Nonlinear controllability and function representation by neural stochastic differential equations
There has been a great deal of recent interest in learning and approximation of functions that can be expressed as expectations of a given nonlinearity with respect to its random internal parameters. Examples of such representations include "infinitely wide" neural nets, where the underlying nonlinearity is given by the activation function of an individual neuron. In this paper, we bring this perspective to function representation by neural stochastic differential equations (SDEs). A neural SDE is an It\^o diffusion process whose drift and diffusion matrix are elements of some parametric families. We show that the ability of a neural SDE to realize nonlinear functions of its initial condition can be related to the problem of optimally steering a certain deterministic dynamical system between two given points in finite time. This auxiliary system is obtained by formally replacing the Brownian motion in the SDE by a deterministic control input. We derive upper and lower bounds on the minimum control effort needed to accomplish this steering; these bounds may be of independent interest in the context of motion planning and deterministic optimal control.
Code (0)
등록된 구현이 없습니다.
Tasks
Motion PlanningMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
On Differential Controllability and Observability Functions
Differential balancing theory for nonlinear model reduction relies on differential controllability and observability functions. In this paper, we further investigate them from two different perspectives. First, we establ…
Kernel Methods for the Approximation of Some Key Quantities of Nonlinear Systems
We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existi…
A deep branching solver for fully nonlinear partial differential equations
We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving g…
Deep LearningUniversal Approximation Power of Deep Residual Neural Networks via Nonlinear Control Theory
In this paper, we explain the universal approximation capabilities of deep residual neural networks through geometric nonlinear control. Inspired by recent work establishing links between residual networks and control sy…
Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement lea…
Deep Reinforcement Learningreinforcement-learningReinforcement LearningReinforcement Learning (RL)