We present a Representer Theorem result for a large class of weak formulation problems. We provide examples of applications of our formulation both in traditional machine learning and numerical methods as well as in new and emerging techniques. Finally we apply our formulation to generalize the multivariate occupation kernel (MOCK) method for learning dynamical systems from data proposing the more general Riesz Occupation Kernel (ROCK) method. Our generalized methods are both more computationally efficient and performant on most of the benchmarks we test against.

The method of occupation kernels has been used to learn ordinary differential equations from data in a non-parametric way. We propose a two-step method for learning the drift and diffusion of a stochastic differential equation given snapshots of the process. In the first step, we learn the drift by applying the occupation kernel algorithm to the expected value of the process. In the second step, we learn the diffusion given the drift using a semi-definite program. Specifically, we learn the diffusion squared as a non-negative function in a RKHS associated with the square of a kernel. We present examples and simulations.

We consider the problem of active learning in the context of spatial sampling for level set estimation (LSE), where the goal is to localize all regions where a function of interest lies above/below a given threshold as quickly as possible. We present a finite-horizon search procedure to perform LSE in one dimension while optimally balancing both the final estimation error and the distance traveled for a fixed number of samples. A tuning parameter is used to trade off between the estimation accuracy and distance traveled. We show that the resulting optimization problem can be solved in closed form and that the resulting policy generalizes existing approaches to this problem. We then show how this approach can be used to perform level set estimation in higher dimensions under the popular Gaussian process model. Empirical results on synthetic data indicate that as the cost of travel increases, our method's ability to treat distance nonmyopically allows it to significantly improve on the state of the art. On real air quality data, our approach achieves roughly one fifth the estimation error at less than half the cost of competing algorithms.

Learning a nonparametric system of ordinary differential equations (ODEs) from $n$ trajectory snapshots in a $d$-dimensional state space requires learning $d$ functions of $d$ variables. Explicit formulations scale quadratically in $d$ unless additional knowledge about system properties, such as sparsity and symmetries, is available. In this work, we propose a linear approach to learning using the implicit formulation provided by vector-valued Reproducing Kernel Hilbert Spaces. By rewriting the ODEs in a weaker integral form, which we subsequently minimize, we derive our learning algorithm. The minimization problem's solution for the vector field relies on multivariate occupation kernel functions associated with the solution trajectories. We validate our approach through experiments on highly nonlinear simulated and real data, where $d$ may exceed 100. We further demonstrate the versatility of the proposed method by learning a nonparametric first order quasilinear partial differential equation.

This volume represents the accepted submissions from the Machine Learning for Health (ML4H) workshop at the conference on Neural Information Processing Systems (NeurIPS) 2018, held on December 8, 2018 in Montreal, Canada.