Identifying Sources and Sinks in the Presence of Multiple Agents with Gaussian Process Vector Calculus

Abstract

In systems of multiple agents, identifying the cause of observed agent dynamics is challenging. Often, these agents operate in diverse, non-stationary environments, where models rely on handcrafted environment-specific features to infer influential regions in the system's surroundings. To overcome the limitations of these inflexible models, we present GP-LAPLACE, a technique for locating sources and sinks from trajectories in time-varying fields. Using Gaussian processes, we jointly infer a spatio-temporal vector field, as well as canonical vector calculus operations on that field. Notably, we do this from only agent trajectories without requiring knowledge of the environment, and also obtain a metric for denoting the significance of inferred causal features in the environment by exploiting our probabilistic method. To evaluate our approach, we apply it to both synthetic and real-world GPS data, demonstrating the applicability of our technique in the presence of multiple agents, as well as its superiority over existing methods.

Gaussian Process Regression for Data Fulfilling Linear Differential Equations with Localized Sources

“…In the 3D case, one would proceed in a similar fashion with spherical Bessel functions, which yields the kernel that was already postulated in [11]. In contrast to the case of Laplace's equation in the previous section, these source-free Helmholtz kernels do not possess singularities at any finite distance from the origin, i.e., no virtual exterior sources in the Mercer kernel (20). As a consequence they provide smoothing regularization on the order of the wavelength λ 0 = 2π/k 0 to reconstructed fields and boundary conditions that may or may not be desired.…”

“…A starting point could be squared exponential kernels for divergence-and curl-free vector fields [18]. Such kernels have been used in [19] to perform statistical reconstruction, and [20] apply them to GPs for source identification in the Laplace/Poisson equation. To model Hamiltonian dynamics in phase-space, vector-valued GPs could possibly be extended to represent not only volume-preserving (divergence-free) maps but retain full symplectic properties, conserving all integrals of motion such as energy or momentum.…”

“…Kernels of the form (20) are known as convolution kernels. Such a kernel is at least positive semidefinite, and positive definiteness follows in the case of linearly independent basis functions φ(x, ζ) [14].…”

“…Such a kernel is at least positive semidefinite, and positive definiteness follows in the case of linearly independent basis functions φ(x, ζ) [14]. For treatment of PDEs, the possible choices of index variables in Equation (18) or Equation (20) include separation constants of analytical solutions, or the frequency variable of an integral transform. In accordance with [10], using basis functions that satisfy the underlying PDE, a probabilistic meshless method (PMM) is constructed.…”

“…Here, all solutions yield finite values at x = 0, so we do not have to exclude any of them a priori. Introducing, again, a diagonal covariance operator in (20) and taking the real part yields…”