei Lu, Mauro Maggioni, Sui Tang}, url = {https://arxiv.org/abs/2007.15174}, year = {2020}, date = {2020-07-15}, journal = {Foundation of Computational Mathematics}, abstract = {We consider stochastic systems of interacting particles or agents, with dynamics determined by an interaction kernel which only depends on pairwise distances. We study the problem of inferring this interaction kernel from observations of the positions of the particles, in either continuous or discrete time, along multiple independent trajectories. We introduce a nonparametric inference approach to this inverse problem, based on a regularized maximum likelihood estimator constrained to suitable hypothesis spaces adaptive to data. We show that a coercivity condition enables us to control the condition number of this problem and prove the consistency of our estimator, and that in fact it converges at a near-optimal learning rate, equal to the min-max rate of 1-dimensional non-parametric regression. In particular, this rate is independent of the dimension of the state space, which is typically very high. We also analyze the discretization errors in the case of discrete-time observations, showing that it is of order 1/2 in terms of the time spacings between observations. This term, when large, dominates the sampling error and the approximation error, preventing convergence of the estimator. Finally, we exhibit an efficient parallel al- gorithm to construct the estimator from data, and we demonstrate the effectiveness of our algorithm with numerical tests on prototype systems including stochastic opinion dynamics and a Lennard-Jones model.}, keywords = {agent-based models, interacting particle systems, Machine learning, statistics, stochastic systems}, pubstate = {forthcoming}, tppubtype = {article} } We consider stochastic systems of interacting particles or agents, with dynamics determined by an interaction kernel which only depends on pairwise distances. We study the problem of inferring this interaction kernel from observations of the positions of the particles, in either continuous or discrete time, along multiple independent trajectories. We introduce a nonparametric inference approach to this inverse problem, based on a regularized maximum likelihood estimator constrained to suitable hypothesis spaces adaptive to data. We show that a coercivity condition enables us to control the condition number of this problem and prove the consistency of our estimator, and that in fact it converges at a near-optimal learning rate, equal to the min-max rate of 1-dimensional non-parametric regression. In particular, this rate is independent of the dimension of the state space, which is typically very high. We also analyze the discretization errors in the case of discrete-time observations, showing that it is of order 1/2 in terms of the time spacings between observations. This term, when large, dominates the sampling error and the approximation error, preventing convergence of the estimator. Finally, we exhibit an efficient parallel al- gorithm to construct the estimator from data, and we demonstrate the effectiveness of our algorithm with numerical tests on prototype systems including stochastic opinion dynamics and a Lennard-Jones model. |

## 2019 |

Maggioni, Mauro; Miller, Jason; Zhong, Ming Data-driven Discovery of Emergent Behaviors in Collective Dynamics Journal Article Physica D: Nonlinear Phenomena, 2019. Links | BibTeX | Tags: agent-based models, interacting particle systems, inverse problems, Machine learning, model reduction, statistics @article{DiscoveryEmergentBehaviors, title = {Data-driven Discovery of Emergent Behaviors in Collective Dynamics}, author = {Mauro Maggioni and Jason Miller and Ming Zhong}, url = {https://arxiv.org/abs/1912.11123}, doi = {https://doi.org/10.1016/j.physd.2020.132542}, year = {2019}, date = {2019-01-01}, journal = {Physica D: Nonlinear Phenomena}, keywords = {agent-based models, interacting particle systems, inverse problems, Machine learning, model reduction, statistics}, pubstate = {published}, tppubtype = {article} } |

Lu, Fei; Zhong, Ming; Tang, Sui; Maggioni, Mauro Nonparametric inference of interaction laws in systems of agents from trajectory data Journal Article Proceedings of the National Academy of Sciences, 116 (29), pp. 14424–14433, 2019, ISSN: 0027-8424. Links | BibTeX | Tags: agent-based models, interacting particle systems, inverse problems, Machine learning, model reduction, statistics @article{LMTZ:AgentsNonParametric, title = {Nonparametric inference of interaction laws in systems of agents from trajectory data}, author = {Fei Lu and Ming Zhong and Sui Tang and Mauro Maggioni}, url = {https://www.pnas.org/content/116/29/14424}, doi = {10.1073/pnas.1822012116}, issn = {0027-8424}, year = {2019}, date = {2019-01-01}, journal = {Proceedings of the National Academy of Sciences}, volume = {116}, number = {29}, pages = {14424–14433}, publisher = {National Academy of Sciences}, keywords = {agent-based models, interacting particle systems, inverse problems, Machine learning, model reduction, statistics}, pubstate = {published}, tppubtype = {article} } |

## 2016 |

Bongini, Mattia; Fornasier, Massimo; Hansen, M; Maggioni, Mauro Inferring Interaction Rules From Observations of Evolutive Systems I: The Variational Approach 2016. Links | BibTeX | Tags: agent-based models, interacting particle systems, Machine learning, statistics @journal{BFHM:LearningInteractionRulesI, title = {Inferring Interaction Rules From Observations of Evolutive Systems I: The Variational Approach}, author = {Mattia Bongini and Massimo Fornasier and M Hansen and Mauro Maggioni}, url = {https://arxiv.org/pdf/1602.00342.pdf}, doi = {https://doi.org/10.1142/S0218202517500208}, year = {2016}, date = {2016-01-01}, journal = {Mathematical Models and Methods in Applied Sciences}, volume = {27}, number = {05}, pages = {909-951}, keywords = {agent-based models, interacting particle systems, Machine learning, statistics}, pubstate = {published}, tppubtype = {journal} } |

## 0000 |

Lu, Fei; Maggioni, Mauro; Tang, Sui Learning interaction kernels in heterogeneous systems of agents from multiple trajectories Journal Article Journ. Mach. Learn. Res., 22 (32), pp. 1–67, 0000. @article{LMT:AgentsHeterogeneous, title = {Learning interaction kernels in heterogeneous systems of agents from multiple trajectories}, author = {Fei Lu and Mauro Maggioni and Sui Tang}, url = {https://jmlr.csail.mit.edu/papers/volume22/19-861/19-861.pdf}, journal = {Journ. Mach. Learn. Res.}, volume = {22}, number = {32}, pages = {1–67}, keywords = {agent-based models, interacting particle systems, inverse problems, Machine learning, model reduction, statistics}, pubstate = {published}, tppubtype = {article} } |

- Lectures at Summer School at Peking University, July 2017.
- PCMI Lectures, Summer 2016: Lecture 1, Lecture 2, Problems/discussion points
- Google Scholar
- Papers on the ArXiv
- Papers on MathsciNet
- Tutorials on diffusion geometry and multiscale analysis on graphs at the MRA Internet Program at IPAM: Part I and Part II.
- Diffusion Geometries, Diffusion Wavelets and Harmonic Analysis of large data sets, IPAM, Multiscale Geometric Analysis Program, Fall 2004.
- Diffusion Geometries, global and multiscale, IPAM, 2005.