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.

我们考虑相互作用粒子或代理的随机系统，其动力学由相互作用内核确定，仅取决于成对距离。我们研究从连续或离散时间沿多个独立轨迹观察粒子位置来推断该相互作用核的问题。我们针对这个逆问题引入了一种非参数推理方法，它基于一个正则化的最大似然估计器，该估计器被约束到适合数据的合适假设空间。我们证明了矫顽力条件使我们能够控制这个问题的条件数并证明我们的估计量的一致性，事实上它以接近最优的学习率收敛，等于一维非参数的最小-最大速率回归。特别是，这个速率与状态空间的维度无关，状态空间的维度通常非常高。我们还分析了离散时间观测的离散化误差，表明就观测之间的时间间隔而言，它是 1/2 的。当这个项很大时，它会支配采样误差和近似误差，阻止估计器的收敛。最后，我们展示了一种有效的并行算法来从数据构建估计器，并且我们通过对原型系统（包括随机意见动态和 Lennard-Jones 模型）的数值测试证明了我们算法的有效性。该项较大时，会支配采样误差和近似误差，从而防止估计器收敛。最后，我们展示了一种有效的并行算法来从数据构建估计器，并且我们通过对原型系统（包括随机意见动态和 Lennard-Jones 模型）的数值测试证明了我们算法的有效性。当这个项很大时，它会支配采样误差和近似误差，阻止估计器的收敛。最后，我们展示了一种有效的并行算法来从数据构建估计器，并且我们通过对原型系统（包括随机意见动态和 Lennard-Jones 模型）的数值测试证明了我们算法的有效性。