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Modeling the complex interactions of systems of particles or agents is a fundamental problem across the sciences, from physics and biology, to economics and social sciences. In this work, we consider second-order, heterogeneous, multivariable models of interacting agents or particles, within simple environments. We describe a nonparametric inference framework to efficiently estimate the latent interaction kernels which drive these dynamical systems. We develop a 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 optimal rates only depends on intrinsic dimension of interactions, which is typically much smaller than the ambient dimension. Our arguments are based on a coercivity condition which ensures that the interaction kernels can be estimated in stable fashion. The numerical algorithm presented to build the estimators is parallelizable, performs well on high-dimensional problems, and its performance is tested on a variety of complex dynamical systems.

 

We consider the regression problem of estimating functions on $ \mathbb{R}^D $ but supported on a $ d $-dimensional manifold $ \mathcal{M} ~~\subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on $ \mathcal{M} $ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $ d $, instead of an unknown manifold embedded in $ \mathbb{R}^D $. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $ D $ and exponential in $ d $. Our work therefore establishes a new framework for regression on low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.

 

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 one-dimensional nonparametric 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 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. 

Abstract Sudden cardiac death from arrhythmia is a major cause of mortality worldwide. In this study, we developed a novel deep learning (DL) approach that blends neural networks and survival analysis to predict patient-specific survival curves from contrast-enhanced cardiac magnetic resonance images and clinical covariates for patients with ischemic heart disease. The DL-predicted survival curves offer accurate predictions at times up to 10 years and allow for estimation of uncertainty in predictions. The performance of this learning architecture was evaluated on multi-center internal validation data and tested on an independent test set, achieving concordance indexes of 0.83 and 0.74 and 10-year integrated Brier scores of 0.12 and 0.14. We demonstrate that our DL approach, with only raw cardiac images as input, outperforms standard survival models constructed using clinical covariates. This technology has the potential to transform clinical decision-making by offering accurate and generalizable predictions of patient-specific survival probabilities of arrhythmic death over time.