Weak Formulation for Solving Inverse Problems in Reproducing Kernel Hilbert Spaces (With Applications to Learning Dynamical Systems)

We combine numerical and machine learning techniques to present a general framework for solving inverse problems using vector valued reproducing kernel Hilbert spaces in a variational formulation. We present this framework in two papers. In the first paper, we present an original state-of-the-art method derived in the context of our general framework for learning dynamical systems. In the second paper, we generalize the method from our first paper to arrive at the framework for solving inverse problems. Then we apply our general framework to the task of learning dynamical systems. In both papers we consider numerous applications of our methods and demonstrate the effectiveness of our methods with experiments on datasets from dynamical systems. Our experiments demonstrate our methods are both computationally efficient and provide compelling performance on the task of learning dynamical systems.