Collaborating Authors

Tang, Sui

Data-driven discovery of interacting particle systems using Gaussian processes Machine Learning

Interacting particle or agent systems that display a rich variety of collection motions are ubiquitous in science and engineering. A fundamental and challenging goal is to understand the link between individual interaction rules and collective behaviors. In this paper, we study the data-driven discovery of distance-based interaction laws in second-order interacting particle systems. We propose a learning approach that models the latent interaction kernel functions as Gaussian processes, which can simultaneously fulfill two inference goals: one is the nonparametric inference of interaction kernel function with the pointwise uncertainty quantification, and the other one is the inference of unknown parameters in the non-collective forces of the system. We formulate learning interaction kernel functions as a statistical inverse problem and provide a detailed analysis of recoverability conditions, establishing that a coercivity condition is sufficient for recoverability. We provide a finite-sample analysis, showing that our posterior mean estimator converges at an optimal rate equal to the one in the classical 1-dimensional Kernel Ridge regression. Numerical results on systems that exhibit different collective behaviors demonstrate efficient learning of our approach from scarce noisy trajectory data.

Learning Theory for Inferring Interaction Kernels in Second-Order Interacting Agent Systems Machine Learning

Modeling the complex interactions of systems of particles or agents is a fundamental scientific and mathematical problem that is studied in diverse fields, ranging from physics and biology, to economics and machine learning. In this work, we describe a very general second-order, heterogeneous, multivariable, interacting agent model, with an environment, that encompasses a wide variety of known systems. We describe an inference framework that uses nonparametric regression and approximation theory based techniques to efficiently derive estimators of the interaction kernels which drive these dynamical systems. We develop a complete learning theory which establishes strong consistency and optimal nonparametric min-max rates of convergence for the estimators, as well as provably accurate predicted trajectories. The estimators exploit the structure of the equations in order to overcome the curse of dimensionality and we describe a fundamental coercivity condition on the inverse problem which ensures that the kernels can be learned and relates to the minimal singular value of the learning matrix. The numerical algorithm presented to build the estimators is parallelizable, performs well on high-dimensional problems, and is demonstrated on complex dynamical systems.

Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories Machine Learning

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 gaps 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 algorithm 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.

Learning interaction kernels in heterogeneous systems of agents from multiple trajectories Machine Learning

Systems of interacting particles or agents have wide applications in many disciplines such as 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 estimators that are 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 based on least squares, 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-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 produce accurate predictions of dynamics in relative large time intervals, even when they are learned from data collected in short time intervals.

Nonparametric inference of interaction laws in systems of agents from trajectory data Machine Learning

Inferring the laws of interaction between particles and agents in complex dynamical systems from observational data is a fundamental challenge in a wide variety of disciplines. We propose a non-parametric statistical learning approach to estimate the governing laws of distance-based interactions, with no reference or assumption about their analytical form, from data consisting trajectories of interacting agents. We demonstrate the effectiveness of our learning approach both by providing theoretical guarantees, and by testing the approach on a variety of prototypical systems in various disciplines. These systems include homogeneous and heterogeneous agents systems, ranging from particle systems in fundamental physics to agent-based systems modeling opinion dynamics under the social influence, prey-predator dynamics, flocking and swarming, and phototaxis in cell dynamics.