| Abstract: |
| The existing AI agents can do many actions, however, it seems that we still have problems with control and prediction next steps in time series. Our aim in this article, being in the spirit of machine learning, extends the applicability of neural ordinary differential equation (Neural CDE), described in \cite{FLN} for neural network modeled by an ordinary differential equations to a case of a mathematical model given by a system of partial differential equations (PDE). That requires new approach for prediction the next step in the given time series. In order to apply Neural CDE methodology we transform our system of PDE (modeling mathematically Alzheimer) to a new system of ODE. It is possible because we can treat an ANN as a function which in turn for a given set of weights determines a point in this space, and learning algorithms differ in the way they traverse it. We replace this space with a family of functions (family of solutions) which conforms well defined mathematical conditions. This allows us to develop an optimal control approach, parameterized by a set of controls and defined as neural controlled differential equations (Neural CDE). Thus we turn the difficult to explain artificial neural network learning process into a well-posed mathematical problem and apply to it a dual dynamic programming idea to formulate an optimization problem subject to Neural CDE. Turning the problem into optimal control problem allows one to define it in a rigorous way; however, it does not say much about the way it may be solved. To overcome theoretical difficulties and solve it in satisfactory and applicable way (our goal is to propose general methodology which is both theoretically correct and practically applicable to various tasks), we follow the methodology of the dual dynamic programming to derive the sufficient optimality conditions in the form of verification theorem and that simplify all practical considerations. We provide and prove verification conditions that should correctly predict the next step in time series. To show applicability of the presented theory we adopt state-of-the-art mathematical model of the Alzheimer disease for Neural CDE to predict state of the disease after 10, 20 years. |
|